Integrand size = 24, antiderivative size = 805 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {-56 b^3 c^3+96 a b^2 c^2 d-189 a^2 b c d^2+77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} (7 b c-19 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^4}+\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^4} \]
1/48*(77*a^3*d^3-189*a^2*b*c*d^2+96*a*b^2*c^2*d-56*b^3*c^3)/a^2/c^3/(-a*d+ b*c)^3/x^(3/2)+1/4*d*(a*d+2*b*c)/a/c/(-a*d+b*c)^2/x^(3/2)/(d*x^2+c)^2+1/2* b/a/(-a*d+b*c)/x^(3/2)/(b*x^2+a)/(d*x^2+c)^2+1/16*d*(-11*a^2*d^2+27*a*b*c* d+8*b^2*c^2)/a/c^2/(-a*d+b*c)^3/x^(3/2)/(d*x^2+c)+1/8*b^(15/4)*(-19*a*d+7* b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(11/4)/(-a*d+b*c)^4*2^(1/ 2)-1/8*b^(15/4)*(-19*a*d+7*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/ a^(11/4)/(-a*d+b*c)^4*2^(1/2)+1/64*d^(11/4)*(77*a^2*d^2-266*a*b*c*d+285*b^ 2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*d+b*c)^4*2^( 1/2)-1/64*d^(11/4)*(77*a^2*d^2-266*a*b*c*d+285*b^2*c^2)*arctan(1+d^(1/4)*2 ^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*d+b*c)^4*2^(1/2)+1/16*b^(15/4)*(-19*a *d+7*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/( -a*d+b*c)^4*2^(1/2)-1/16*b^(15/4)*(-19*a*d+7*b*c)*ln(a^(1/2)+x*b^(1/2)+a^( 1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/(-a*d+b*c)^4*2^(1/2)+1/128*d^(11/4) *(77*a^2*d^2-266*a*b*c*d+285*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4) *2^(1/2)*x^(1/2))/c^(15/4)/(-a*d+b*c)^4*2^(1/2)-1/128*d^(11/4)*(77*a^2*d^2 -266*a*b*c*d+285*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^( 1/2))/c^(15/4)/(-a*d+b*c)^4*2^(1/2)
Time = 1.57 (sec) , antiderivative size = 521, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {1}{192} \left (-\frac {4 \left (-56 b^4 c^3 x^2 \left (c+d x^2\right )^2-32 a b^3 c^2 \left (c-3 d x^2\right ) \left (c+d x^2\right )^2+a^4 d^3 \left (32 c^2+121 c d x^2+77 d^2 x^4\right )+3 a^2 b^2 c d \left (32 c^3+32 c^2 d x^2-67 c d^2 x^4-63 d^3 x^6\right )+a^3 b d^2 \left (-96 c^3-265 c^2 d x^2-68 c d^2 x^4+77 d^3 x^6\right )\right )}{a^2 c^3 (-b c+a d)^3 x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {24 \sqrt {2} b^{15/4} (7 b c-19 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{11/4} (b c-a d)^4}+\frac {3 \sqrt {2} d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{15/4} (b c-a d)^4}+\frac {24 \sqrt {2} b^{15/4} (-7 b c+19 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{11/4} (b c-a d)^4}-\frac {3 \sqrt {2} d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{15/4} (b c-a d)^4}\right ) \]
((-4*(-56*b^4*c^3*x^2*(c + d*x^2)^2 - 32*a*b^3*c^2*(c - 3*d*x^2)*(c + d*x^ 2)^2 + a^4*d^3*(32*c^2 + 121*c*d*x^2 + 77*d^2*x^4) + 3*a^2*b^2*c*d*(32*c^3 + 32*c^2*d*x^2 - 67*c*d^2*x^4 - 63*d^3*x^6) + a^3*b*d^2*(-96*c^3 - 265*c^ 2*d*x^2 - 68*c*d^2*x^4 + 77*d^3*x^6)))/(a^2*c^3*(-(b*c) + a*d)^3*x^(3/2)*( a + b*x^2)*(c + d*x^2)^2) + (24*Sqrt[2]*b^(15/4)*(7*b*c - 19*a*d)*ArcTan[( Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(11/4)*(b*c - a*d)^4) + (3*Sqrt[2]*d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Arc Tan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(15/4)*(b *c - a*d)^4) + (24*Sqrt[2]*b^(15/4)*(-7*b*c + 19*a*d)*ArcTanh[(Sqrt[2]*a^( 1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(11/4)*(b*c - a*d)^4) - ( 3*Sqrt[2]*d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTanh[(Sqrt[ 2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(15/4)*(b*c - a*d)^ 4))/192
Time = 1.32 (sec) , antiderivative size = 745, normalized size of antiderivative = 0.93, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {368, 972, 25, 1049, 27, 1049, 1053, 27, 1020, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {1}{x^2 \left (b x^2+a\right )^2 \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 \left (\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int -\frac {15 b d x^2+7 b c-4 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {15 b d x^2+7 b c-4 a d}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {4 \left (14 b^2 c^2-16 a b d c+11 a^2 d^2+11 b d (2 b c+a d) x^2\right )}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {14 b^2 c^2-16 a b d c+11 a^2 d^2+11 b d (2 b c+a d) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3+7 b d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right ) x^2}{x^2 \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\int \frac {3 \left (56 b^4 c^4-96 