Integrand size = 24, antiderivative size = 718 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {(b c+2 a d) \sqrt {x}}{4 b (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {(7 b c+17 a d) \sqrt {x}}{16 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\sqrt [4]{a} b^{3/4} (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} (b c-a d)^4}-\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4}+\frac {\left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{3/4} \sqrt [4]{d} (b c-a d)^4} \]
1/8*a^(1/4)*b^(3/4)*(7*a*d+5*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4) )/(-a*d+b*c)^4*2^(1/2)-1/8*a^(1/4)*b^(3/4)*(7*a*d+5*b*c)*arctan(1+b^(1/4)* 2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^4*2^(1/2)-1/64*(5*a^2*d^2+70*a*b*c*d+2 1*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(1/4)/(-a*d +b*c)^4*2^(1/2)+1/64*(5*a^2*d^2+70*a*b*c*d+21*b^2*c^2)*arctan(1+d^(1/4)*2^ (1/2)*x^(1/2)/c^(1/4))/c^(3/4)/d^(1/4)/(-a*d+b*c)^4*2^(1/2)+1/16*a^(1/4)*b ^(3/4)*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2)) /(-a*d+b*c)^4*2^(1/2)-1/16*a^(1/4)*b^(3/4)*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1 /2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/(-a*d+b*c)^4*2^(1/2)-1/128*(5*a^2*d^2 +70*a*b*c*d+21*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^(3/4)/d^(1/4)/(-a*d+b*c)^4*2^(1/2)+1/128*(5*a^2*d^2+70*a*b*c*d+21*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^(3/4)/d^(1/ 4)/(-a*d+b*c)^4*2^(1/2)+1/4*(2*a*d+b*c)*x^(1/2)/b/(-a*d+b*c)^2/(d*x^2+c)^2 +1/2*a*x^(1/2)/b/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^2+1/16*(17*a*d+7*b*c)*x^(1 /2)/(-a*d+b*c)^3/(d*x^2+c)
Time = 1.44 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.53 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {\frac {4 (b c-a d) \sqrt {x} \left (b^2 c x^2 \left (11 c+7 d x^2\right )+a^2 d \left (5 c+9 d x^2\right )+a b \left (19 c^2+28 c d x^2+17 d^2 x^4\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^2}+8 \sqrt {2} \sqrt [4]{a} b^{3/4} (5 b c+7 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} \left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{3/4} \sqrt [4]{d}}-8 \sqrt {2} \sqrt [4]{a} b^{3/4} (5 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )+\frac {\sqrt {2} \left (21 b^2 c^2+70 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{3/4} \sqrt [4]{d}}}{64 (b c-a d)^4} \]
((4*(b*c - a*d)*Sqrt[x]*(b^2*c*x^2*(11*c + 7*d*x^2) + a^2*d*(5*c + 9*d*x^2 ) + a*b*(19*c^2 + 28*c*d*x^2 + 17*d^2*x^4)))/((a + b*x^2)*(c + d*x^2)^2) + 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a ^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/( c^(3/4)*d^(1/4)) - 8*Sqrt[2]*a^(1/4)*b^(3/4)*(5*b*c + 7*a*d)*ArcTanh[(Sqrt [2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] + (Sqrt[2]*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt [c] + Sqrt[d]*x)])/(c^(3/4)*d^(1/4)))/(64*(b*c - a*d)^4)
Time = 1.00 (sec) , antiderivative size = 648, normalized size of antiderivative = 0.90, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {368, 970, 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.
