Integrand size = 19, antiderivative size = 845 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
1/2*e*(-5*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1/2*(c*d*x+a*e)/a/( a*e^2+c*d^2)/(c*x^2+a)/(e*x+d)^(1/2)+1/8*c^(1/4)*e*arctanh((-c^(1/4)*2^(1/ 2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+ c*d^2)^(1/2))^(1/2))*(c^(3/2)*d^3+13*a*d*e^2*c^(1/2)+(-5*a*e^2+c*d^2)*(a*e ^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1 /2))^(1/2)-1/8*c^(1/4)*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2) +(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(c^(3/ 2)*d^3+13*a*d*e^2*c^(1/2)+(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c *d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/16*c^(1/4)*e*l n((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^( 1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c^(3/2)*d^3+13*a*d*e^2*c^(1/2)-(-5*a*e^2 +c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)+(a*e ^2+c*d^2)^(1/2))^(1/2)+1/16*c^(1/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/ 2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c ^(3/2)*d^3+13*a*d*e^2*c^(1/2)-(-5*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))/a/(a*e ^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 1.83 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.40 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \left (-4 a^2 e^3+c^2 d^2 x (d+e x)+a c e \left (2 d^2+d e x-5 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x} \left (a+c x^2\right )}+\frac {i \left (2 c d+5 i \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (2 c d-5 i \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \]
((2*Sqrt[a]*(-4*a^2*e^3 + c^2*d^2*x*(d + e*x) + a*c*e*(2*d^2 + d*e*x - 5*e ^2*x^2)))/((c*d^2 + a*e^2)^2*Sqrt[d + e*x]*(a + c*x^2)) + (I*(2*c*d + (5*I )*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e *x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/((Sqrt[c]*d + I*Sqrt[a]*e)^2*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - (I*(2*c*d - (5*I)*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqr t[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)]) /((Sqrt[c]*d - I*Sqrt[a]*e)^2*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(4*a^(3 /2))
Time = 1.49 (sec) , antiderivative size = 911, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {496, 27, 655, 27, 654, 27, 1483, 27, 1142, 25, 27, 1083, 219, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right )^2 (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {\int -\frac {2 c d^2+3 c e x d+5 a e^2}{2 (d+e x)^{3/2} \left (c x^2+a\right )}dx}{2 a \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 c d^2+3 c e x d+5 a e^2}{(d+e x)^{3/2} \left (c x^2+a\right )}dx}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (2 d \left (c d^2+4 a e^2\right )+e \left (c d^2-5 a e^2\right ) x\right )}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{a e^2+c d^2}+\frac {2 e \left (c d^2-5 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \int \frac {2 d \left (c d^2+4 a e^2\right )+e \left (c d^2-5 a e^2\right ) x}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{a e^2+c d^2}+\frac {2 e \left (c d^2-5 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {2 c \int \frac {e \left (d \left (c d^2+13 a e^2\right )+\left (c d^2-5 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}+\frac {2 e \left (c d^2-5 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 c e \int \frac {d \left (c d^2+13 a e^2\right )+\left (c d^2-5 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}+\frac {2 e \left (c d^2-5 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {2 c e \left (\frac {\int \frac {\sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d^2+13 a e^2\right )+\sqrt [4]{c} \left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d^2+13 a e^2\right )+\sqrt [4]{c} \left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}+\frac {2 e \left (c d^2-5 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 c e \left (\frac {\int \frac {\sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d^2+13 a e^2\right )+\sqrt [4]{c} \left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d^2+13 a e^2\right )+\sqrt [4]{c} \left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}+\frac {2 e \left (c d^2-5 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) \sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}+\frac {\frac {2 e \left (c d^2-5 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 c e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}+\frac {\frac {2 e \left (c d^2-5 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 c e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}-\frac {1}{2} \sqrt [4]{c} \left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}+\frac {\frac {2 e \left (c d^2-5 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 c e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}-\frac {\left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {\left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}+\frac {\frac {2 e \left (c d^2-5 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 c e \left (\frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {c}}-\frac {\left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {c}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}+\frac {\frac {2 e \left (c d^2-5 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 c e \left (\frac {-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )}+\frac {\frac {2 e \left (c d^2-5 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 c e \left (\frac {\frac {1}{2} \sqrt [4]{c} \left (\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}-d \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (c d^3+13 a e^2 d-\frac {\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
