Integrand size = 19, antiderivative size = 781 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\frac {4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2-\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {e \left (2 c^{3/2} d^3+2 a \sqrt {c} d e^2+\left (3 c d^2-a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{7/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
2/3*e*(e*x+d)^(3/2)/c+4*d*e*(e*x+d)^(1/2)/c-1/2*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))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)-(-a*e^2+3*c*d^2)*(a*e ^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d ^2)^(1/2))^(1/2)+1/2*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))*(2*c^(3/ 2)*d^3+2*a*d*e^2*c^(1/2)-(-a*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^( 1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)+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))*(2*c^(3/2)*d^3+2*a*d*e^2*c^(1/2)+(-a*e^2+3*c* d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+( a*e^2+c*d^2)^(1/2))^(1/2)-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))*(2*c^(3/ 2)*d^3+2*a*d*e^2*c^(1/2)+(-a*e^2+3*c*d^2)*(a*e^2+c*d^2)^(1/2))/c^(7/4)*2^( 1/2)/(a*e^2+c*d^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 = 0.85 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.32 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\frac {2 \sqrt {c} e \sqrt {d+e x} (7 d+e x)+\frac {3 i \left (\sqrt {c} d+i \sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (\sqrt {c} d-i \sqrt {a} e\right )^3 \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{3 c^{3/2}} \]
(2*Sqrt[c]*e*Sqrt[d + e*x]*(7*d + e*x) + ((3*I)*(Sqrt[c]*d + I*Sqrt[a]*e)^ 3*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I *Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - ((3*I)*(Sqrt[ c]*d - I*Sqrt[a]*e)^3*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(Sqrt[a]*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c] *e]))/(3*c^(3/2))
Time = 1.53 (sec) , antiderivative size = 724, normalized size of antiderivative = 0.93, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {481, 653, 27, 654, 25, 27, 1483, 27, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx\) |
| \(\Big \downarrow \) 481 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (c d^2+2 c e x d-a e^2\right )}{c x^2+a}dx}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x\right )}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{c}+4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {d \left (c d^2-3 a e^2\right )+e \left (3 c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (c x^2+a\right )}dx+4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 \int -\frac {e \left (2 d \left (c d^2+a e^2\right )-\left (3 c d^2-a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}+4 d e \sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 d e \sqrt {d+e x}-2 \int \frac {e \left (2 d \left (c d^2+a e^2\right )-\left (3 c d^2-a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 d e \sqrt {d+e x}-2 e \int \frac {2 d \left (c d^2+a e^2\right )-\left (3 c d^2-a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\int \frac {\sqrt {c d^2+a e^2} \left (2 \sqrt {2} d \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (2 \sqrt {c d^2+a e^2} d+\frac {3 c d^2-a e^2}{\sqrt {c}}\right ) \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {c d^2+a e^2} \left (2 \sqrt {2} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} d+\sqrt [4]{c} \left (2 \sqrt {c d^2+a e^2} d+\frac {3 c d^2-a e^2}{\sqrt {c}}\right ) \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} d \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} d+\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} d \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} d+\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}-2 e \left (\frac {-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {1}{2} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\frac {1}{2} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c}}+\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt [4]{c}}+\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 e (d+e x)^{3/2}}{3 c}+\frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (3 c d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}+\frac {\left (3 c d^2+2 \sqrt {c} \sqrt {c d^2+a e^2} d-a e^2\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 d e \sqrt {d+e x}-2 e \left (\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {1}{2} \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {1}{2} \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{c}+\frac {2 e (d+e x)^{3/2}}{3 c}\) |
