\(\int \frac {1}{x (c+d x) (-a c^2+2 a d^2 x^2)^{3/4}} \, dx\) [1662]

Optimal result

Integrand size = 31, antiderivative size = 513 \[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{-a c^2+2 a d^2 x^2}}\right )}{2 a^{3/4} c^{5/2}}+\frac {\arctan \left (\frac {\sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{a^{3/4} c^{5/2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} a^{3/4} c^{5/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} a^{3/4} c^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{a^{3/4} c^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c} \sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt [4]{a} d x}\right )}{2 a^{3/4} c^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt {a} c+\sqrt {-a c^2+2 a d^2 x^2}}\right )}{\sqrt {2} a^{3/4} c^{5/2}}-\frac {\sqrt {\frac {a d^2 x^2}{\left (\sqrt {a} c+\sqrt {-a c^2+2 a d^2 x^2}\right )^2}} \left (\sqrt {a} c+\sqrt {-a c^2+2 a d^2 x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-a c^2+2 a d^2 x^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{\sqrt {2} a^{5/4} c^{5/2} d x} \] Output:

1/2*arctan(a^(1/4)*d*x/c^(1/2)/(2*a*d^2*x^2-a*c^2)^(1/4))/a^(3/4)/c^(5/2)+ 
arctan((2*a*d^2*x^2-a*c^2)^(1/4)/a^(1/4)/c^(1/2))/a^(3/4)/c^(5/2)-1/2*arct 
an(1-2^(1/2)*(2*a*d^2*x^2-a*c^2)^(1/4)/a^(1/4)/c^(1/2))*2^(1/2)/a^(3/4)/c^ 
(5/2)+1/2*arctan(1+2^(1/2)*(2*a*d^2*x^2-a*c^2)^(1/4)/a^(1/4)/c^(1/2))*2^(1 
/2)/a^(3/4)/c^(5/2)+arctanh((2*a*d^2*x^2-a*c^2)^(1/4)/a^(1/4)/c^(1/2))/a^( 
3/4)/c^(5/2)-1/2*arctanh(1/a^(1/4)/d/x*c^(1/2)*(2*a*d^2*x^2-a*c^2)^(1/4))/ 
a^(3/4)/c^(5/2)+1/2*arctanh(2^(1/2)*a^(1/4)*c^(1/2)*(2*a*d^2*x^2-a*c^2)^(1 
/4)/(a^(1/2)*c+(2*a*d^2*x^2-a*c^2)^(1/2)))*2^(1/2)/a^(3/4)/c^(5/2)-1/2*(a* 
d^2*x^2/(a^(1/2)*c+(2*a*d^2*x^2-a*c^2)^(1/2))^2)^(1/2)*(a^(1/2)*c+(2*a*d^2 
*x^2-a*c^2)^(1/2))*InverseJacobiAM(2*arctan((2*a*d^2*x^2-a*c^2)^(1/4)/a^(1 
/4)/c^(1/2)),1/2*2^(1/2))*2^(1/2)/a^(5/4)/c^(5/2)/d/x
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.67 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\frac {\left (c^2-2 d^2 x^2\right )^{3/4} \left (4 \sqrt {2} c^{5/2} \sqrt {\frac {d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (-i,\arcsin \left (\frac {\sqrt [4]{c^2-2 d^2 x^2}}{\sqrt [4]{c^2}}\right ),-1\right )+4 \sqrt {2} c^{5/2} \sqrt {\frac {d^2 x^2}{c^2}} \operatorname {EllipticPi}\left (i,\arcsin \left (\frac {\sqrt [4]{c^2-2 d^2 x^2}}{\sqrt [4]{c^2}}\right ),-1\right )-\left (c^2\right )^{3/4} d x \left (4 \arctan \left (\frac {\sqrt [4]{c^2-2 d^2 x^2}}{\sqrt {c}}\right )+4 \text {arctanh}\left (\frac {\sqrt [4]{c^2-2 d^2 x^2}}{\sqrt {c}}\right )+\sqrt {2} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c^2-2 d^2 x^2}}{\sqrt {c}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c^2-2 d^2 x^2}}{\sqrt {c}}\right )-\log \left (c-\sqrt {2} \sqrt {c} \sqrt [4]{c^2-2 d^2 x^2}+\sqrt {c^2-2 d^2 x^2}\right )+\log \left (c+\sqrt {2} \sqrt {c} \sqrt [4]{c^2-2 d^2 x^2}+\sqrt {c^2-2 d^2 x^2}\right )\right )\right )\right )}{4 c^{5/2} \left (c^2\right )^{3/4} d x \left (-a \left (c^2-2 d^2 x^2\right )\right )^{3/4}} \] Input:

