Integrand size = 24, antiderivative size = 739 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^4}-\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^4} \]
-3/8*b^(11/4)*(-5*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/ 4)/(-a*d+b*c)^4*2^(1/2)+3/8*b^(11/4)*(-5*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2) *x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^4*2^(1/2)-3/64*d^(7/4)*(7*a^2*d^2-30* a*b*c*d+55*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a *d+b*c)^4*2^(1/2)+3/64*d^(7/4)*(7*a^2*d^2-30*a*b*c*d+55*b^2*c^2)*arctan(1+ d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^4*2^(1/2)-3/16*b^(11/ 4)*(-5*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7 /4)/(-a*d+b*c)^4*2^(1/2)+3/16*b^(11/4)*(-5*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a ^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/(-a*d+b*c)^4*2^(1/2)-3/128*d^(7/4) *(7*a^2*d^2-30*a*b*c*d+55*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^(11/4)/(-a*d+b*c)^4*2^(1/2)+3/128*d^(7/4)*(7*a^2*d^2-30*a *b*c*d+55*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 ^(11/4)/(-a*d+b*c)^4*2^(1/2)+1/4*d*(a*d+2*b*c)*x^(1/2)/a/c/(-a*d+b*c)^2/(d *x^2+c)^2+1/2*b*x^(1/2)/a/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^2+1/16*d*(-7*a^2* d^2+23*a*b*c*d+8*b^2*c^2)*x^(1/2)/a/c^2/(-a*d+b*c)^3/(d*x^2+c)
Time = 1.69 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {1}{64} \left (-\frac {4 \sqrt {x} \left (8 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (11 c+7 d x^2\right )+a b^2 c d^2 x^2 \left (27 c+23 d x^2\right )+a^2 b d^2 \left (27 c^2+12 c d x^2-7 d^2 x^4\right )\right )}{a c^2 (-b c+a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {24 \sqrt {2} b^{11/4} (-b c+5 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4} (b c-a d)^4}-\frac {3 \sqrt {2} d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{11/4} (b c-a d)^4}+\frac {24 \sqrt {2} b^{11/4} (b c-5 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4} (b c-a d)^4}+\frac {3 \sqrt {2} d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{11/4} (b c-a d)^4}\right ) \]
((-4*Sqrt[x]*(8*b^3*c^2*(c + d*x^2)^2 - a^3*d^3*(11*c + 7*d*x^2) + a*b^2*c *d^2*x^2*(27*c + 23*d*x^2) + a^2*b*d^2*(27*c^2 + 12*c*d*x^2 - 7*d^2*x^4))) /(a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + (24*Sqrt[2]*b^(11/4) *(-(b*c) + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sq rt[x])])/(a^(7/4)*(b*c - a*d)^4) - (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b *c*d + 7*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sq rt[x])])/(c^(11/4)*(b*c - a*d)^4) + (24*Sqrt[2]*b^(11/4)*(b*c - 5*a*d)*Arc Tanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(7/4)*(b *c - a*d)^4) + (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Ar cTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(11/4)* (b*c - a*d)^4))/64
Time = 1.09 (sec) , antiderivative size = 681, normalized size of antiderivative = 0.92, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {368, 931, 25, 1024, 27, 1024, 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.
| \(\displaystyle \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {1}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle 2 \left (\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int -\frac {11 b d x^2+3 b c-4 a d}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {11 b d x^2+3 b c-4 a d}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {4 \left (6 b^2 c^2-16 a b d c+7 a^2 d^2+7 b d (2 b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {6 b^2 c^2-16 a b d c+7 a^2 d^2+7 b d (2 b c+a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {3 \left (8 b^3 c^3-32 a b^2 d c^2+23 a^2 b d^2 c-7 a^3 d^3+b d \left (8 b^2 c^2+23 a b d c-7 a^2 d^2\right ) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \int \frac {8 b^3 c^3-32 a b^2 d c^2+23 a^2 b d^2 c-7 a^3 d^3+b d \left (8 b^2 c^2+23 a b d c-7 a^2 d^2\right ) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \left (\frac {a d^2 \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}+\frac {8 b^3 c^2 (b c-5 a d) \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}\right )}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \left (\frac {a d^2 \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^3 c^2 (b c-5 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}\right )}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \left (\frac {a d^2 \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^3 c^2 (b c-5 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}\right )}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \left (\frac {a d^2 \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^3 c^2 (b c-5 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}\right )}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \left (\frac {a d^2 \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^3 c^2 (b c-5 a d) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}\right )}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {\sqrt {x} b}{4 a (b c-a d) \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d \sqrt {x} (2 b c+a d)}{2 c (b c-a d) \left (d x^2+c\right )^2}+\frac {\frac {d \sqrt {x} \left (8 b^2 c^2+23 a b d c-7 a^2 d^2\right )}{4 c (b c-a d) \left (d x^2+c\right )}+\frac {3 \left (\frac {8 c^2 (b