Integrand size = 19, antiderivative size = 736 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
-2/3*e/(a*e^2+c*d^2)/(e*x+d)^(3/2)-4*c*d*e/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1 /2*c^(3/4)*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))*(3*c*d^2-a*e^2-2* d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(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)-1/2*c^(3/4)*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))*(3*c*d^2-a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(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)-1/4*c^(3/4)*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))*(3*c*d^2-a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(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)+1/4*c^(3/4)*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))*(3*c*d^2-a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2) ^(1/2))/(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 = 1.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=-\frac {2 e \left (a e^2+c d (7 d+6 e x)\right )}{3 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {\sqrt {c} \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \left (-i \sqrt {c} d+\sqrt {a} e\right )^3}+\frac {\sqrt {c} \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \left (i \sqrt {c} d+\sqrt {a} e\right )^3} \]
(-2*e*(a*e^2 + c*d*(7*d + 6*e*x)))/(3*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (Sqrt[c]*Sqrt[-(c*d) - 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[a]*((-I)*Sq rt[c]*d + Sqrt[a]*e)^3) + (Sqrt[c]*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*ArcT an[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[ a]*e)])/(Sqrt[a]*(I*Sqrt[c]*d + Sqrt[a]*e)^3)
Time = 1.17 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {482, 655, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right ) (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 482 |
| \(\displaystyle \frac {c \int \frac {d-e x}{(d+e x)^{3/2} \left (c x^2+a\right )}dx}{a e^2+c d^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {c d^2-2 c e x d-a e^2}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{a e^2+c d^2}-\frac {4 d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {c \left (\frac {2 \int \frac {e \left (3 c d^2-2 c (d+e x) d-a e^2\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}-\frac {4 d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {2 e \int \frac {3 c d^2-2 c (d+e x) d-a e^2}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}-\frac {4 d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\int \frac {\sqrt {2} \left (3 c d^2-a e^2\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-a e^2\right )+\sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}-\frac {4 d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\int \frac {\sqrt {2} \left (3 c d^2-a e^2\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-a e^2\right )+\sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}-\frac {4 d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {4 d e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{c d^2+a e^2}-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {4 d e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{c d^2+a e^2}-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {4 d e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{c d^2+a e^2}-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {4 d e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{c d^2+a e^2}-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {4 d e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{c d^2+a e^2}-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {c \left (\frac {2 e \left (\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {1}{2} \sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {4 d e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\right )}{c d^2+a e^2}-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}\) |
(-2*e)/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + (c*((-4*d*e)/((c*d^2 + a*e^2) *Sqrt[d + e*x]) + (2*e*((-((c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]* (3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(c^(1/4)*(-((S qrt[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]])])/Sqrt[Sqrt[c]*d - Sqrt[c *d^2 + a*e^2]]) - (c^(1/4)*(3*c*d^2 - a*e^2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + 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]]) + (-((c^(1/4)*Sqrt[S qrt[c]*d + Sqrt[c*d^2 + a*e^2]]*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*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 ]])])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*(3*c*d^2 - a*e^2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + 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]])))/(c*d^2 + a*e^2)))/(c*d^2 + a*e^2)
3.7.23.3.1 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), x_Symbol] :> Simp[d*((c + d*x)^(n + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[b/(b*c^2 + a*d^2) I nt[(c + d*x)^(n + 1)*((c - d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[n, -1]
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 = 2.62 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.08
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {\left (e x +d \right )^{\frac {3}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\frac {\left (e x +d \right )^{\frac {3}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\left (\frac {2 \sqrt {e^{2} a +c \,d^{2}}\, \left (6 x c d e +e^{2} a +7 c \,d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}{3}+\left (\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (a c \,e^{2}-3 c^{2} d^{2}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right ) \left (e x +d \right )^{\frac {3}{2}}\right ) e^{2} a}{\left (e^{2} a +c \,d^{2}\right )^{\frac {5}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \left (e x +d \right )^{\frac {3}{2}} a e}\) | \(793\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2547\) |
| default | \(\text {Expression too large to display}\) | \(2547\) |
-1/(a*e^2+c*d^2)^(5/2)/(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)*(-1/4*(e*x+d)^(3/2)*(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)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2) *((e^2*a-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2)-3*c*d^2)*((a*e^2+c*d^2)*c)^(1/2)- d*e^2*a*c+3*c^2*d^3+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^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)^(3/2)*(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)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*((e^2*a-2*d*c^(1/2) *(a*e^2+c*d^2)^(1/2)-3*c*d^2)*((a*e^2+c*d^2)*c)^(1/2)-d*e^2*a*c+3*c^2*d^3+ 2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d^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))+(2/3*(a*e^2+c*d^2)^ (1/2)*(6*c*d*e*x+a*e^2+7*c*d^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)+(arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c* d^2)*c)^(1/2)+2*c*d)^(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))-arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2 )*c)^(1/2)+2*c*d)^(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)))*(a*c*e^2-3*c^2*d^2+2*c^(3/2)*(a*e^2+c*d^2)^(1/2)*d) *(e*x+d)^(3/2))*e^2*a)/(e*x+d)^(3/2)/a/e
Leaf count of result is larger than twice the leaf count of optimal. 5149 vs. \(2 (597) = 1194\).
Time = 0.40 (sec) , antiderivative size = 5149, normalized size of antiderivative = 7.00 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1215 vs. \(2 (597) = 1194\).
