Integrand size = 19, antiderivative size = 663 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
-2*e/(a*e^2+c*d^2)/(e*x+d)^(1/2)+1/2*c^(1/4)*e*arctanh((-c^(1/4)*2^(1/2)*( e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^ 2)^(1/2))^(1/2))*(2*d*c^(1/2)-(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(3/2)*2^( 1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/2*c^(1/4)*e*arctanh((c^(1/4)* 2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a *e^2+c*d^2)^(1/2))^(1/2))*(2*d*c^(1/2)-(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^ (3/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/4*c^(1/4)*e*ln((e*x+ d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a *e^2+c*d^2)^(1/2))^(1/2))*(2*d*c^(1/2)+(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^ (3/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/4*c^(1/4)*e*ln((e*x+ d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a *e^2+c*d^2)^(1/2))^(1/2))*(2*d*c^(1/2)+(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^ (3/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.37 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {i \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \left (-i \sqrt {c} d+\sqrt {a} e\right )^2}-\frac {i \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \left (i \sqrt {c} d+\sqrt {a} e\right )^2} \]
(-2*e)/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (I*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c ]*e]*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[a]*((-I)*Sqrt[c]*d + Sqrt[a]*e)^2) - (I*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(Sqrt[a]*(I*Sqrt[c]*d + Sqrt[a]*e)^2)
Time = 1.10 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {482, 654, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right ) (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 482 |
| \(\displaystyle \frac {c \int \frac {d-e x}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 c \int \frac {e (d-e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c e \int \frac {d-e x}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {2 c e \left (\frac {\int \frac {2 \sqrt {2} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (2 d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} d+\sqrt [4]{c} \left (2 d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c e \left (\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} d+\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c e \left (\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} d \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {2 \sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} d+\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2 c e \left (\frac {\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {1}{2} \left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {1}{2} \left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c e \left (\frac {\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {1}{2} \left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {1}{2} \left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c e \left (\frac {\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}+\frac {\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2 c e \left (\frac {\frac {\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt {2} \left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\sqrt {2} \left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt [4]{c}}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 c e \left (\frac {\frac {\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (2 \sqrt {c} d+\sqrt {c d^2+a e^2}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt [4]{c}}-\frac {\left (2 \sqrt {c} d-\sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}-\frac {2 e}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 c e \left (\frac {-\frac {\left (2 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {1}{2} \left (\sqrt {a e^2+c d^2}+2 \sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\frac {1}{2} \left (\sqrt {a e^2+c d^2}+2 \sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )-\frac {\left (2 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}}{2 \sqrt {2} c^{3/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}-\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\) |
(-2*e)/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (2*c*e*((-(((2*Sqrt[c]*d - Sqrt[c *d^2 + a*e^2])*Sqrt[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]]) - ((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[c*d^2 + a*e^2]*Sqrt[Sqr t[c]*d + Sqrt[c*d^2 + a*e^2]]) + (-(((2*Sqrt[c]*d - Sqrt[c*d^2 + a*e^2])*S qrt[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[Sqr t[c]*d - Sqrt[c*d^2 + a*e^2]])])/Sqrt[Sqrt[c]*d - Sqrt[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[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])))/(c*d^2 + a*e^2)
3.7.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d*((c + d*x)^(n + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[b/(b*c^2 + a*d^2) I nt[(c + d*x)^(n + 1)*((c - d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[n, -1]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 2.56 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {\left (\left (2 d \sqrt {c}+\sqrt {e^{2} a +c \,d^{2}}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-c d \sqrt {e^{2} a +c \,d^{2}}-2 c^{\frac {3}{2}} d^{2}\right ) \sqrt {e x +d}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+\frac {\left (\left (2 d \sqrt {c}+\sqrt {e^{2} a +c \,d^{2}}\right ) \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-c d \sqrt {e^{2} a +c \,d^{2}}-2 c^{\frac {3}{2}} d^{2}\right ) \sqrt {e x +d}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}\, \ln \left (\left (e x +d \right ) \sqrt {c}+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}+\sqrt {e^{2} a +c \,d^{2}}\right )}{4}+e^{2} \left (-2 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \sqrt {c}+\sqrt {e x +d}\, \left (\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}}\right )\right ) \left (\sqrt {e^{2} a +c \,d^{2}}\, c -2 c^{\frac {3}{2}} d \right )\right ) a}{\sqrt {e x +d}\, \sqrt {4 \sqrt {e^{2} a +c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (e^{2} a +c \,d^{2}\right ) c}-2 c d}\, \left (e^{2} a +c \,d^{2}\right )^{\frac {3}{2}} \sqrt {c}\, a e}\) | \(722\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2096\) |
| default | \(\text {Expression too large to display}\) | \(2096\) |
1/(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)*(-1/4*((2*d*c^(1/2)+(a*e^2+c*d^2)^(1/2))*((a*e^2+c*d^2)*c)^(1/ 2)-c*d*(a*e^2+c*d^2)^(1/2)-2*c^(3/2)*d^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)*(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*d*c^(1/2)+(a*e^2+c*d^2)^(1/2 ))*((a*e^2+c*d^2)*c)^(1/2)-c*d*(a*e^2+c*d^2)^(1/2)-2*c^(3/2)*d^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 )*(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*(-2*(a*e ^2+c*d^2)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2 *c*d)^(1/2)*c^(1/2)+(e*x+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*e^2+c*d^2)^(1/2)*c-2*c^(3/2)*d))*a)/(a*e^ 2+c*d^2)^(3/2)/c^(1/2)/a/e
Leaf count of result is larger than twice the leaf count of optimal. 2863 vs. \(2 (528) = 1056\).