| \(\displaystyle \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^4}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int \frac {a c-(4 b c+7 a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {\int \frac {4 b c \left (3 a c-7 (b c+2 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \int \frac {3 a c-7 (b c+2 a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\int \frac {c \left (a (19 b c+5 a d)-3 b (7 b c+17 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\int \frac {a (19 b c+5 a d)-3 b (7 b c+17 a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b (b c-a d) \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac {\frac {b \left (\frac {\frac {8 a b (5 b c+7 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 c-a d}-\frac {\left (21 b^2 c^2+70 a b d c+5 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}}{4 (b c-a d)}-\frac {(7 b c+17 a d) \sqrt {x}}{4 (b c-a d) \left (d x^2+c\right )}\right )}{2 (b c-a d)}-\frac {(b c+2 a d) \sqrt {x}}{2 (b c-a d) \left (d x^2+c\right )^2}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b (b c-a d) \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac {\frac {b \left (\frac {\frac {8 a b (5 b c+7 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 c-a d}-\frac {\left (21 b^2 c^2+70 a b d c+5 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}}{4 (b c-a d)}-\frac {(7 b c+17 a d) \sqrt {x}}{4 (b c-a d) \left (d x^2+c\right )}\right )}{2 (b c-a d)}-\frac {(b c+2 a d) \sqrt {x}}{2 (b c-a d) \left (d x^2+c\right )^2}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {a \sqrt {x}}{4 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\frac {b \left (\frac {\frac {8 a b (7 a d+5 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {\left (5 a^2 d^2+70 a b c d+21 b^2 c^2\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}}{4 (b c-a d)}-\frac {\sqrt {x} (17 a d+7 b c)}{4 \left (c+d x^2\right ) (b c-a d)}\right )}{2 (b c-a d)}-\frac {\sqrt {x} (2 a d+b c)}{2 \left (c+d x^2\right )^2 (b c-a d)}}{4 b (b c-a d)}\right )\) |
2*((a*Sqrt[x])/(4*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (-1/2*((b*c + 2*a*d)*Sqrt[x])/((b*c - a*d)*(c + d*x^2)^2) + (b*(-1/4*((7*b*c + 17*a*d)* Sqrt[x])/((b*c - a*d)*(c + d*x^2)) + ((8*a*b*(5*b*c + 7*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)*Sqrt[x] + Sqrt[b]*x]/(Sqr t[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq rt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(b*c - a*d) - ((21*b^2 *c^2 + 70*a*b*c*d + 5*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]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sq rt[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))/(4*(b*c - a*d))))/(2*(b*c - a*d)))/(4*b *(b*c - a*d)))
3.5.96.3.1 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[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) ^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) /(b*n*(b*c - a*d)*(p + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d *x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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[((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 = 6.99 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.51
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{3}+\frac {1}{16} a b c \,d^{2}+\frac {7}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}+6 a b c d -11 b^{2} c^{2}\right ) \sqrt {x}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+70 a b c d +21 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{\left (a d -b c \right )^{4}}-\frac {2 a b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}\) | \(364\) |
| default | \(\frac {\frac {2 \left (\left (-\frac {9}{32} a^{2} d^{3}+\frac {1}{16} a b c \,d^{2}+\frac {7}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}-\frac {c \left (5 a^{2} d^{2}+6 a b c d -11 b^{2} c^{2}\right ) \sqrt {x}}{32}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}+70 a b c d +21 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c}}{\left (a d -b c \right )^{4}}-\frac {2 a b \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (7 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{4}}\) | \(364\) |
2/(a*d-b*c)^4*(((-9/32*a^2*d^3+1/16*a*b*c*d^2+7/32*b^2*c^2*d)*x^(5/2)-1/32 *c*(5*a^2*d^2+6*a*b*c*d-11*b^2*c^2)*x^(1/2))/(d*x^2+c)^2+1/256*(5*a^2*d^2+ 70*a*b*c*d+21*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* a*b/(a*d-b*c)^4*((1/4*a*d-1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(7*a*d+5*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. \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.19 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b c + 7 \, a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, b c + 7 \, a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, b c + 7 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b c + 7 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a b}{16 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac {{\left (7 \, b^{2} c d + 17 \, a b d^{2}\right )} x^{\frac {9}{2}} + {\left (11 \, b^{2} c^{2} + 28 \, a b c d + 9 \, a^{2} d^{2}\right )} x^{\frac {5}{2}} + {\left (19 \, a b c^{2} + 5 \, a^{2} c d\right )} \sqrt {x}}{16 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (21 \, b^{2} c^{2} + 70 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} \]
-1/16*(2*sqrt(2)*(5*b*c + 7*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)*(5*b*c + 7*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)*(5*b*c + 7*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)*(5*b*c + 7*a*d)*log(-sqrt(2)* a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*a*b/(b^4 *c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + 1/16 *((7*b^2*c*d + 17*a*b*d^2)*x^(9/2) + (11*b^2*c^2 + 28*a*b*c*d + 9*a^2*d^2) *x^(5/2) + (19*a*b*c^2 + 5*a^2*c*d)*sqrt(x))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b ^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2 *d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3 *d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2) + 1/128*(2*sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*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))) + 2*sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*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))) + sqrt(2)*(21*b^2*c^2 + 70*a*b*c*d + 5*a^2*d^2)*log(s qrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4))...
Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (562) = 1124\).
