Integrand size = 19, antiderivative size = 905 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (a+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a+c x^2\right )}+\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (6 c d^2+8 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4-2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^2 d^4+11 a c d^2 e^2+5 a^2 e^4-2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (3 c d^2+4 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
-1/4*(-c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(c*x^2+a)^2-1/16*(a*e*(5*a*e^2+7*c*d^2 )-2*c*d*(2*a*e^2+3*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c^2/(c*x^2+a)+1/64*e*arctan h((-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))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4+ d*(8*a*e^2+6*c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/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)-1/64*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))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4+d*(8*a*e^ 2+6*c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/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)-1/128*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))*(6*c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4-2*d*(4*a*e^2+3*c*d^2)*c^(1 /2)*(a*e^2+c*d^2)^(1/2))/a^2/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)+1/128*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))*(6* c^2*d^4+11*a*c*d^2*e^2+5*a^2*e^4-2*d*(4*a*e^2+3*c*d^2)*c^(1/2)*(a*e^2+c*d^ 2)^(1/2))/a^2/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 = 3.98 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.41 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (-5 a^3 e^3+6 c^3 d^3 x^3+a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )\right )}{\left (a+c x^2\right )^2}+\frac {\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \left (12 i c d^2+18 \sqrt {a} \sqrt {c} d e-5 i a e^2\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-c d-i \sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \left (-12 i c d^2+18 \sqrt {a} \sqrt {c} d e+5 i a e^2\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c^2} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(-5*a^3*e^3 + 6*c^3*d^3*x^3 + a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) - a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)))/(a + c*x^2 )^2 + ((Sqrt[c]*d + I*Sqrt[a]*e)^2*((12*I)*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e - (5*I)*a*e^2)*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)^2*((-12*I)*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + (5*I)*a*e^2)* ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*S qrt[a]*e)])/Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c^2)
Time = 1.65 (sec) , antiderivative size = 894, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {495, 27, 684, 27, 654, 27, 1483, 27, 1142, 25, 27, 1083, 219, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (6 c d^2+c e x d+5 a e^2\right )}{2 \left (c x^2+a\right )^2}dx}{4 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (6 c d^2+c e x d+5 a e^2\right )}{\left (c x^2+a\right )^2}dx}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )+2 c d e \left (3 c d^2+4 a e^2\right ) x}{2 \sqrt {d+e x} \left (c x^2+a\right )}dx}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (5 a e^2+7 c d^2\right )-2 c d x \left (2 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )+2 c d e \left (3 c d^2+4 a e^2\right ) x}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{4 a c}-\frac {\sqrt {d+e x} \left (a e \left (5 a e^2+7 c d^2\right )-2 c d x \left (2 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int \frac {e \left (\left (c d^2+a e^2\right ) \left (6 c d^2+5 a e^2\right )+2 c d \left (3 c d^2+4 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (5 a e^2+7 c d^2\right )-2 c d x \left (2 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \int \frac {\left (c d^2+a e^2\right ) \left (6 c d^2+5 a e^2\right )+2 c d \left (3 c d^2+4 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (5 a e^2+7 c d^2\right )-2 c d x \left (2 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {e \left (\frac {\int \frac {\sqrt {c d^2+a e^2} \left (\sqrt {2} \sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {c d^2+a e^2} \left (\sqrt {2} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c d^2+5 a e^2\right )+\sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (5 a e^2+7 c d^2\right )-2 c d x \left (2 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \left (\frac {\int \frac {\sqrt {2} \sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c d^2+5 a e^2\right )+\sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (5 a e^2+7 c d^2\right )-2 c d x \left (2 a e^2+3 c d^2\right )\right )}{2 a c \left (a+c x^2\right )}}{8 a c}-\frac {(d+e x)^{5/2} (a e-c d x)}{4 a c \left (a+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {1}{2} \sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{2 a c \left (c x^2+a\right )}}{8 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {1}{2} \sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{2 a c \left (c