a b^3 d c^3-96 a^2 b^2 d^2 c^2+189 a^3 b d^3 c-77 a^4 d^4+b d \left (56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3\right ) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{3 a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\int \frac {56 b^4 c^4-96 a b^3 d c^3-96 a^2 b^2 d^2 c^2+189 a^3 b d^3 c-77 a^4 d^4+b d \left (56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3\right ) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\frac {a^2 d^3 \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}+\frac {8 b^4 c^3 (7 b c-19 a d) \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\frac {a^2 d^3 \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^4 c^3 (7 b c-19 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\frac {a^2 d^3 \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^4 c^3 (7 b c-19 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\frac {a^2 d^3 \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^4 c^3 (7 b c-19 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {-\frac {\frac {a^2 d^3 \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^4 c^3 (7 b c-19 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}+\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {b}{4 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d (2 b c+a d)}{2 c (b c-a d) x^{3/2} \left (d x^2+c\right )^2}+\frac {\frac {d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{4 c (b c-a d) x^{3/2} \left (d x^2+c\right )}+\frac {-\frac {56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{3 a c x^{3/2}}-\frac {\frac {8 c^3 (7 b c-19 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right ) b^4}{b c-a d}+\frac {a^2 d^3 \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}}{4 c (b c-a d)}}{2 c (b c-a d)}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {b}{4 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d (2 b c+a d)}{2 c (b c-a d) x^{3/2} \left (d x^2+c\right )^2}+\frac {\frac {d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{4 c (b c-a d) x^{3/2} \left (d x^2+c\right )}+\frac {-\frac {56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{3 a c x^{3/2}}-\frac {\frac {8 c^3 (7 b c-19 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right ) b^4}{b c-a d}+\frac {a^2 d^3 \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}}{4 c (b c-a d)}}{2 c (b c-a d)}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {b}{4 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d (2 b c+a d)}{2 c (b c-a d) x^{3/2} \left (d x^2+c\right )^2}+\frac {\frac {d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{4 c (b c-a d) x^{3/2} \left (d x^2+c\right )}+\frac {-\frac {56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{3 a c x^{3/2}}-\frac {\frac {8 c^3 (7 b c-19 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}\right ) b^4}{b c-a d}+\frac {a^2 d^3 \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}}{a c}}{4 c (b c-a d)}}{2 c (b c-a d)}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {d \left (-11 a^2 d^2+27 a b c d+8 b^2 c^2\right )}{4 c x^{3/2} \left (c+d x^2\right ) (b c-a d)}+\frac {-\frac {\frac {a^2 d^3 \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^4 c^3 (7 b c-19 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}}{a c}-\frac {-77 a^3 d^3+189 a^2 b c d^2-96 a b^2 c^2 d+56 b^3 c^3}{3 a c x^{3/2}}}{4 c (b c-a d)}}{2 c (b c-a d)}+\frac {d (a d+2 b c)}{2 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b}{4 a x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
2*(b/(4*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)^2) + ((d*(2*b*c + a* d))/(2*c*(b*c - a*d)*x^(3/2)*(c + d*x^2)^2) + ((d*(8*b^2*c^2 + 27*a*b*c*d - 11*a^2*d^2))/(4*c*(b*c - a*d)*x^(3/2)*(c + d*x^2)) + (-1/3*(56*b^3*c^3 - 96*a*b^2*c^2*d + 189*a^2*b*c*d^2 - 77*a^3*d^3)/(a*c*x^(3/2)) - ((8*b^4*c^ 3*(7*b*c - 19*a*d)*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt [2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqr t[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b ^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt [2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*S qrt[a])))/(b*c - a*d) + (a^2*d^3*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)* ((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4) )) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4 )))/(2*Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq rt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4) *Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c])))/(b*c - a* d))/(a*c))/(4*c*(b*c - a*d)))/(2*c*(b*c - a*d)))/(4*a*(b*c - a*d)))