(2*e*(d + e*x)^(3/2))/(3*c) + (4*d*e*Sqrt[d + e*x] - 2*e*(((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)*(-((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]] - ((3*c*d^2 - a*e^2 + 2*Sqrt[c]*d*S qrt[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 ]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + ((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])*ArcTa nh[(c^(1/4)*((Sqrt[2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])/c^(1/4) + 2*S qrt[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]] + ((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]*c^(3/4)* Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])))/c
3.7.18.3.1 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)/(b*(n - 1))), x] + Simp[1/b Int[(c + d*x)^(n - 2)*(Simp[b *c^2 - a*d^2 + 2*b*c*d*x, x]/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 1]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
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[((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 = 5.66 (sec) , antiderivative size = 739, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\frac {\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \left (\left (e^{2} a -2 d \sqrt {c}\, \sqrt {e^{2} a +c \,d^{2}}-3 c \,d^{2}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-d \,e^{2} a c +3 c^{2} d^{3}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d^{2}\right ) \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+e^{2} \left (\frac {14 \sqrt {e x +d}\, \left (\frac {e x}{7}+d \right ) c^{\frac {3}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}{3}+\left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (a c \,e^{2}-3 c^{2} d^{2}+2 c^{\frac {3}{2}} \sqrt {e^{2} a +c \,d^{2}}\, d \right )\right ) a}{c^{\frac {5}{2}} \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, a e}\) | \(739\) |
| risch | \(\text {Expression too large to display}\) | \(1594\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1599\) |
| default | \(\text {Expression too large to display}\) | \(1599\) |
1/c^(5/2)*(-1/4*(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)*(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*(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 )*(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))+e^2*(14/3*(e*x+d)^(1/2)*(1/7*e*x+d)*c^(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)+(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))*a)/(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/e
Leaf count of result is larger than twice the leaf count of optimal. 1641 vs. \(2 (634) = 1268\).
Time = 0.61 (sec) , antiderivative size = 1641, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\text {Too large to display} \]
-1/6*(3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25* c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^ 3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) + (10*a*c^4*d^5*e^2 - 20*a^2*c^3*d^3* e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2 - a^2*c^5*e^2)*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a* c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 + a*c^3*sqrt(-(25*c^4 *d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^ 4*e^10)/(a*c^7)))/(a*c^3))) - 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2* d*e^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4* e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))*log((5*c^4*d^8*e - 1 4*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*sqrt(e*x + d) - (10*a*c^4*d ^5*e^2 - 20*a^2*c^3*d^3*e^4 + 2*a^3*c^2*d*e^6 + (a*c^6*d^2 - a^2*c^5*e^2)* sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c *d^2*e^8 + a^4*e^10)/(a*c^7)))*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e ^4 + a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^3))) + 3*c*sqrt(-(c^2*d^5 - 10*a*c*d^3*e^2 + 5*a^2*d*e^4 - a*c^3*sqrt(-(25*c^4*d^8*e^2 - 100*a*c^3*d^ 6*e^4 + 110*a^2*c^2*d^4*e^6 - 20*a^3*c*d^2*e^8 + a^4*e^10)/(a*c^7)))/(a*c^ 3))*log((5*c^4*d^8*e - 14*a^2*c^2*d^4*e^5 - 8*a^3*c*d^2*e^7 + a^4*e^9)*...