Integrate[1/(x*(c + d*x)*(-(a*c^2) + 2*a*d^2*x^2)^(3/4)),x]
 

Output:

((c^2 - 2*d^2*x^2)^(3/4)*(4*Sqrt[2]*c^(5/2)*Sqrt[(d^2*x^2)/c^2]*EllipticPi 
[-I, ArcSin[(c^2 - 2*d^2*x^2)^(1/4)/(c^2)^(1/4)], -1] + 4*Sqrt[2]*c^(5/2)* 
Sqrt[(d^2*x^2)/c^2]*EllipticPi[I, ArcSin[(c^2 - 2*d^2*x^2)^(1/4)/(c^2)^(1/ 
4)], -1] - (c^2)^(3/4)*d*x*(4*ArcTan[(c^2 - 2*d^2*x^2)^(1/4)/Sqrt[c]] + 4* 
ArcTanh[(c^2 - 2*d^2*x^2)^(1/4)/Sqrt[c]] + Sqrt[2]*(-2*ArcTan[1 - (Sqrt[2] 
*(c^2 - 2*d^2*x^2)^(1/4))/Sqrt[c]] + 2*ArcTan[1 + (Sqrt[2]*(c^2 - 2*d^2*x^ 
2)^(1/4))/Sqrt[c]] - Log[c - Sqrt[2]*Sqrt[c]*(c^2 - 2*d^2*x^2)^(1/4) + Sqr 
t[c^2 - 2*d^2*x^2]] + Log[c + Sqrt[2]*Sqrt[c]*(c^2 - 2*d^2*x^2)^(1/4) + Sq 
rt[c^2 - 2*d^2*x^2]]))))/(4*c^(5/2)*(c^2)^(3/4)*d*x*(-(a*(c^2 - 2*d^2*x^2) 
))^(3/4))
 

Rubi [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.30, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.645, Rules used = {621, 311, 232, 351, 354, 97, 73, 755, 27, 756, 216, 219, 761, 1476, 1082, 217, 1479, 25, 27, 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.

\(\displaystyle \int \frac {1}{x (c+d x) \left (2 a d^2 x^2-a c^2\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 621

\(\displaystyle c \int \frac {1}{x \left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx-d \int \frac {1}{\left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx\)

\(\Big \downarrow \) 311

\(\displaystyle c \int \frac {1}{x \left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx-d \left (\frac {d^2 \int \frac {x^2}{\left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx}{c^2}+\frac {\int \frac {1}{\left (2 a d^2 x^2-a c^2\right )^{3/4}}dx}{c^2}\right )\)

\(\Big \downarrow \) 232

\(\displaystyle c \int \frac {1}{x \left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx-d \left (\frac {d^2 \int \frac {x^2}{\left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx}{c^2}+\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}\right )\)

\(\Big \downarrow \) 351

\(\displaystyle c \int \frac {1}{x \left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} c \int \frac {1}{x^2 \left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx^2-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 97