3.6.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^n), x], x ] - Simp[(d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b , c, d, e, f, n}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 2.80 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.48
| method | result | size |
| derivativedivides | \(-\frac {2 d^{3} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {23}{16} a b c \,d^{2}+\frac {31}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-54 a b c d +35 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-266 a b c d +285 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{c^{3} \left (a d -b c \right )^{4}}-\frac {2}{3 a^{2} c^{3} x^{\frac {3}{2}}}+\frac {2 b^{4} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (19 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} \left (a d -b c \right )^{4}}\) | \(385\) |
| default | \(-\frac {2 d^{3} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {23}{16} a b c \,d^{2}+\frac {31}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-54 a b c d +35 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-266 a b c d +285 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{c^{3} \left (a d -b c \right )^{4}}-\frac {2}{3 a^{2} c^{3} x^{\frac {3}{2}}}+\frac {2 b^{4} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (19 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} \left (a d -b c \right )^{4}}\) | \(385\) |
| risch | \(-\frac {2}{3 a^{2} c^{3} x^{\frac {3}{2}}}-\frac {-\frac {2 c^{3} b^{4} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (19 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}+\frac {2 a^{2} d^{3} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {23}{16} a b c \,d^{2}+\frac {31}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-54 a b c d +35 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-266 a b c d +285 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{\left (a d -b c \right )^{4}}}{c^{3} a^{2}}\) | \(394\) |
-2*d^3/c^3/(a*d-b*c)^4*(((15/32*a^2*d^3-23/16*a*b*c*d^2+31/32*b^2*c^2*d)*x ^(5/2)+1/32*c*(19*a^2*d^2-54*a*b*c*d+35*b^2*c^2)*x^(1/2))/(d*x^2+c)^2+1/25 6*(77*a^2*d^2-266*a*b*c*d+285*b^2*c^2)*(c/d)^(1/4)/c*2^(1/2)*(ln((x+(c/d)^ (1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1 /2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4) *x^(1/2)-1)))-2/3/a^2/c^3/x^(3/2)+2*b^4/a^2/(a*d-b*c)^4*((1/4*a*d-1/4*b*c) *x^(1/2)/(b*x^2+a)+1/32*(19*a*d-7*b*c)*(a/b)^(1/4)/a*2^(1/2)*(ln((x+(a/b)^ (1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1 /2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4) *x^(1/2)-1)))
Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 1064, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/16*(2*sqrt(2)*(7*b*c - 19*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1 /4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt (b))) + 2*sqrt(2)*(7*b*c - 19*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^ (1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sq rt(b))) + sqrt(2)*(7*b*c - 19*a*d)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + s qrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(7*b*c - 19*a*d)*log(-sqrt (2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*b^4/ (a^2*b^4*c^4 - 4*a^3*b^3*c^3*d + 6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d ^4) - 1/48*(32*a*b^3*c^5 - 96*a^2*b^2*c^4*d + 96*a^3*b*c^3*d^2 - 32*a^4*c^ 2*d^3 + (56*b^4*c^3*d^2 - 96*a*b^3*c^2*d^3 + 189*a^2*b^2*c*d^4 - 77*a^3*b* d^5)*x^6 + (112*b^4*c^4*d - 160*a*b^3*c^3*d^2 + 201*a^2*b^2*c^2*d^3 + 68*a ^3*b*c*d^4 - 77*a^4*d^5)*x^4 + (56*b^4*c^5 - 32*a*b^3*c^4*d - 96*a^2*b^2*c ^3*d^2 + 265*a^3*b*c^2*d^3 - 121*a^4*c*d^4)*x^2)/((a^2*b^4*c^6*d^2 - 3*a^3 *b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^(15/2) + (2*a^2*b^4*c^ 7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5) *x^(11/2) + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5 *d^3 - 2*a^6*c^4*d^4)*x^(7/2) + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c ^6*d^2 - a^6*c^5*d^3)*x^(3/2)) - 1/128*(2*sqrt(2)*(285*b^2*c^2*d^3 - 266*a *b*c*d^4 + 77*a^2*d^5)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqr t(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + ...
