Integrand size = 19, antiderivative size = 920 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\frac {(a e+c d x) \sqrt {d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-2 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
1/4*(c*d*x+a*e)*(e*x+d)^(1/2)/a/(a*e^2+c*d^2)/(c*x^2+a)^2+1/16*(a*e*(7*a*e ^2+c*d^2)+6*c*d*(2*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)/a^2/(a*e^2+c*d^2)^2/(c*x^ 2+a)+3/64*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))*(2*c^2*d^4+5*a*c*d ^2*e^2+7*a^2*e^4+2*d*(2*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(1 /4)/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-3/64 *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))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a ^2*e^4+2*d*(2*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(1/4)/(a*e^2 +c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-3/128*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))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4-2*d*(2*a*e^ 2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(1/4)/(a*e^2+c*d^2)^(5/2)*2^(1 /2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+3/128*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))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4-2*d*(2*a*e^2+c*d^2)*c^(1/2)*( a*e^2+c*d^2)^(1/2))/a^2/c^(1/4)/(a*e^2+c*d^2)^(5/2)*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 = 2.06 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (11 a^3 e^3+6 c^3 d^3 x^3+a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {3 i \left (4 c d^2+10 i \sqrt {a} \sqrt {c} d e-7 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (4 c d^2-10 i \sqrt {a} \sqrt {c} d e-7 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(11*a^3*e^3 + 6*c^3*d^3*x^3 + a^2*c*e*(5*d^2 + 1 6*d*e*x + 7*e^2*x^2) + a*c^2*d*x*(10*d^2 + d*e*x + 12*e^2*x^2)))/((c*d^2 + a*e^2)^2*(a + c*x^2)^2) + ((3*I)*(4*c*d^2 + (10*I)*Sqrt[a]*Sqrt[c]*d*e - 7*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]*e)^2*Sqrt[-(c*d) - I*Sqrt[a]* Sqrt[c]*e]) - ((3*I)*(4*c*d^2 - (10*I)*Sqrt[a]*Sqrt[c]*d*e - 7*a*e^2)*ArcT an[(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)^2*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/ (32*a^(5/2))
Time = 1.92 (sec) , antiderivative size = 1001, normalized size of antiderivative = 1.09, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {496, 27, 686, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right )^3 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac {\int -\frac {6 c d^2+5 c e x d+7 a e^2}{2 \sqrt {d+e x} \left (c x^2+a\right )^2}dx}{4 a \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 c d^2+5 c e x d+7 a e^2}{\sqrt {d+e x} \left (c x^2+a\right )^2}dx}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (6 c d x \left (2 a e^2+c d^2\right )+a e \left (7 a e^2+c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {\int -\frac {3 c \left (4 c^2 d^4+9 a c e^2 d^2+2 c e \left (c d^2+2 a e^2\right ) x d+7 a^2 e^4\right )}{2 \sqrt {d+e x} \left (c x^2+a\right )}dx}{2 a c \left (a e^2+c d^2\right )}}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \int \frac {4 c^2 d^4+9 a c e^2 d^2+2 c e \left (c d^2+2 a e^2\right ) x d+7 a^2 e^4}{\sqrt {d+e x} \left (c x^2+a\right )}dx}{4 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (6 c d x \left (2 a e^2+c d^2\right )+a e \left (7 a e^2+c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {3 \int \frac {e \left (2 c^2 d^4+5 a c e^2 d^2+2 c \left (c d^2+2 a e^2\right ) (d+e x) d+7 a^2 e^4\right )}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (6 c d x \left (2 a e^2+c d^2\right )+a e \left (7 a e^2+c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 e \int \frac {2 c^2 d^4+5 a c e^2 d^2+2 c \left (c d^2+2 a e^2\right ) (d+e x) d+7 a^2 e^4}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (6 c d x \left (2 a e^2+c d^2\right )+a e \left (7 a e^2+c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {3 e \left (\frac {\int \frac {\sqrt {2} \left (2 c^2 d^4+5 a c e^2 d^2+7 a^2 e^4\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+7 a^2 e^4\right )+\sqrt [4]{c} \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (6 c d x \left (2 a e^2+c d^2\right )+a e \left (7 a e^2+c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 e \left (\frac {\int \frac {\sqrt {2} \left (2 c^2 d^4+5 a c e^2 d^2+7 a^2 e^4\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt [4]{c} \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+7 a^2 e^4\right )+\sqrt [4]{c} \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (6 c d x \left (2 a e^2+c d^2\right )+a e \left (7 a e^2+c d^2\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}}{8 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {3 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) 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 )}{2 a \left (c d^2+a e^2\right )}}{8 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {3 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) 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 )}{2 a \left (c d^2+a e^2\right )}}{8 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {3 e \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}+\frac {\left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {3 e \left (\frac {\frac {\left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) d+7 a^2 e^4\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {3 e \left (\frac {\frac {\left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) 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 (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) 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 )}{2 a \left (c d^2+a e^2\right )}}{8 a \left (c d^2+a e^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{2 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )}+\frac {3 e \left (\frac {-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) d+7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {1}{2} \sqrt [4]{c} \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )-\frac {\sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4+5 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (2 c d^2+4 a e^2\right ) d+7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{2 a \left (c d^2+a e^2\right )}}{8 a \left (c d^2+a e^2\right )}\) |
((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 + a*e^2)*(a + c*x^2)^2) + ((Sqrt [d + e*x]*(a*e*(c*d^2 + 7*a*e^2) + 6*c*d*(c*d^2 + 2*a*e^2)*x))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (3*e*((-((c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a *e^2]]*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e ^2]*(2*c*d^2 + 4*a*e^2))*ArcTanh[(c^(1/4)*(-((Sqrt[2]*Sqrt[Sqrt[c]*d + Sqr t[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*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*( c*d^2 + 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]]*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(2*c*d^2 + 4*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*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 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]])))/(2*a*(c*d^2 + a*e^2)))/(8*a*(c*d^2 + a*e^...