(a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*(a + c*x^2)) + ((2*e*(c*d ^2 - 5*a*e^2))/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (2*c*e*((-((Sqrt[Sqrt[c]* d + Sqrt[c*d^2 + a*e^2]]*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*a* e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(c^(1/4)*(-((Sqrt[2]*Sqrt[Sqrt[c]*d + Sq rt[c*d^2 + a*e^2]])/c^(1/4)) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])) + (c^(1/4)*(((c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])/Sqrt[c] - d*(c*d^2 + 1 3*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[ c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/2)/(2*Sqrt[2]*Sqrt[c]* Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (-((Sqrt[Sqrt [c]*d + Sqrt[c*d^2 + a*e^2]]*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(c^(1/4)*((Sqrt[2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])/c^(1/4) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])) + (c^(1/4)*(c*d^3 + 13*a*d*e^2 - ((c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])/ Sqrt[c])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c *d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/2)/(2*Sqrt[2]*Sqrt[c]*S qrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])))/(c*d^2 + a*e^2 ))/(4*a*(c*d^2 + a*e^2))
3.7.36.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) ), x] + Simp[1/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 3.74 (sec) , antiderivative size = 1059, normalized size of antiderivative = 1.25
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1059\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3080\) |
| default | \(\text {Expression too large to display}\) | \(3080\) |
-5/4/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2) /(e*x+d)^(1/2)/c^(1/2)*(1/4*(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d )^(1/2)*(((e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*(1/13*a*(13 *e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2+1/13*x^2*c^(5/2)*d^2)*d)*((a*e^2+c*d ^2)*c)^(1/2)-(c*(e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*d*(1/ 13*a*(13*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)+1/13*c^(7/2)*d^2*x^2))*d)*(4 *(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*ln((e* x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+ c*d^2)^(1/2))-1/4*(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*(( (e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*(1/13*a*(13*e^2*x^2+d ^2)*c^(3/2)+c^(1/2)*a^2*e^2+1/13*x^2*c^(5/2)*d^2)*d)*((a*e^2+c*d^2)*c)^(1/ 2)-(c*(e^2*a-1/5*c*d^2)*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)+13/5*d*(1/13*a*(13*e ^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)+1/13*c^(7/2)*d^2*x^2))*d)*(4*(a*e^2+c* d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*ln((e*x+d)*c^(1/ 2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/ 2))+(8/5*(-1/2*e*(-5/2*x^2*e^2+1/2*d*e*x+d^2)*a*c^(3/2)-1/4*d^2*x*(e*x+d)* c^(5/2)+a^2*e^3*c^(1/2))*(a*e^2+c*d^2)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2 )-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)+(e*x+d)^(1/2)*(c*(e^2*a-1/5*c*d^2 )*(c*x^2+a)*(a*e^2+c*d^2)^(1/2)-13/5*d*(1/13*a*(13*e^2*x^2+d^2)*c^(5/2)+a^ 2*e^2*c^(3/2)+1/13*c^(7/2)*d^2*x^2))*e*(arctan((2*c^(1/2)*(e*x+d)^(1/2)...
Leaf count of result is larger than twice the leaf count of optimal. 5698 vs. \(2 (691) = 1382\).
Time = 1.15 (sec) , antiderivative size = 5698, normalized size of antiderivative = 6.74 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1397 vs. \(2 (691) = 1382\).
Time = 0.43 (sec) , antiderivative size = 1397, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c ) - (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^ 2*c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e ^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*a rctan(sqrt(e*x + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 + s qrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (a*c^3*d^6 + 3*a^2*c ^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a ^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*d^8*e + 4*a^3*c^3*d^6*e^3 + 6*a^4*c^2*d^4*e^5 + 4*a^5*c*d^2*e^7 + a^6*e^9 + sqrt (-a*c)*a*c^4*d^9 + 4*sqrt(-a*c)*a^2*c^3*d^7*e^2 + 6*sqrt(-a*c)*a^3*c^2*d^5 *e^4 + 4*sqrt(-a*c)*a^4*c*d^3*e^6 + sqrt(-a*c)*a^5*d*e^8)*sqrt(-c^2*d - sq rt(-a*c)*c*e)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/4*((a*c^2* d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) + (sqrt(-a *c)*c^3*d^7*e + 15*sqrt(-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*ab s(c) + 2*(a*c^6*d^12*e + 8*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4* c^3*d^6*e^7 + 17*a^5*c^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*arctan(sqrt(e *x + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt((a*c^3*d ^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^...
Time = 12.38 (sec) , antiderivative size = 8777, normalized size of antiderivative = 10.39 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
atan((((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 15 4*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2 *e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2 )*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c ^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6* c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4 *e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c ^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080 *a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^1 4 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^ 3*e^20) + 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^1 7*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^ 9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232* a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^ 13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11* d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c ^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11* c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^...