c-5 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^3}{b c-a d}+\frac {a d^2 \left (55 b^2 c^2-30 a b d c+7 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}\right )}{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 {\sqrt {x} b}{4 a (b c-a d) \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d \sqrt {x} (2 b c+a d)}{2 c (b c-a d) \left (d x^2+c\right )^2}+\frac {\frac {d \sqrt {x} \left (8 b^2 c^2+23 a b d c-7 a^2 d^2\right )}{4 c (b c-a d) \left (d x^2+c\right )}+\frac {3 \left (\frac {8 c^2 (b c-5 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^3}{b c-a d}+\frac {a d^2 \left (55 b^2 c^2-30 a b d c+7 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}\right )}{4 c (b c-a d)}}{2 c (b c-a d)}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\sqrt {x} b}{4 a (b c-a d) \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d \sqrt {x} (2 b c+a d)}{2 c (b c-a d) \left (d x^2+c\right )^2}+\frac {\frac {d \sqrt {x} \left (8 b^2 c^2+23 a b d c-7 a^2 d^2\right )}{4 c (b c-a d) \left (d x^2+c\right )}+\frac {3 \left (\frac {8 c^2 (b c-5 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}\right ) b^3}{b c-a d}+\frac {a d^2 \left (55 b^2 c^2-30 a b d c+7 a^2 d^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )}{4 c (b c-a d)}}{2 c (b c-a d)}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {3 \left (\frac {a d^2 \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}+\frac {8 b^3 c^2 (b c-5 a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}\right )}{4 c (b c-a d)}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b \sqrt {x}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
2*((b*Sqrt[x])/(4*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + ((d*(2*b*c + a*d)*Sqrt[x])/(2*c*(b*c - a*d)*(c + d*x^2)^2) + ((d*(8*b^2*c^2 + 23*a*b*c* d - 7*a^2*d^2)*Sqrt[x])/(4*c*(b*c - a*d)*(c + d*x^2)) + (3*((8*b^3*c^2*(b* c - 5*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)]/(Sqrt[2]*a^( 1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*S qrt[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*Sqrt[a])) )/(b*c - a*d) + (a*d^2*(55*b^2*c^2 - 30*a*b*c*d + 7*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] + Sqrt[d]*x]/(Sqr t[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sq rt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c])))/(b*c - a*d)))/(4*c*(b* c - a*d)))/(2*c*(b*c - a*d)))/(4*a*(b*c - a*d)))
3.5.100.3.1 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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 = 2.87 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.51
| method | result | size |
| derivativedivides | \(\frac {2 d^{2} \left (\frac {\frac {d \left (7 a^{2} d^{2}-30 a b c d +23 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-38 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (7 a^{2} d^{2}-30 a b c d +55 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{4}}-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {3 \left (5 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{4}}\) | \(375\) |
| default | \(\frac {2 d^{2} \left (\frac {\frac {d \left (7 a^{2} d^{2}-30 a b c d +23 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-38 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (7 a^{2} d^{2}-30 a b c d +55 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{4}}-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {3 \left (5 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{4}}\) | \(375\) |
2*d^2/(a*d-b*c)^4*((1/32*d*(7*a^2*d^2-30*a*b*c*d+23*b^2*c^2)/c^2*x^(5/2)+1 /32*(11*a^2*d^2-38*a*b*c*d+27*b^2*c^2)/c*x^(1/2))/(d*x^2+c)^2+3/256*(7*a^2 *d^2-30*a*b*c*d+55*b^2*c^2)/c^3*(c/d)^(1/4)*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*a rctan(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*b^3/(a*d-b*c)^4*(1/4*(a*d-b*c)/a*x^(1/2)/(b*x^2+a)+3/32*(5*a*d-b*c) /a^2*(a/b)^(1/4)*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. \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 951, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
3/16*(2*sqrt(2)*(b*c - 5*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)*(b*c - 5*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))) + sqrt(2)*(b*c - 5*a*d)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b*c - 5*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^3/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4) + 1/16*((8* b^3*c^2*d^2 + 23*a*b^2*c*d^3 - 7*a^2*b*d^4)*x^(9/2) + (16*b^3*c^3*d + 27*a *b^2*c^2*d^2 + 12*a^2*b*c*d^3 - 7*a^3*d^4)*x^(5/2) + (8*b^3*c^4 + 27*a^2*b *c^2*d^2 - 11*a^3*c*d^3)*sqrt(x))/(a^2*b^3*c^7 - 3*a^3*b^2*c^6*d + 3*a^4*b *c^5*d^2 - a^5*c^4*d^3 + (a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^ 3*d^4 - a^4*b*c^2*d^5)*x^6 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5*d^2 + 3*a^3*b^ 2*c^4*d^3 + a^4*b*c^3*d^4 - a^5*c^2*d^5)*x^4 + (a*b^4*c^7 - a^2*b^3*c^6*d - 3*a^3*b^2*c^5*d^2 + 5*a^4*b*c^4*d^3 - 2*a^5*c^3*d^4)*x^2) + 3/128*(2*sqr t(2)*(55*b^2*c^2*d^2 - 30*a*b*c*d^3 + 7*a^2*d^4)*arctan(1/2*sqrt(2)*(sqrt( 2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sq rt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(55*b^2*c^2*d^2 - 30*a*b*c*d^3 + 7*a^2*d^ 4)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt( sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(55*b^2*c^2...
