Integrand size = 19, antiderivative size = 887 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
-1/2*d*e*(e*x+d)^(3/2)/a/c-1/2*(-c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(c*x^2+a)-1/ 2*e*(-5*a*e^2+c*d^2)*(e*x+d)^(1/2)/a/c^2+1/8*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^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4+d*(13*a*e^2+c*d^2)*c^(1/ 2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4)*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)-1/8*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^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4+d*(13*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2) ^(1/2))/a/c^(9/4)*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)-1/16*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^2*d^4-4*a*c*d^2*e^2- 5*a^2*e^4-d*(13*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4)*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)+1/16*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^2*d^4-4*a*c*d^2*e^2-5*a^2*e^4-d*(13*a*e^2+c* d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4)*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 = 1.75 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.36 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+4 e^2 x^2\right )\right )}{a+c x^2}+\frac {\left (\sqrt {c} d+i \sqrt {a} e\right )^3 \left (2 i \sqrt {c} d+5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-c d-i \sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {c} d-i \sqrt {a} e\right )^3 \left (-2 i \sqrt {c} d+5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2} c^2} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + 4*e^2*x^2)))/(a + c*x^2) + ((Sqrt[c]*d + I*Sqrt[a]*e)^3*((2*I)*Sqrt[c]* d + 5*Sqrt[a]*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]*Sqrt[c]*e] + ((Sqrt[c ]*d - I*Sqrt[a]*e)^3*((-2*I)*Sqrt[c]*d + 5*Sqrt[a]*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]*Sqrt[c]*e])/(4*a^(3/2)*c^2)
Time = 1.62 (sec) , antiderivative size = 856, normalized size of antiderivative = 0.97, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {495, 27, 653, 27, 653, 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.
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (2 c d^2-3 c e x d+5 a e^2\right )}{2 \left (c x^2+a\right )}dx}{2 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (2 c d^2-3 c e x d+5 a e^2\right )}{c x^2+a}dx}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {\frac {\int \frac {c \sqrt {d+e x} \left (2 d \left (c d^2+4 a e^2\right )-e \left (c d^2-5 a e^2\right ) x\right )}{c x^2+a}dx}{c}-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} \left (2 d \left (c d^2+4 a e^2\right )-e \left (c d^2-5 a e^2\right ) x\right )}{c x^2+a}dx-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle \frac {\frac {\int \frac {\left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )+c d e \left (c d^2+13 a e^2\right ) x}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{c}-\frac {2 e \sqrt {d+e x} \left (c d^2-5 a e^2\right )}{c}-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {2 \int \frac {e \left (\left (c d^2-5 a e^2\right ) \left (c d^2+a e^2\right )+c d \left (c d^2+13 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} \left (c d^2-5 a e^2\right )}{c}-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 e \int \frac {\left (c d^2-5 a e^2\right ) \left (c d^2+a e^2\right )+c d \left (c d^2+13 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} \left (c d^2-5 a e^2\right )}{c}-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {2 e \left (\frac {\int \frac {\sqrt {c d^2+a e^2} \left (\sqrt {2} \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {c d^2+a e^2} \left (\sqrt {2} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d^2-5 a e^2\right )+\sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (c d^2-5 a e^2\right )}{c}-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 e \left (\frac {\int \frac {\sqrt {2} \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c d^2-5 a e^2\right )+\sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (c d^2-5 a e^2\right )}{c}-2 d e (d+e x)^{3/2}}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-2 d e (d+e x)^{3/2}-\frac {2 e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{c}+\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}}{4 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-2 d e (d+e x)^{3/2}-\frac {2 e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{c}+\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}}{4 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-2 