Integrand size = 19, antiderivative size = 930 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
1/6*e*(-7*a*e^2+3*c*d^2)/a/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)+1/2*(c*d*x+a*e)/a /(a*e^2+c*d^2)/(e*x+d)^(3/2)/(c*x^2+a)+1/2*c*d*e*(-19*a*e^2+c*d^2)/a/(a*e^ 2+c*d^2)^3/(e*x+d)^(1/2)+1/8*c^(3/4)*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)^( 1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2) )^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4+d*(-19*a*e^2+c*d^2)*c^(1/2)*(a* e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^( 1/2))^(1/2)-1/8*c^(3/4)*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2 )+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2* d^4+34*a*c*d^2*e^2-7*a^2*e^4+d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/ 2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/ 16*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d )^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a ^2*e^4-d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7 /2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/16*c^(3/4)*e*ln((e*x+d )*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a* e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4-d*(-19*a*e^2+c* d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*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 = 3.13 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \left (4 a^3 e^5-3 c^3 d^3 x (d+e x)^2+a^2 c e^3 \left (55 d^2+54 d e x+7 e^2 x^2\right )+a c^2 d e \left (-9 d^3-9 d^2 e x+61 d e^2 x^2+57 e^3 x^3\right )\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {3 \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (-2 i c d+7 \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^4}+\frac {3 \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (2 i c d+7 \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^4}}{12 a^{3/2}} \]
((-2*Sqrt[a]*(4*a^3*e^5 - 3*c^3*d^3*x*(d + e*x)^2 + a^2*c*e^3*(55*d^2 + 54 *d*e*x + 7*e^2*x^2) + a*c^2*d*e*(-9*d^3 - 9*d^2*e*x + 61*d*e^2*x^2 + 57*e^ 3*x^3)))/((c*d^2 + a*e^2)^3*(d + e*x)^(3/2)*(a + c*x^2)) + (3*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*((-2*I)*c*d + 7*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)])/(Sqr t[c]*d + I*Sqrt[a]*e)^4 + (3*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*((2*I)*c*d + 7*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[c]*d - I*Sqrt[a]*e)^4)/(12*a^(3/ 2))
Time = 2.06 (sec) , antiderivative size = 1013, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {496, 27, 655, 27, 655, 654, 27, 1483, 27, 1142, 25, 27, 1083, 219, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right )^2 (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac {\int -\frac {2 c d^2+5 c e x d+7 a e^2}{2 (d+e x)^{5/2} \left (c x^2+a\right )}dx}{2 a \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 c d^2+5 c e x d+7 a e^2}{(d+e x)^{5/2} \left (c x^2+a\right )}dx}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {\frac {\int \frac {c \left (2 d \left (c d^2+6 a e^2\right )+e \left (3 c d^2-7 a e^2\right ) x\right )}{(d+e x)^{3/2} \left (c x^2+a\right )}dx}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \int \frac {2 d \left (c d^2+6 a e^2\right )+e \left (3 c d^2-7 a e^2\right ) x}{(d+e x)^{3/2} \left (c x^2+a\right )}dx}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {\frac {c \left (\frac {\int \frac {2 c^2 d^4+15 a c e^2 d^2+c e \left (c d^2-19 a e^2\right ) x d-7 a^2 e^4}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{a e^2+c d^2}+\frac {2 d e \left (c d^2-19 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 \int \frac {e \left (c^2 d^4+34 a c e^2 d^2+c \left (c d^2-19 a e^2\right ) (d+e x) d-7 a^2 e^4\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}+\frac {2 d e \left (c d^2-19 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 e \int \frac {c^2 d^4+34 a c e^2 d^2+c \left (c d^2-19 a e^2\right ) (d+e x) d-7 a^2 e^4}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{a e^2+c d^2}+\frac {2 d e \left (c d^2-19 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 e \left (\frac {\int \frac {\sqrt {2} \left (c^2 d^4+34 a c e^2 d^2-7 a^2 e^4\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2-7 a^2 e^4\right )+\sqrt [4]{c} \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}+\frac {2 d e \left (c d^2-19 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c \left (\frac {2 e \left (\frac {\int \frac {\sqrt {2} \left (c^2 d^4+34 a c e^2 d^2-7 a^2 e^4\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2-7 a^2 e^4\right )+\sqrt [4]{c} \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{a e^2+c d^2}+\frac {2 d e \left (c d^2-19 a e^2\right )}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{a e^2+c d^2}+\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}}{4 a \left (a e^2+c d^2\right )}+\frac {a e+c d x}{2 a \left (a+c x^2\right ) (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 d e \left (c d^2-19 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\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 (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\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 (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 d e \left (c d^2-19 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\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 (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\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 (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 d e \left (c d^2-19 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 d e \left (c d^2-19 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 e \left (\frac {\frac {\left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 d e \left (c d^2-19 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 e \left (\frac {\frac {\left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\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^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\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^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {\frac {2 e \left (3 c d^2-7 a e^2\right )}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 d e \left (c d^2-19 a e^2\right )}{\left (c d^2+a e^2\right ) \sqrt {d+e x}}+\frac {2 e \left (\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {1}{2} \sqrt [4]{c} \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{c d^2+a e^2}\right )}{c d^2+a e^2}}{4 a \left (c d^2+a e^2\right )}\) |
(a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*(d + e*x)^(3/2)*(a + c*x^2)) + ((2*e*(3 *c*d^2 - 7*a*e^2))/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + (c*((2*d*e*(c*d^2 - 19*a*e^2))/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (2*e*((-((c^(1/4)*Sqrt[Sqr t[c]*d + Sqrt[c*d^2 + a*e^2]]*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqrt [c]*d*(c*d^2 - 19*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^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c* d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^( 1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e* x)])/2)/(2*Sqrt[2]*Sqrt[c]*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (-((c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqrt[c]*d*(c*d^2 - 19*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^2*d^4 + 34*a*c*d^2* e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sq rt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]* Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/2)/(2*Sqrt[2]*Sqrt[c]*Sqrt[c*d^2 + a*e ^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])))/(c*d^2 + a*e^2)))/(c*d^2 +...