x^2+a\right )}}{8 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{2 a c \left (c x^2+a\right )}}{8 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {e \left (\frac {-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )-\frac {\left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )-\frac {\left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{2 a c \left (c x^2+a\right )}}{8 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {e \left (\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{2 a c \left (c x^2+a\right )}}{8 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {e \left (\frac {\frac {1}{2} \sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {1}{2} \sqrt [4]{c} \left (2 \sqrt {c} d \left (3 c d^2+4 a e^2\right )-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a c}-\frac {\sqrt {d+e x} \left (a e \left (7 c d^2+5 a e^2\right )-2 c d \left (3 c d^2+2 a e^2\right ) x\right )}{2 a c \left (c x^2+a\right )}}{8 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{4 a c \left (c x^2+a\right )^2}\) |
-1/4*((a*e - c*d*x)*(d + e*x)^(5/2))/(a*c*(a + c*x^2)^2) + (-1/2*(Sqrt[d + e*x]*(a*e*(7*c*d^2 + 5*a*e^2) - 2*c*d*(3*c*d^2 + 2*a*e^2)*x))/(a*c*(a + c *x^2)) + (e*((-((c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*(6*c^(3/2)* d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*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[Sqr t[c]*d - Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*(2*Sqrt[c]*d*(3*c*d^2 + 4*a*e^2) - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*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]]*(6*c^(3/2)*d^3 + 8*a*Sq rt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*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)*(2*Sqrt[c]*d*(3*c*d^2 + 4*a*e^2) - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqr t[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]])))/(2*a*c))/(8*a* c)
3.7.43.3.1 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[((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)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
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 = 3.14 (sec) , antiderivative size = 1283, normalized size of antiderivative = 1.42
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1283\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2333\) |
| default | \(\text {Expression too large to display}\) | \(2333\) |
-3/16/(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)*(-((a^2*(5/3*x^2*e^2+d^2)*c^(9/2)+5/6*c^(7/2)*a^3*e^2+x^2*c^(11 /2)*((5/6*x^2*e^2+2*d^2)*a+c*d^2*x^2))*(a*e^2+c*d^2)^(1/2)+4/3*(e^2*a+3/4* c*d^2)*c^4*d*(c*x^2+a)^2)*e^2*a*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))-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)*(((-2*x^2*(5/12*x^2*e^2+d^2)*a*c^(9/2)-a^2*(5/ 3*x^2*e^2+d^2)*c^(7/2)-c^(11/2)*d^2*x^4-5/6*c^(5/2)*a^3*e^2)*(a*e^2+c*d^2) ^(1/2)+4/3*(e^2*a+3/4*c*d^2)*c^3*d*(c*x^2+a)^2)*((a*e^2+c*d^2)*c)^(1/2)+d* ((a^2*(5/3*x^2*e^2+d^2)*c^(9/2)+5/6*c^(7/2)*a^3*e^2+x^2*c^(11/2)*((5/6*x^2 *e^2+2*d^2)*a+c*d^2*x^2))*(a*e^2+c*d^2)^(1/2)-4/3*(e^2*a+3/4*c*d^2)*c^4*d* (c*x^2+a)^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*(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*x^2*(5/12*x^2*e^2+d^2)*a*c^(9/2)-a^2*(5/3*x^2*e^2+d^2)*c^(7/2)-c^(11 /2)*d^2*x^4-5/6*c^(5/2)*a^3*e^2)*(a*e^2+c*d^2)^(1/2)+4/3*(e^2*a+3/4*c*d^2) *c^3*d*(c*x^2+a)^2)*((a*e^2+c*d^2)*c)^(1/2)+d*((a^2*(5/3*x^2*e^2+d^2)*c^(9 /2)+5/6*c^(7/2)*a^3*e^2+x^2*c^(11/2)*((5/6*x^2*e^2+2*d^2)*a+c*d^2*x^2))*(a *e^2+c*d^2)^(1/2)-4/3*(e^2*a+3/4*c*d^2)*c^4*d*(c*x^2+a)^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...
Leaf count of result is larger than twice the leaf count of optimal. 1751 vs. \(2 (762) = 1524\).
Time = 0.44 (sec) , antiderivative size = 1751, normalized size of antiderivative = 1.93 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/64*((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c ^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^ 4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*log((302 4*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250*a^3*c*d ^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d) + (126*a^3*c^4*d^4*e^6 + 255*a^4*c^3 *d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*sqrt(-(4 41*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt(-(144 *c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4 *sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/( a^5*c^4))) - (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441 *c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))/(a^5*c^4))*l og((3024*c^4*d^8*e^5 + 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 + 6250* a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(e*x + d) - (126*a^3*c^4*d^4*e^6 + 255* a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 + 13*a^6*c^7*d*e^2)*s qrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqr t(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + a^5*c^4*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c ^9)))/(a^5*c^4))) + (a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(-(144*c^3 *d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - a^5*c^4*...