Time = 0.56 (sec) , antiderivative size = 1278, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/2*b^4*sqrt(x)/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3 )*(b*x^2 + a)) - 1/4*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3)^(1/4)*a*b^3*d)*ar ctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a ^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*sqrt( 2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) - 1/4*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3 )^(1/4)*a*b^3*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/ b)^(1/4))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b ^2*c^2*d^2 - 4*sqrt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) - 1/32*(285*(c*d^3)^ (1/4)*b^2*c^2*d^2 - 266*(c*d^3)^(1/4)*a*b*c*d^3 + 77*(c*d^3)^(1/4)*a^2*d^4 )*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt( 2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2) *a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) - 1/32*(285*(c*d^3)^(1/4)*b^2*c^2*d^ 2 - 266*(c*d^3)^(1/4)*a*b*c*d^3 + 77*(c*d^3)^(1/4)*a^2*d^4)*arctan(-1/2*sq rt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^8 - 4* sqrt(2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) - 1/8*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3)^(1/4)*a*b ^3*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^4*c^ 4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*sqrt(2)*a^6*b* c*d^3 + sqrt(2)*a^7*d^4) + 1/8*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3)^(1/4)*a *b^3*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*...
Time = 23.05 (sec) , antiderivative size = 180372, normalized size of antiderivative = 224.06 \[ \int \frac {1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((x^(1/2)*(857712418202478182400*a^18*b^48*c^62*d^11 - 28925330217666 430894080*a^19*b^47*c^61*d^12 + 465808355868544602210304*a^20*b^46*c^60*d^ 13 - 4772189938359453553262592*a^21*b^45*c^59*d^14 + 349820765298262334012 12928*a^22*b^44*c^58*d^15 - 195811106815542077297786880*a^23*b^43*c^57*d^1 6 + 873231122236416493313064960*a^24*b^42*c^56*d^17 - 32015883183408887393 56606464*a^25*b^41*c^55*d^18 + 9904866981547362725832687616*a^26*b^40*c^54 *d^19 - 26475613142538536817178705920*a^27*b^39*c^53*d^20 + 62528004036875 405150857986048*a^28*b^38*c^52*d^21 - 133143680796215491474489344000*a^29* b^37*c^51*d^22 + 259595474982835164713400139776*a^30*b^36*c^50*d^23 - 4671 06577738876991145070559232*a^31*b^35*c^49*d^24 + 7753210968231093026749352 50944*a^32*b^34*c^48*d^25 - 1179424943892680059222782640128*a^33*b^33*c^47 *d^26 + 1629690593600095833823295569920*a^34*b^32*c^46*d^27 - 202814334571 9314676074795761664*a^35*b^31*c^45*d^28 + 2257905973104023956972306956288* a^36*b^30*c^44*d^29 - 2237449183565830435563494178816*a^37*b^29*c^43*d^30 + 1966204854457469918399988498432*a^38*b^28*c^42*d^31 - 152764940604836662 1262568488960*a^39*b^27*c^41*d^32 + 1046409458758522347995126562816*a^40*b ^26*c^40*d^33 - 629956523592774331698776113152*a^41*b^25*c^39*d^34 + 33206 5764335584004230153764864*a^42*b^24*c^38*d^35 - 15254319696813365092271574 2208*a^43*b^23*c^37*d^36 + 60699171433471101739298979840*a^44*b^22*c^36*d^ 37 - 20757436699772395749793333248*a^45*b^21*c^35*d^38 + 60378259517970...