\(\displaystyle \frac {1}{2} c \left (\frac {d^2 \int \frac {1}{\left (c^2-d^2 x^2\right ) \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx^2}{c^2}+\frac {\int \frac {1}{x^2 \left (2 a d^2 x^2-a c^2\right )^{3/4}}dx^2}{c^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} c \left (\frac {2 \int \frac {1}{\frac {c^2}{2}-\frac {x^8}{2 a}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2}+\frac {2 \int \frac {1}{\frac {x^8}{2 a d^2}+\frac {c^2}{2 d^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {1}{2} c \left (\frac {2 \int \frac {1}{\frac {c^2}{2}-\frac {x^8}{2 a}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2}+\frac {2 \left (\frac {\int \frac {2 a d^2 \left (\sqrt {a} c-x^4\right )}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {a} c}+\frac {\int \frac {2 a d^2 \left (x^4+\sqrt {a} c\right )}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {a} c}\right )}{a c^2 d^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} c \left (\frac {2 \int \frac {1}{\frac {c^2}{2}-\frac {x^8}{2 a}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2}+\frac {2 \left (\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} d^2 \int \frac {x^4+\sqrt {a} c}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {a} c-x^4}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} \int \frac {1}{x^4+\sqrt {a} c}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2}+\frac {2 \left (\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} d^2 \int \frac {x^4+\sqrt {a} c}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {a} c-x^4}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}+\frac {2 \left (\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} d^2 \int \frac {x^4+\sqrt {a} c}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} d^2 \int \frac {x^4+\sqrt {a} c}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}+\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}\right )-d \left (\frac {\sqrt {2} \sqrt {\frac {d^2 x^2}{c^2}} \int \frac {1}{\sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{a c^2 d^2 x}+\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} d^2 \int \frac {x^4+\sqrt {a} c}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}+\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}\right )-d \left (\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right )^2}} \left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} d^2 x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} d^2 \left (\frac {1}{2} \int \frac {1}{x^4+\sqrt {a} c-\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}+\frac {1}{2} \int \frac {1}{x^4+\sqrt {a} c+\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}\right )}{c}+\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}+\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}\right )-d \left (\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right )^2}} \left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} d^2 x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} d^2 \left (\frac {\int \frac {1}{-x^4-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\int \frac {1}{-x^4-1}d\left (\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}\right )}{c}+\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}\right )}{a c^2 d^2}+\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}\right )-d \left (\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right )^2}} \left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} d^2 x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} d^2 \int \frac {\sqrt {a} c-x^4}{x^8+a c^2}d\sqrt [4]{2 a d^2 x^2-a c^2}}{c}+\frac {\sqrt {a} d^2 \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}\right )}{c}\right )}{a c^2 d^2}+\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}\right )-d \left (\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right )^2}} \left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} d^2 x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}+\frac {2 \left (\frac {\sqrt {a} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}\right ) d^2}{c}+\frac {\sqrt {a} \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{2 a d^2 x^2-a c^2}}{x^4+\sqrt {a} c-\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}\right )}{x^4+\sqrt {a} c+\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}\right ) d^2}{c}\right )}{a c^2 d^2}\right )-d \left (\frac {\left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right ) d^2}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {a} c+\sqrt {2 a d^2 x^2-a c^2}\right )^2}} \left (\sqrt {a} c+\sqrt {2 a d^2 x^2-a c^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1} d^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}+\frac {2 \left (\frac {\sqrt {a} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}\right ) d^2}{c}+\frac {\sqrt {a} \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{2 a d^2 x^2-a c^2}}{x^4+\sqrt {a} c-\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}\right )}{x^4+\sqrt {a} c+\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}\right ) d^2}{c}\right )}{a c^2 d^2}\right )-d \left (\frac {\left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right ) d^2}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {a} c+\sqrt {2 a d^2 x^2-a c^2}\right )^2}} \left (\sqrt {a} c+\sqrt {2 a d^2 x^2-a c^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1} d^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt {a} d^2 \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {c}-2 \sqrt [4]{2 a d^2 x^2-a c^2}}{x^4+\sqrt {a} c-\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {c}+\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{x^4+\sqrt {a} c+\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}d\sqrt [4]{2 a d^2 x^2-a c^2}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{c}+\frac {\sqrt {a} d^2 \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}\right )}{c}\right )}{a c^2 d^2}+\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}\right )-d \left (\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right )^2}} \left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} d^2 x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{2} c \left (\frac {2 \left (\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{c^{3/2}}\right )}{a c^2}+\frac {2 \left (\frac {\sqrt {a} d^2 \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {c}}\right )}{c}+\frac {\sqrt {a} d^2 \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}+\sqrt {a} c+x^4\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}+\sqrt {a} c+x^4\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}\right )}{c}\right )}{a c^2 d^2}\right )-d \left (\frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}-\frac {\arctan \left (\frac {\sqrt [4]{a} d x}{\sqrt {c} \sqrt [4]{2 a d^2 x^2-a c^2}}\right )}{2 a^{3/4} \sqrt {c} d^3}\right )}{c^2}+\frac {\sqrt {\frac {d^2 x^2}{c^2}} \sqrt {\frac {a d^2 x^2}{\left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right )^2}} \left (\sqrt {2 a d^2 x^2-a c^2}+\sqrt {a} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{2 a d^2 x^2-a c^2}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{a^{5/4} c^{5/2} d^2 x \sqrt {\frac {2 a d^2 x^2-a c^2}{a c^2}+1}}\right )\)