Time = 0.37 (sec) , antiderivative size = 1215, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=-\frac {{\left (2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}^{2} a c d e {\left | c \right |} + {\left (3 \, \sqrt {-a c} c^{3} d^{6} e + 5 \, \sqrt {-a c} a c^{2} d^{4} e^{3} + \sqrt {-a c} a^{2} c d^{2} e^{5} - \sqrt {-a c} a^{3} e^{7}\right )} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |} {\left | c \right |} - {\left (c^{6} d^{11} e + 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} - 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} + \sqrt {{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c^{4} d^{8} e + 4 \, a^{2} c^{3} d^{6} e^{3} + 6 \, a^{3} c^{2} d^{4} e^{5} + 4 \, a^{4} c d^{2} e^{7} + a^{5} e^{9} + \sqrt {-a c} c^{4} d^{9} + 4 \, \sqrt {-a c} a c^{3} d^{7} e^{2} + 6 \, \sqrt {-a c} a^{2} c^{2} d^{5} e^{4} + 4 \, \sqrt {-a c} a^{3} c d^{3} e^{6} + \sqrt {-a c} a^{4} d e^{8}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |}} - \frac {{\left (2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}^{2} a c d e {\left | c \right |} - {\left (3 \, \sqrt {-a c} c^{3} d^{6} e + 5 \, \sqrt {-a c} a c^{2} d^{4} e^{3} + \sqrt {-a c} a^{2} c d^{2} e^{5} - \sqrt {-a c} a^{3} e^{7}\right )} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |} {\left | c \right |} - {\left (c^{6} d^{11} e + 3 \, a c^{5} d^{9} e^{3} + 2 \, a^{2} c^{4} d^{7} e^{5} - 2 \, a^{3} c^{3} d^{5} e^{7} - 3 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} - \sqrt {{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c^{4} d^{8} e + 4 \, a^{2} c^{3} d^{6} e^{3} + 6 \, a^{3} c^{2} d^{4} e^{5} + 4 \, a^{4} c d^{2} e^{7} + a^{5} e^{9} - \sqrt {-a c} c^{4} d^{9} - 4 \, \sqrt {-a c} a c^{3} d^{7} e^{2} - 6 \, \sqrt {-a c} a^{2} c^{2} d^{5} e^{4} - 4 \, \sqrt {-a c} a^{3} c d^{3} e^{6} - \sqrt {-a c} a^{4} d e^{8}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |}} - \frac {2 \, {\left (6 \, {\left (e x + d\right )} c d e + c d^{2} e + a e^{3}\right )}}{3 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]
-(2*(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)^2*a*c*d*e*abs(c) + (3*sqrt(-a*c) *c^3*d^6*e + 5*sqrt(-a*c)*a*c^2*d^4*e^3 + sqrt(-a*c)*a^2*c*d^2*e^5 - sqrt( -a*c)*a^3*e^7)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*abs(c) - (c^6*d^11 *e + 3*a*c^5*d^9*e^3 + 2*a^2*c^4*d^7*e^5 - 2*a^3*c^3*d^5*e^7 - 3*a^4*c^2*d ^3*e^9 - a^5*c*d*e^11)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^3*d^5 + 2*a*c ^2*d^3*e^2 + a^2*c*d*e^4 + sqrt((c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^ 2 - (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(c^3*d^4 + 2*a *c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a* c^4*d^8*e + 4*a^2*c^3*d^6*e^3 + 6*a^3*c^2*d^4*e^5 + 4*a^4*c*d^2*e^7 + a^5* e^9 + sqrt(-a*c)*c^4*d^9 + 4*sqrt(-a*c)*a*c^3*d^7*e^2 + 6*sqrt(-a*c)*a^2*c ^2*d^5*e^4 + 4*sqrt(-a*c)*a^3*c*d^3*e^6 + sqrt(-a*c)*a^4*d*e^8)*sqrt(-c^2* d - sqrt(-a*c)*c*e)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)) - (2*(c^2*d^ 4*e + 2*a*c*d^2*e^3 + a^2*e^5)^2*a*c*d*e*abs(c) - (3*sqrt(-a*c)*c^3*d^6*e + 5*sqrt(-a*c)*a*c^2*d^4*e^3 + sqrt(-a*c)*a^2*c*d^2*e^5 - sqrt(-a*c)*a^3*e ^7)*abs(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5)*abs(c) - (c^6*d^11*e + 3*a*c^ 5*d^9*e^3 + 2*a^2*c^4*d^7*e^5 - 2*a^3*c^3*d^5*e^7 - 3*a^4*c^2*d^3*e^9 - a^ 5*c*d*e^11)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 - sqrt((c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^ 6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(c^3*d^4 + 2*a*c^2*d^2*e^ 2 + a^2*c*e^4)))/(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c^4*d^8*...
Time = 11.62 (sec) , antiderivative size = 7908, normalized size of antiderivative = 10.74 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
- ((2*e)/(3*(a*e^2 + c*d^2)) + (4*c*d*e*(d + e*x))/(a*e^2 + c*d^2)^2)/(d + e*x)^(3/2) - atan((((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16 *e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*(-a^3*c^3)^(1/2 ) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c ^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4 *e^6)))^(1/2)*(96*a*c^13*d^18*e^3 - (d + e*x)^(1/2)*(-(a^2*e^5*(-a^3*c^3)^ (1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a ^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^ 10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2 *d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13* d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^1 0*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8* c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^ 4*e^21 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^ 12*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6* e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(-a^3*c^3) ^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(- a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^...