Leaf count of result is larger than twice the leaf count of optimal. 1253 vs. \(2 (583) = 1166\).
Time = 0.54 (sec) , antiderivative size = 1253, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
1/2*b^3*sqrt(x)/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*( b*x^2 + a)) + 3/4*((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a*b^2*d)*arctan(1 /2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^4 *c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5 *b*c*d^3 + sqrt(2)*a^6*d^4) + 3/4*((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a *b^2*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4)) /(sqrt(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^ 2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^4) + 3/32*(55*(c*d^3)^(1/4)*b^2* c^2*d - 30*(c*d^3)^(1/4)*a*b*c*d^2 + 7*(c*d^3)^(1/4)*a^2*d^3)*arctan(1/2*s qrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^7 - 4 *sqrt(2)*a*b^3*c^6*d + 6*sqrt(2)*a^2*b^2*c^5*d^2 - 4*sqrt(2)*a^3*b*c^4*d^3 + sqrt(2)*a^4*c^3*d^4) + 3/32*(55*(c*d^3)^(1/4)*b^2*c^2*d - 30*(c*d^3)^(1 /4)*a*b*c*d^2 + 7*(c*d^3)^(1/4)*a^2*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d )^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^7 - 4*sqrt(2)*a*b^3*c^6*d + 6*sqrt(2)*a^2*b^2*c^5*d^2 - 4*sqrt(2)*a^3*b*c^4*d^3 + sqrt(2)*a^4*c^3*d ^4) + 3/8*((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a*b^2*d)*log(sqrt(2)*sqrt (x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3* c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^ 4) - 3/8*((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a*b^2*d)*log(-sqrt(2)*sqrt (x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b...
Time = 23.40 (sec) , antiderivative size = 150312, normalized size of antiderivative = 203.40 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
atan(-((((158640570309279744*a^62*d^62 + 461689330549653504*b^62*c^62 + 11 43142782440942075904*a^2*b^60*c^60*d^2 - 25023561715791219916800*a^3*b^59* c^59*d^3 + 392117365329126217482240*a^4*b^58*c^58*d^4 - 469019849064388682 4751104*a^5*b^57*c^57*d^5 + 44594910394380994297724928*a^6*b^56*c^56*d^6 - 346602278587137521765842944*a^7*b^55*c^55*d^7 + 2247504424575830750669045 760*a^8*b^54*c^54*d^8 - 12350275985199266166472704000*a^9*b^53*c^53*d^9 + 58231240117103771404688424960*a^10*b^52*c^52*d^10 - 2380225223137141762882 22085120*a^11*b^51*c^51*d^11 + 851128269824272461500629647360*a^12*b^50*c^ 50*d^12 - 2685471663425998106604003655680*a^13*b^49*c^49*d^13 + 7544170129 817035367585352253440*a^14*b^48*c^48*d^14 - 190680743185073013668351500615 68*a^15*b^47*c^47*d^15 + 43925200681264313454548679131136*a^16*b^46*c^46*d ^16 - 93701324613150775962838140715008*a^17*b^45*c^45*d^17 + 1884640418061 98255158575413329920*a^18*b^44*c^44*d^18 - 3634827683906392986791393309491 20*a^19*b^43*c^43*d^19 + 679593524406433989867498790453248*a^20*b^42*c^42* d^20 - 1234226492432831870920084030488576*a^21*b^41*c^41*d^21 + 2166299333 940469885543144979693568*a^22*b^40*c^40*d^22 - 364988050828568851765026499 8543360*a^23*b^39*c^39*d^23 + 5882337238786870089625427666534400*a^24*b^38 *c^38*d^24 - 9084025233921418993848385529708544*a^25*b^37*c^37*d^25 + 1351 7918768320685624871901691117568*a^26*b^36*c^36*d^26 - 19498271125182229871 738826673618944*a^27*b^35*c^35*d^27 + 273150464430696567053626240715980...