3.7.37.3.1 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*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[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), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) ), x] + Simp[1/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 4.36 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.25
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1162\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3541\) |
| default | \(\text {Expression too large to display}\) | \(3541\) |
7/4/(a*e^2+c*d^2)^(7/2)/(e*x+d)^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a *e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(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)*(e*x+d)^(3/2)*(2*((a*e^2+c*d^2)*c)^(1/2 )+2*c*d)^(1/2)*((-19/7*(-1/19*a*(-19*e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2- 1/19*x^2*c^(5/2)*d^2)*d*(a*e^2+c*d^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7* c^2*d^4)*(c*x^2+a))*((a*e^2+c*d^2)*c)^(1/2)-d*(-19/7*d*(-1/19*a*(-19*e^2*x ^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)-1/19*c^(7/2)*d^2*x^2)*(a*e^2+c*d^2)^(1/2)+ (a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^2*d^4)*c*(c*x^2+a)))*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)*( e*x+d)^(3/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*((-19/7*(-1/19*a*(-19 *e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2-1/19*x^2*c^(5/2)*d^2)*d*(a*e^2+c*d^2 )^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^2*d^4)*(c*x^2+a))*((a*e^2+c*d^2)*c )^(1/2)-d*(-19/7*d*(-1/19*a*(-19*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)-1/19 *c^(7/2)*d^2*x^2)*(a*e^2+c*d^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^2*d^ 4)*c*(c*x^2+a)))*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))+(-8/21*(-3/4*d^3*x*(e*x+d)^2*c^3-9/4* e*d*a*(-19/3*e^3*x^3-61/9*d*e^2*x^2+d^2*e*x+d^3)*c^2+55/4*(7/55*x^2*e^2+54 /55*d*e*x+d^2)*e^3*a^2*c+a^3*e^5)*(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)+(e*x+d)^(3/2)*(19/7*d...
Leaf count of result is larger than twice the leaf count of optimal. 8281 vs. \(2 (780) = 1560\).
Time = 3.29 (sec) , antiderivative size = 8281, normalized size of antiderivative = 8.90 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2014 vs. \(2 (780) = 1560\).
Time = 0.51 (sec) , antiderivative size = 2014, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
-1/4*((a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 + a^4*e^7)^2*(c^2 *d^3*e - 19*a*c*d*e^3)*abs(c) - (sqrt(-a*c)*c^5*d^10*e + 37*sqrt(-a*c)*a*c ^4*d^8*e^3 + 98*sqrt(-a*c)*a^2*c^3*d^6*e^5 + 82*sqrt(-a*c)*a^3*c^2*d^4*e^7 + 13*sqrt(-a*c)*a^4*c*d^2*e^9 - 7*sqrt(-a*c)*a^5*e^11)*abs(-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)*abs(c) + (2*a*c^9*d^17*e + 27*a^2*c^8*d^15*e^3 + 113*a^3*c^7*d^13*e^5 + 223*a^4*c^6*d^11*e^7 + 225*a ^5*c^5*d^9*e^9 + 97*a^6*c^4*d^7*e^11 - 13*a^7*c^3*d^5*e^13 - 27*a^8*c^2*d^ 3*e^15 - 7*a^9*c*d*e^17)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^4*d^7 + 3 *a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6 + sqrt((a*c^4*d^7 + 3*a ^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)^2 - (a*c^4*d^8 + 4*a^2*c ^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 + 3 *a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/(a*c^4*d^6 + 3*a^2*c^3 *d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/((a^2*c^6*d^12*e + 6*a^3*c^5*d ^10*e^3 + 15*a^4*c^4*d^8*e^5 + 20*a^5*c^3*d^6*e^7 + 15*a^6*c^2*d^4*e^9 + 6 *a^7*c*d^2*e^11 + a^8*e^13 - sqrt(-a*c)*a*c^6*d^13 - 6*sqrt(-a*c)*a^2*c^5* d^11*e^2 - 15*sqrt(-a*c)*a^3*c^4*d^9*e^4 - 20*sqrt(-a*c)*a^4*c^3*d^7*e^6 - 15*sqrt(-a*c)*a^5*c^2*d^5*e^8 - 6*sqrt(-a*c)*a^6*c*d^3*e^10 - sqrt(-a*c)* a^7*d*e^12)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-a*c^3*d^6*e - 3*a^2*c^2*d^4 *e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/4*((a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 + a^4*e^7)^2*(c^2*d^3*e - 19*a*c*d*e^3)*abs(c) + (sq...
Time = 13.90 (sec) , antiderivative size = 12390, normalized size of antiderivative = 13.32 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
atan((((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040* a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 2 82560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^1 4*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 10494 40*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) + (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2 ) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^ 6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9 *c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7 *d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35* a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2* d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 83 7*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c *d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^3 2 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^2 7*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a ^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*...