Input:

Int[1/(x*(c + d*x)*(-(a*c^2) + 2*a*d^2*x^2)^(3/4)),x]
 

Output:

-(d*((d^2*(-1/2*ArcTan[(a^(1/4)*d*x)/(Sqrt[c]*(-(a*c^2) + 2*a*d^2*x^2)^(1/ 
4))]/(a^(3/4)*Sqrt[c]*d^3) + ArcTanh[(a^(1/4)*d*x)/(Sqrt[c]*(-(a*c^2) + 2* 
a*d^2*x^2)^(1/4))]/(2*a^(3/4)*Sqrt[c]*d^3)))/c^2 + (Sqrt[(d^2*x^2)/c^2]*Sq 
rt[(a*d^2*x^2)/(Sqrt[a]*c + Sqrt[-(a*c^2) + 2*a*d^2*x^2])^2]*(Sqrt[a]*c + 
Sqrt[-(a*c^2) + 2*a*d^2*x^2])*EllipticF[2*ArcTan[(-(a*c^2) + 2*a*d^2*x^2)^ 
(1/4)/(a^(1/4)*Sqrt[c])], 1/2])/(a^(5/4)*c^(5/2)*d^2*x*Sqrt[1 + (-(a*c^2) 
+ 2*a*d^2*x^2)/(a*c^2)]))) + (c*((2*((a^(1/4)*ArcTan[(-(a*c^2) + 2*a*d^2*x 
^2)^(1/4)/(a^(1/4)*Sqrt[c])])/c^(3/2) + (a^(1/4)*ArcTanh[(-(a*c^2) + 2*a*d 
^2*x^2)^(1/4)/(a^(1/4)*Sqrt[c])])/c^(3/2)))/(a*c^2) + (2*((Sqrt[a]*d^2*(-( 
ArcTan[1 - (Sqrt[2]*(-(a*c^2) + 2*a*d^2*x^2)^(1/4))/(a^(1/4)*Sqrt[c])]/(Sq 
rt[2]*a^(1/4)*Sqrt[c])) + ArcTan[1 + (Sqrt[2]*(-(a*c^2) + 2*a*d^2*x^2)^(1/ 
4))/(a^(1/4)*Sqrt[c])]/(Sqrt[2]*a^(1/4)*Sqrt[c])))/c + (Sqrt[a]*d^2*(-1/2* 
Log[Sqrt[a]*c + x^4 - Sqrt[2]*a^(1/4)*Sqrt[c]*(-(a*c^2) + 2*a*d^2*x^2)^(1/ 
4)]/(Sqrt[2]*a^(1/4)*Sqrt[c]) + Log[Sqrt[a]*c + x^4 + Sqrt[2]*a^(1/4)*Sqrt 
[c]*(-(a*c^2) + 2*a*d^2*x^2)^(1/4)]/(2*Sqrt[2]*a^(1/4)*Sqrt[c])))/c))/(a*c 
^2*d^2)))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