Time = 0.33 (sec) , antiderivative size = 2863, normalized size of antiderivative = 4.32 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
-1/2*((c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 + (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^ 3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d ^2*e^10 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a ^4*e^6))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 - 2 *a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8) *sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c ^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 + (a*c^3*d^ 6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4 *d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^ 7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6))) - (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^2 + (a* c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(-(9*c^3*d^4* e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a ^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^1 0 + a^7*e^12)))/(a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6 ))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2 - 2*a^...
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 659, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 \, e}{{\left (c d^{2} + a e^{2}\right )} \sqrt {e x + d}} + \frac {{\left ({\left (c d^{2} e + a e^{3}\right )}^{2} a e {\left | c \right |} + 2 \, {\left (\sqrt {-a c} c d^{3} e + \sqrt {-a c} a d e^{3}\right )} {\left | -c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (c^{3} d^{6} e + 2 \, a c^{2} d^{4} e^{3} + a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} + a c d e^{2} + \sqrt {{\left (c^{2} d^{3} + a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}}}{c^{2} d^{2} + a c e^{2}}}}\right )}{{\left (a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} - \sqrt {-a c} c^{2} d^{5} - 2 \, \sqrt {-a c} a c d^{3} e^{2} - \sqrt {-a c} a^{2} d e^{4}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -c d^{2} e - a e^{3} \right |}} + \frac {{\left ({\left (c d^{2} e + a e^{3}\right )}^{2} a e {\left | c \right |} - 2 \, {\left (\sqrt {-a c} c d^{3} e + \sqrt {-a c} a d e^{3}\right )} {\left | -c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (c^{3} d^{6} e + 2 \, a c^{2} d^{4} e^{3} + a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d^{3} + a c d e^{2} - \sqrt {{\left (c^{2} d^{3} + a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}}}{c^{2} d^{2} + a c e^{2}}}}\right )}{{\left (a c^{2} d^{4} e + 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} + \sqrt {-a c} c^{2} d^{5} + 2 \, \sqrt {-a c} a c d^{3} e^{2} + \sqrt {-a c} a^{2} d e^{4}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -c d^{2} e - a e^{3} \right |}} \]
-2*e/((c*d^2 + a*e^2)*sqrt(e*x + d)) + ((c*d^2*e + a*e^3)^2*a*e*abs(c) + 2 *(sqrt(-a*c)*c*d^3*e + sqrt(-a*c)*a*d*e^3)*abs(-c*d^2*e - a*e^3)*abs(c) - (c^3*d^6*e + 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*abs(c))*arctan(sqrt(e*x + d) /sqrt(-(c^2*d^3 + a*c*d*e^2 + sqrt((c^2*d^3 + a*c*d*e^2)^2 - (c^2*d^4 + 2* a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 + a*c*e^2)))/(c^2*d^2 + a*c*e^2)))/((a*c^2 *d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5 - sqrt(-a*c)*c^2*d^5 - 2*sqrt(-a*c)*a*c *d^3*e^2 - sqrt(-a*c)*a^2*d*e^4)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-c*d^2* e - a*e^3)) + ((c*d^2*e + a*e^3)^2*a*e*abs(c) - 2*(sqrt(-a*c)*c*d^3*e + sq rt(-a*c)*a*d*e^3)*abs(-c*d^2*e - a*e^3)*abs(c) - (c^3*d^6*e + 2*a*c^2*d^4* e^3 + a^2*c*d^2*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d^3 + a*c*d*e ^2 - sqrt((c^2*d^3 + a*c*d*e^2)^2 - (c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(c ^2*d^2 + a*c*e^2)))/(c^2*d^2 + a*c*e^2)))/((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5 + sqrt(-a*c)*c^2*d^5 + 2*sqrt(-a*c)*a*c*d^3*e^2 + sqrt(-a*c)*a^2 *d*e^4)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(-c*d^2*e - a*e^3))
Time = 10.51 (sec) , antiderivative size = 4471, normalized size of antiderivative = 6.74 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
- atan((((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6* e^4 + 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c* d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2* e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^8*d^9*e^3 - (d + e*x)^(1/2)*(-(a* c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2)) /(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)* (64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7 *d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a* c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2)) /(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)* 1i + ((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8) - (-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e ^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(-a^3* c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3 *d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 6 4*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6 *d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*...