\[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{a + c x^{2}}\, dx \]
\[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{c x^{2} + a} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.54 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=-\frac {{\left (\sqrt {-a c} c^{4} d^{4} e - 3 \, \sqrt {-a c} a c^{3} d^{2} e^{3} + {\left (3 \, \sqrt {-a c} a c d^{2} e - \sqrt {-a c} a^{2} e^{3}\right )} c^{2} e^{2} - 2 \, {\left (a c^{3} d^{3} e + a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {-a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {-a c} c^{4} d^{4} e - 3 \, \sqrt {-a c} a c^{3} d^{2} e^{3} + {\left (3 \, \sqrt {-a c} a c d^{2} e - \sqrt {-a c} a^{2} e^{3}\right )} c^{2} e^{2} + 2 \, {\left (a c^{3} d^{3} e + a^{2} c^{2} d e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} + a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {-a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {e x + d} c^{2} d e\right )}}{3 \, c^{3}} \]
-(sqrt(-a*c)*c^4*d^4*e - 3*sqrt(-a*c)*a*c^3*d^2*e^3 + (3*sqrt(-a*c)*a*c*d^ 2*e - sqrt(-a*c)*a^2*e^3)*c^2*e^2 - 2*(a*c^3*d^3*e + a^2*c^2*d*e^3)*abs(c) *abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d + sqrt(c^8*d^2 - (c^4*d^2 + a*c ^3*e^2)*c^4))/c^4))/((a*c^4*d - sqrt(-a*c)*a*c^3*e)*sqrt(-c^2*d - sqrt(-a* c)*c*e)*abs(e)) + (sqrt(-a*c)*c^4*d^4*e - 3*sqrt(-a*c)*a*c^3*d^2*e^3 + (3* sqrt(-a*c)*a*c*d^2*e - sqrt(-a*c)*a^2*e^3)*c^2*e^2 + 2*(a*c^3*d^3*e + a^2* c^2*d*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^4*d - sqrt(c^8*d^2 - (c^4*d^2 + a*c^3*e^2)*c^4))/c^4))/((a*c^4*d + sqrt(-a*c)*a*c^3*e)*sqrt( -c^2*d + sqrt(-a*c)*c*e)*abs(e)) + 2/3*((e*x + d)^(3/2)*c^2*e + 6*sqrt(e*x + d)*c^2*d*e)/c^3
Time = 0.48 (sec) , antiderivative size = 3481, normalized size of antiderivative = 4.46 \[ \int \frac {(d+e x)^{5/2}}{a+c x^2} \, dx=\text {Too large to display} \]
(2*e*(d + e*x)^(3/2))/(3*c) - atan((a^3*e^8*(d + e*x)^(1/2)*((e^5*(-a^3*c^ 7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3 ) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/ (2*a*c^6))^(1/2)*32i)/((16*a^4*e^11)/c^2 + 64*a^2*d^4*e^7 - 80*c^2*d^8*e^3 - (160*a^3*d^2*e^9)/c + 160*a*c*d^6*e^5 - (160*d^5*e^6*(-a^3*c^7)^(1/2))/ c^3 - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^4 + (32*a^2*d*e^10*(-a^3*c^7)^(1/ 2))/c^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/(a*c^2)) - (d^5*e^3*(-a^3*c^7)^(1 /2)*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1/2))/(4*c^7) - d^5/(4*a*c) + (5*d^3 *e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + (5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^ 5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a*c^6))^(1/2)*160i)/((16*a^5*e^11)/c - 160*a^4*d^2*e^9 - 80*a*c^3*d^8*e^3 + 64*a^3*c*d^4*e^7 + 160*a^2*c^2*d^6* e^5 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/c - (160*a*d^5*e^6*(-a^3*c^7)^(1/2))/ c^2 + (32*a^3*d*e^10*(-a^3*c^7)^(1/2))/c^4 - (288*a^2*d^3*e^8*(-a^3*c^7)^( 1/2))/c^3) + (d^3*e^5*(-a^3*c^7)^(1/2)*(d + e*x)^(1/2)*((e^5*(-a^3*c^7)^(1 /2))/(4*c^7) - d^5/(4*a*c) + (5*d^3*e^2)/(2*c^2) - (5*a*d*e^4)/(4*c^3) + ( 5*d^4*e*(-a^3*c^7)^(1/2))/(4*a^2*c^5) - (5*d^2*e^3*(-a^3*c^7)^(1/2))/(2*a* c^6))^(1/2)*320i)/(16*a^4*e^11 - 80*c^4*d^8*e^3 + 160*a*c^3*d^6*e^5 - 160* a^3*c*d^2*e^9 + 64*a^2*c^2*d^4*e^7 + (160*d^7*e^4*(-a^3*c^7)^(1/2))/a - (1 60*d^5*e^6*(-a^3*c^7)^(1/2))/c - (288*a*d^3*e^8*(-a^3*c^7)^(1/2))/c^2 + (3 2*a^2*d*e^10*(-a^3*c^7)^(1/2))/c^3) - (a*d*e^7*(-a^3*c^7)^(1/2)*(d + e*...