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])
 

rule 232

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[2*(Sqrt[(-b)*(x^2/a)]/( 
b*x))   Subst[Int[1/Sqrt[1 - x^4/a], x], x, (a + b*x^2)^(1/4)], x] /; FreeQ 
[{a, b}, x] && NegQ[a]
 

rule 311

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[1/c   Int[1/(a + b*x^2)^(3/4), x], x] - Simp[d/c   Int[x^2/((a + b*x^2)^( 
3/4)*(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0]
 

rule 351

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] : 
> Simp[(-b/(Sqrt[2]*a*d*Rt[-b^2/a, 4]^3))*ArcTan[(Rt[-b^2/a, 4]*x)/(Sqrt[2] 
*(a + b*x^2)^(1/4))], x] + Simp[(b/(Sqrt[2]*a*d*Rt[-b^2/a, 4]^3))*ArcTanh[( 
Rt[-b^2/a, 4]*x)/(Sqrt[2]*(a + b*x^2)^(1/4))], x] /; FreeQ[{a, b, c, d}, x] 
 && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 621

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[c   Int[x^m*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] - Simp[d   Int[ 
x^(m + 1)*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, m, 
p}, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

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]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [F]

\[\int \frac {1}{x \left (d x +c \right ) \left (2 a \,d^{2} x^{2}-a \,c^{2}\right )^{\frac {3}{4}}}d x\]

Input:

int(1/x/(d*x+c)/(2*a*d^2*x^2-a*c^2)^(3/4),x)
 

Output:

int(1/x/(d*x+c)/(2*a*d^2*x^2-a*c^2)^(3/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\text {Timed out} \] Input:

integrate(1/x/(d*x+c)/(2*a*d^2*x^2-a*c^2)^(3/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\int \frac {1}{x \left (a \left (- c^{2} + 2 d^{2} x^{2}\right )\right )^{\frac {3}{4}} \left (c + d x\right )}\, dx \] Input:

integrate(1/x/(d*x+c)/(2*a*d**2*x**2-a*c**2)**(3/4),x)
 

Output:

Integral(1/(x*(a*(-c**2 + 2*d**2*x**2))**(3/4)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (2 \, a d^{2} x^{2} - a c^{2}\right )}^{\frac {3}{4}} {\left (d x + c\right )} x} \,d x } \] Input:

integrate(1/x/(d*x+c)/(2*a*d^2*x^2-a*c^2)^(3/4),x, algorithm="maxima")
 

Output:

integrate(1/((2*a*d^2*x^2 - a*c^2)^(3/4)*(d*x + c)*x), x)
 

Giac [F]

\[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (2 \, a d^{2} x^{2} - a c^{2}\right )}^{\frac {3}{4}} {\left (d x + c\right )} x} \,d x } \] Input:

integrate(1/x/(d*x+c)/(2*a*d^2*x^2-a*c^2)^(3/4),x, algorithm="giac")
 

Output:

integrate(1/((2*a*d^2*x^2 - a*c^2)^(3/4)*(d*x + c)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\int \frac {1}{x\,{\left (2\,a\,d^2\,x^2-a\,c^2\right )}^{3/4}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/(x*(2*a*d^2*x^2 - a*c^2)^(3/4)*(c + d*x)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(x*(2*a*d^2*x^2 - a*c^2)^(3/4)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {1}{x (c+d x) \left (-a c^2+2 a d^2 x^2\right )^{3/4}} \, dx=\frac {\int \frac {1}{\left (2 d^{2} x^{2}-c^{2}\right )^{\frac {3}{4}} c x +\left (2 d^{2} x^{2}-c^{2}\right )^{\frac {3}{4}} d \,x^{2}}d x}{a^{\frac {3}{4}}} \] Input:

int(1/x/(d*x+c)/(2*a*d^2*x^2-a*c^2)^(3/4),x)
 

Output:

int(1/(( - c**2 + 2*d**2*x**2)**(3/4)*c*x + ( - c**2 + 2*d**2*x**2)**(3/4) 
*d*x**2),x)/a**(3/4)