d e (d+e x)^{3/2}-\frac {2 e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{c}+\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}}{4 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-2 d e (d+e x)^{3/2}-\frac {2 e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{c}+\frac {2 e \left (\frac {\frac {\left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}}{4 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-2 d e (d+e x)^{3/2}-\frac {2 e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{c}+\frac {2 e \left (\frac {\frac {\left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}}{4 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-2 d e (d+e x)^{3/2}-\frac {2 e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{c}+\frac {2 e \left (\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {1}{2} \sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-\sqrt {c} d \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c}}{4 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}\) |
-1/2*((a*e - c*d*x)*(d + e*x)^(5/2))/(a*c*(a + c*x^2)) + ((-2*e*(c*d^2 - 5 *a*e^2)*Sqrt[d + e*x])/c - 2*d*e*(d + e*x)^(3/2) + (2*e*((-((c^(1/4)*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[Sqr t[c]*d - Sqrt[c*d^2 + a*e^2]])])/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 + 13*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 [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^(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]])])/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 + 13*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[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])))/c)/(4*a*c)
3.7.31.3.1 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*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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 = 3.24 (sec) , antiderivative size = 1015, normalized size of antiderivative = 1.14
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1015\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2224\) |
| default | \(\text {Expression too large to display}\) | \(2224\) |
| risch | \(\text {Expression too large to display}\) | \(2226\) |
1/4/(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)/ c^(11/2)*(1/4*(((-x^2*c^(9/2)*d^2-a*(-5*e^2*x^2+d^2)*c^(7/2)+5*c^(5/2)*a^2 *e^2)*(a*e^2+c*d^2)^(1/2)+13*c^3*(e^2*a+1/13*c*d^2)*d*(c*x^2+a))*((a*e^2+c *d^2)*c)^(1/2)+((a*(-5*e^2*x^2+d^2)*c^(9/2)+c^(11/2)*d^2*x^2-5*a^2*e^2*c^( 7/2))*(a*e^2+c*d^2)^(1/2)-13*c^4*(e^2*a+1/13*c*d^2)*d*(c*x^2+a))*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)*(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*(((-x^2*c^(9/2)*d^2- a*(-5*e^2*x^2+d^2)*c^(7/2)+5*c^(5/2)*a^2*e^2)*(a*e^2+c*d^2)^(1/2)+13*c^3*( e^2*a+1/13*c*d^2)*d*(c*x^2+a))*((a*e^2+c*d^2)*c)^(1/2)+((a*(-5*e^2*x^2+d^2 )*c^(9/2)+c^(11/2)*d^2*x^2-5*a^2*e^2*c^(7/2))*(a*e^2+c*d^2)^(1/2)-13*c^4*( e^2*a+1/13*c*d^2)*d*(c*x^2+a))*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)*(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*(-3*(-4/3*x^2*e^2+d*e*x+d^2)*e*a*c^(9/2)+c^(11/2)*d^3 *x+5*a^2*e^3*c^(7/2))*(e*x+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))-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...
Leaf count of result is larger than twice the leaf count of optimal. 2091 vs. \(2 (737) = 1474\).
Time = 0.47 (sec) , antiderivative size = 2091, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
-1/8*((a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c *d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d ^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3 *c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2* c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*s qrt(e*x + d) + (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2* e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4 *d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e ^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a ^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c ^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/ (a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^ 2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8 *e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4 *d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^ 11 - 625*a^5*e^13)*sqrt(e*x + d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^ 6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 + 4*a^4*c^7*d* e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^1 0 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + ...