Time = 0.55 (sec) , antiderivative size = 1193, normalized size of antiderivative = 1.66 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/4*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt( 2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^4*c^4 - 4*sqrt(2)*a*b^ 3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + sqrt(2)*a^4* d^4) - 1/4*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4)*a*d)*arctan(-1/2*sqrt(2) *(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^4*c^4 - 4*sqrt( 2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + sqrt( 2)*a^4*d^4) + 1/32*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt( x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^5*d - 4*sqrt(2)*a*b^3*c^4*d^2 + 6*sqrt(2)* a^2*b^2*c^3*d^3 - 4*sqrt(2)*a^3*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) + 1/32*(21* (c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)*a^2*d^2 )*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt (2)*b^4*c^5*d - 4*sqrt(2)*a*b^3*c^4*d^2 + 6*sqrt(2)*a^2*b^2*c^3*d^3 - 4*sq rt(2)*a^3*b*c^2*d^4 + sqrt(2)*a^4*c*d^5) - 1/8*(5*(a*b^3)^(1/4)*b*c + 7*(a *b^3)^(1/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2) *b^4*c^4 - 4*sqrt(2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a ^3*b*c*d^3 + sqrt(2)*a^4*d^4) + 1/8*(5*(a*b^3)^(1/4)*b*c + 7*(a*b^3)^(1/4) *a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^4*c^4 - 4*sqrt(2)*a*b^3*c^3*d + 6*sqrt(2)*a^2*b^2*c^2*d^2 - 4*sqrt(2)*a^3*b*c*d^3 + sqrt(2)*a^4*d^4) + 1/64*(21*(c*d^3)^(1/4)*b^2*c^2 + 70*(c*d^3)^(1/4)...
Time = 12.53 (sec) , antiderivative size = 48950, normalized size of antiderivative = 68.18 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
2*atan(-((((((1473515*a^9*b^7*c*d^10)/2048 - (4375*a^10*b^6*d^11)/8192 + ( 972405*a^2*b^14*c^8*d^3)/8192 + (3824793*a^3*b^13*c^7*d^4)/2048 + (1156047 9*a^4*b^12*c^6*d^5)/1024 + (69456793*a^5*b^11*c^5*d^6)/2048 + (218830061*a ^6*b^10*c^4*d^7)/4096 + (84943363*a^7*b^9*c^3*d^8)/2048 + (6507125*a^8*b^8 *c^2*d^9)/512)*1i)/(a^13*d^13 - b^13*c^13 - 78*a^2*b^11*c^11*d^2 + 286*a^3 *b^10*c^10*d^3 - 715*a^4*b^9*c^9*d^4 + 1287*a^5*b^8*c^8*d^5 - 1716*a^6*b^7 *c^7*d^6 + 1716*a^7*b^6*c^6*d^7 - 1287*a^8*b^5*c^5*d^8 + 715*a^9*b^4*c^4*d ^9 - 286*a^10*b^3*c^3*d^10 + 78*a^11*b^2*c^2*d^11 + 13*a*b^12*c^12*d - 13* a^12*b*c*d^12) + (-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^ 2 + 30664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3* c^3*d^5 + 745500*a^6*b^2*c^2*d^6 + 2593080*a*b^7*c^7*d + 35000*a^7*b*c*d^7 )/(16777216*b^16*c^19*d + 16777216*a^16*c^3*d^17 - 268435456*a*b^15*c^18*d ^2 - 268435456*a^15*b*c^4*d^16 + 2013265920*a^2*b^14*c^17*d^3 - 9395240960 *a^3*b^13*c^16*d^4 + 30534533120*a^4*b^12*c^15*d^5 - 73282879488*a^5*b^11* c^14*d^6 + 134351945728*a^6*b^10*c^13*d^7 - 191931351040*a^7*b^9*c^12*d^8 + 215922769920*a^8*b^8*c^11*d^9 - 191931351040*a^9*b^7*c^10*d^10 + 1343519 45728*a^10*b^6*c^9*d^11 - 73282879488*a^11*b^5*c^8*d^12 + 30534533120*a^12 *b^4*c^7*d^13 - 9395240960*a^13*b^3*c^6*d^14 + 2013265920*a^14*b^2*c^5*d^1 5))^(3/4)*(((-(625*a^8*d^8 + 194481*b^8*c^8 + 13150620*a^2*b^6*c^6*d^2 + 3 0664200*a^3*b^5*c^5*d^3 + 30250150*a^4*b^4*c^4*d^4 + 7301000*a^5*b^3*c^...