3.7.47.3.1 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)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Timed out.
\[\int \frac {1}{\left (c \,x^{2}+a \right )^{3} \sqrt {e x +d}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 5779 vs. \(2 (774) = 1548\).
Time = 3.93 (sec) , antiderivative size = 5779, normalized size of antiderivative = 6.28 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{3} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1664 vs. \(2 (774) = 1548\).
Time = 0.42 (sec) , antiderivative size = 1664, normalized size of antiderivative = 1.81 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
-3/32*(2*(a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(sqrt(-a*c)*c*d^3*e + 2*sqrt(-a*c)*a*d*e^3)*abs(c) + (2*a^2*c^4*d^8*e + 9*a^3*c^3*d^6*e^3 + 1 9*a^4*c^2*d^4*e^5 + 19*a^5*c*d^2*e^7 + 7*a^6*e^9)*abs(a^2*c^2*d^4*e + 2*a^ 3*c*d^2*e^3 + a^4*e^5)*abs(c) + (4*sqrt(-a*c)*a^3*c^6*d^13*e + 25*sqrt(-a* c)*a^4*c^5*d^11*e^3 + 67*sqrt(-a*c)*a^5*c^4*d^9*e^5 + 98*sqrt(-a*c)*a^6*c^ 3*d^7*e^7 + 82*sqrt(-a*c)*a^7*c^2*d^5*e^9 + 37*sqrt(-a*c)*a^8*c*d^3*e^11 + 7*sqrt(-a*c)*a^9*d*e^13)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4 + sqrt((a^2*c^3*d^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 + a^ 5*e^6)*(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 + 2*a^ 3*c^2*d^2*e^2 + a^4*c*e^4)))/((a^4*c^5*d^9 + 4*a^5*c^4*d^7*e^2 + 6*a^6*c^3 *d^5*e^4 + 4*a^7*c^2*d^3*e^6 + a^8*c*d*e^8 - sqrt(-a*c)*a^4*c^4*d^8*e - 4* sqrt(-a*c)*a^5*c^3*d^6*e^3 - 6*sqrt(-a*c)*a^6*c^2*d^4*e^5 - 4*sqrt(-a*c)*a ^7*c*d^2*e^7 - sqrt(-a*c)*a^8*e^9)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(a^2*c ^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)) + 3/32*(2*(a^2*c^2*d^4*e + 2*a^3*c* d^2*e^3 + a^4*e^5)^2*(c^2*d^3*e + 2*a*c*d*e^3)*abs(c) + (2*sqrt(-a*c)*a*c^ 4*d^8*e + 9*sqrt(-a*c)*a^2*c^3*d^6*e^3 + 19*sqrt(-a*c)*a^3*c^2*d^4*e^5 + 1 9*sqrt(-a*c)*a^4*c*d^2*e^7 + 7*sqrt(-a*c)*a^5*e^9)*abs(a^2*c^2*d^4*e + 2*a ^3*c*d^2*e^3 + a^4*e^5)*abs(c) + (4*a^3*c^7*d^13*e + 25*a^4*c^6*d^11*e^3 + 67*a^5*c^5*d^9*e^5 + 98*a^6*c^4*d^7*e^7 + 82*a^7*c^3*d^5*e^9 + 37*a^8*...
Time = 12.64 (sec) , antiderivative size = 9035, normalized size of antiderivative = 9.82 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 + 18432*a^6*c^6*d^6* e^5 + 38912*a^7*c^5*d^4*e^7 + 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^ 6*c^4*d^8 + 4*a^9*c*d^2*e^6 + 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - (( d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6* c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*( -a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096* (a^15*c*e^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4 096*a^5*c^8*d^9*e^2 + 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 + 1638 4*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c ^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15 *c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^1 5*c)^(1/2)))/(4096*(a^15*c*e^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10* a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 + 612*a*c^6*d^6*e^4 + 10 89*a^2*c^5*d^4*e^6 + 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4* a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5* d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^ 4 + 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*...