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 596, normalized size of antiderivative = 0.67 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e^{3}}{c^{2}} + \frac {{\left ({\left (c^{2} d^{3} e + 13 \, a c d e^{3}\right )} a^{2} e^{2} {\left | c \right |} - {\left (\sqrt {-a c} c^{2} d^{4} e - 4 \, \sqrt {-a c} a c d^{2} e^{3} - 5 \, \sqrt {-a c} a^{2} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} + {\left (2 \, a c^{3} d^{5} e + 9 \, a^{2} c^{2} d^{3} e^{3} - 5 \, a^{3} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left ({\left (\sqrt {-a c} c d^{3} e + 13 \, \sqrt {-a c} a d e^{3}\right )} a^{2} e^{2} {\left | c \right |} - {\left (a c^{2} d^{4} e - 4 \, a^{2} c d^{2} e^{3} - 5 \, a^{3} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} + {\left (2 \, \sqrt {-a c} a c^{2} d^{5} e + 9 \, \sqrt {-a c} a^{2} c d^{3} e^{3} - 5 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d + \sqrt {-a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - \sqrt {e x + d} c^{2} d^{4} e - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d e^{3} + \sqrt {e x + d} a^{2} e^{5}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )} a c^{2}} \]
2*sqrt(e*x + d)*e^3/c^2 + 1/4*((c^2*d^3*e + 13*a*c*d*e^3)*a^2*e^2*abs(c) - (sqrt(-a*c)*c^2*d^4*e - 4*sqrt(-a*c)*a*c*d^2*e^3 - 5*sqrt(-a*c)*a^2*e^5)* abs(a)*abs(c)*abs(e) + (2*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 - 5*a^3*c*d*e^5) *abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^3*d + sqrt(a^2*c^6*d^2 - (a*c^3*d ^2 + a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a^2*c^3*e + sqrt(-a*c)*a*c^3*d)*sqrt (-c^2*d - sqrt(-a*c)*c*e)*abs(a)*abs(e)) + 1/4*((sqrt(-a*c)*c*d^3*e + 13*s qrt(-a*c)*a*d*e^3)*a^2*e^2*abs(c) - (a*c^2*d^4*e - 4*a^2*c*d^2*e^3 - 5*a^3 *e^5)*abs(a)*abs(c)*abs(e) + (2*sqrt(-a*c)*a*c^2*d^5*e + 9*sqrt(-a*c)*a^2* c*d^3*e^3 - 5*sqrt(-a*c)*a^3*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a* c^3*d - sqrt(a^2*c^6*d^2 - (a*c^3*d^2 + a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((a ^2*c^3*d + sqrt(-a*c)*a^2*c^2*e)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a)*abs( e)) + 1/2*((e*x + d)^(3/2)*c^2*d^3*e - sqrt(e*x + d)*c^2*d^4*e - 3*(e*x + d)^(3/2)*a*c*d*e^3 + sqrt(e*x + d)*a^2*e^5)/(((e*x + d)^2*c - 2*(e*x + d)* c*d + c*d^2 + a*e^2)*a*c^2)
Time = 0.89 (sec) , antiderivative size = 4192, normalized size of antiderivative = 4.73 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
(((a^2*e^5 - c^2*d^4*e)*(d + e*x)^(1/2))/(2*a) + ((c^2*d^3*e - 3*a*c*d*e^3 )*(d + e*x)^(3/2))/(2*a))/(c^3*(d + e*x)^2 + c^3*d^2 + a*c^2*e^2 - 2*c^3*d *(d + e*x)) - atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/( 16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7* (-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8 ) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((885*d^5*e^9)/ 2 + (491*a*d^3*e^11)/(2*c) + (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 + (35*c^2*d^9*e^5)/(2*a^2) + (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^7) - (335 *d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/ (a^4*c^5) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^4) + (35*d^8*e^6*(-a^9*c ^9)^(1/2))/(4*a^6*c^3)) + (d^3*e^7*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105* d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/ (64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9* c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/ 2)*308i)/((35*a^2*c^5*d^9*e^5)/2 + (329*a^3*c^4*d^7*e^7)/2 + (885*a^4*c^3* d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 + (125*a^2*e^14*(- a^9*c^9)^(1/2))/(4*c^4) + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^2) - (204*d^4 *e^10*(-a^9*c^9)^(1/2))/c^2 - (335*a*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^3) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a*c)) + (d^5*e^5*(-a^9*c^9)^(1/2)*(d + e*x )^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3...