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{3}} \,d x } \]
Time = 0.42 (sec) , antiderivative size = 719, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=\frac {{\left (2 \, {\left (3 \, a c^{2} d^{3} e + 4 \, a^{2} c d e^{3}\right )} e^{2} {\left | c \right |} - {\left (6 \, \sqrt {-a c} c^{2} d^{4} e + 11 \, \sqrt {-a c} a c d^{2} e^{3} + 5 \, \sqrt {-a c} a^{2} e^{5}\right )} {\left | c \right |} {\left | e \right |} + {\left (12 \, c^{3} d^{5} e + 19 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, {\left (a^{3} c^{3} e + \sqrt {-a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {{\left (2 \, {\left (3 \, a c^{2} d^{3} e + 4 \, a^{2} c d e^{3}\right )} e^{2} {\left | c \right |} + {\left (6 \, \sqrt {-a c} c^{2} d^{4} e + 11 \, \sqrt {-a c} a c d^{2} e^{3} + 5 \, \sqrt {-a c} a^{2} e^{5}\right )} {\left | c \right |} {\left | e \right |} + {\left (12 \, c^{3} d^{5} e + 19 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, {\left (a^{3} c^{3} e - \sqrt {-a c} a^{2} c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} + \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {e x + d} c^{3} d^{6} e + 8 \, {\left (e x + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} - 23 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} + 32 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} - 17 \, \sqrt {e x + d} a c^{2} d^{4} e^{3} - 9 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 14 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 16 \, \sqrt {e x + d} a^{2} c d^{2} e^{5} - 5 \, \sqrt {e x + d} a^{3} e^{7}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )}^{2} a^{2} c^{2}} \]
1/32*(2*(3*a*c^2*d^3*e + 4*a^2*c*d*e^3)*e^2*abs(c) - (6*sqrt(-a*c)*c^2*d^4 *e + 11*sqrt(-a*c)*a*c*d^2*e^3 + 5*sqrt(-a*c)*a^2*e^5)*abs(c)*abs(e) + (12 *c^3*d^5*e + 19*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*abs(c))*arctan(sqrt(e*x + d )/sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 + a^3*c^2*e^2)*a^2*c^ 3))/(a^2*c^3)))/((a^3*c^3*e + sqrt(-a*c)*a^2*c^3*d)*sqrt(-c^2*d - sqrt(-a* c)*c*e)*abs(e)) + 1/32*(2*(3*a*c^2*d^3*e + 4*a^2*c*d*e^3)*e^2*abs(c) + (6* sqrt(-a*c)*c^2*d^4*e + 11*sqrt(-a*c)*a*c*d^2*e^3 + 5*sqrt(-a*c)*a^2*e^5)*a bs(c)*abs(e) + (12*c^3*d^5*e + 19*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*abs(c))*a rctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d - sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 + a ^3*c^2*e^2)*a^2*c^3))/(a^2*c^3)))/((a^3*c^3*e - sqrt(-a*c)*a^2*c^3*d)*sqrt (-c^2*d + sqrt(-a*c)*c*e)*abs(e)) + 1/16*(6*(e*x + d)^(7/2)*c^3*d^3*e - 18 *(e*x + d)^(5/2)*c^3*d^4*e + 18*(e*x + d)^(3/2)*c^3*d^5*e - 6*sqrt(e*x + d )*c^3*d^6*e + 8*(e*x + d)^(7/2)*a*c^2*d*e^3 - 23*(e*x + d)^(5/2)*a*c^2*d^2 *e^3 + 32*(e*x + d)^(3/2)*a*c^2*d^3*e^3 - 17*sqrt(e*x + d)*a*c^2*d^4*e^3 - 9*(e*x + d)^(5/2)*a^2*c*e^5 + 14*(e*x + d)^(3/2)*a^2*c*d*e^5 - 16*sqrt(e* x + d)*a^2*c*d^2*e^5 - 5*sqrt(e*x + d)*a^3*e^7)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + a*e^2)^2*a^2*c^2)
Time = 10.29 (sec) , antiderivative size = 2569, normalized size of antiderivative = 2.84 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
((e*(3*c*d^3 + 4*a*d*e^2)*(d + e*x)^(7/2))/(8*a^2) + ((d + e*x)^(3/2)*(7*a ^2*d*e^5 + 9*c^2*d^5*e + 16*a*c*d^3*e^3))/(8*a^2*c) - ((d + e*x)^(1/2)*(5* a^3*e^7 + 6*c^3*d^6*e + 17*a*c^2*d^4*e^3 + 16*a^2*c*d^2*e^5))/(16*a^2*c^2) - (e*(d + e*x)^(5/2)*(9*a^2*e^4 + 18*c^2*d^4 + 23*a*c*d^2*e^2))/(16*a^2*c ))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 + 2*a*c*e^2)*(d + e*x )^2 - (4*c^2*d^3 + 4*a*c*d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 + 2*a*c*d^ 2*e^2) - 2*atanh((25*e^10*(d + e*x)^(1/2)*(- (9*d^7)/(256*a^5*c) - (105*d* e^6)/(4096*a^2*c^4) - (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a ^4*c^2) - (25*e^7*(-a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(-a^15*c ^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^5*e^9)/(2048*a^3) + (325*d* e^13)/(2048*a*c^2) + (63*c*d^7*e^7)/(512*a^4) + (449*d^3*e^11)/(1024*a^2*c ) + (125*e^14*(-a^15*c^9)^(1/2))/(2048*a^8*c^7) + (95*d^2*e^12*(-a^15*c^9) ^(1/2))/(512*a^9*c^6) + (381*d^4*e^10*(-a^15*c^9)^(1/2))/(2048*a^10*c^5) + (63*d^6*e^8*(-a^15*c^9)^(1/2))/(1024*a^11*c^4))) + (21*d^2*e^8*(d + e*x)^ (1/2)*(- (9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) - (385*d^3*e^4)/ (4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(-a^15*c^9)^(1/2)) /(4096*a^9*c^9) - (21*d^2*e^5*(-a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/( 32*((325*d*e^13)/(2048*c^3) + (63*d^7*e^7)/(512*a^3) + (449*d^3*e^11)/(102 4*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) + (125*e^14*(-a^15*c^9)^(1/2))/(2048 *a^7*c^8) + (95*d^2*e^12*(-a^15*c^9)^(1/2))/(512*a^8*c^7) + (381*d^4*e^...