\(\int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx\) [94]

Optimal result

Integrand size = 27, antiderivative size = 719 \[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{2 a x^2}+\frac {(3 b d-a e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{4 a^2 d x}-\frac {\sqrt {b^2-4 a c} (3 b d-a e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{4 \sqrt {2} a^2 d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {3 \sqrt {b^2-4 a c} (b d-a e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{2 \sqrt {2} a^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} \left (3 b^2 d^2-2 a b d e-a \left (4 c d^2+a e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} a^2 \left (b+\sqrt {b^2-4 a c}\right ) d \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-1/2*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/a/x^2+1/4*(-a*e+3*b*d)*(e*x+d)^(1/2 
)*(c*x^2+b*x+a)^(1/2)/a^2/d/x-1/8*(-4*a*c+b^2)^(1/2)*(-a*e+3*b*d)*(e*x+d)^ 
(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4 
*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+ 
b^2)^(1/2))*e))^(1/2))*2^(1/2)/a^2/d/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/ 
2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)+3/4*(-4*a*c+b^2)^(1/2)*(-a*e+b*d)*(c*(e* 
x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2 
))^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2* 
(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))*2^(1/2)/a^2/ 
(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-1/2*(-4*a*c+b^2)^(1/2)*(3*b^2*d^2-2*a*b* 
d*e-a*(a*e^2+4*c*d^2))*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)* 
(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticPi(1/2*(1+(2*c*x+b)/(-4*a*c+ 
b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2)),(-2* 
(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))*2^(1/2)/a^2/ 
(b+(-4*a*c+b^2)^(1/2))/d/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.08 (sec) , antiderivative size = 6774, normalized size of antiderivative = 9.42 \[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[Sqrt[d + e*x]/(x^3*Sqrt[a + b*x + c*x^2]),x]
 

Output:

Result too large to show
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1462\) vs. \(2(719)=1438\).

Time = 2.96 (sec) , antiderivative size = 1462, normalized size of antiderivative = 2.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {1285, 2154, 1282, 2154, 25, 27, 1172, 321, 1269, 1172, 321, 327, 1279, 187, 413, 413, 412}

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.

\(\displaystyle \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1285

\(\displaystyle -\frac {\int \frac {c e x^2+2 (c d+b e) x+3 b d-a e}{x^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{2 a x^2}\)

\(\Big \downarrow \) 2154

\(\displaystyle -\frac {(3 b d-a e) \int \frac {1}{x^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+\int \frac {2 c d+2 b e+c e x}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{2 a x^2}\)

\(\Big \downarrow \) 1282

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 (b e+c d) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+(3 b d-a e) \left (-\frac {(a e+b d) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx-c e \int \frac {x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{2 a d}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{a d x}\right )+\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}}}{4 a}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{2 a x^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {2 (b e+c d) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+(3 b d-a e) \left (-\frac {(a e+b d) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx-c e \left (\frac {\int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {d \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{2 a d}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{a d x}\right )+\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}}}{4 a}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{2 a x^2}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}+2 (c d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+(3 b d-a e) \left (-\frac {(b d+a e) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}+2 (c d+b e) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+(3 b d-a e) \left (-\frac {(b d+a e) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {(3 b d-a e) \left (-\frac {(a e+b d) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} d \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{2 a d}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{a d x}\right )+2 (b e+c d) \int \frac {1}{x \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx+\frac {2 \sqrt {2} e \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}}}{4 a}-\frac {\sqrt {d+e x} \sqrt {a+b x+c x^2}}{2 a x^2}\)

\(\Big \downarrow \) 1279

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}+\frac {2 (c d+b e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int \frac {1}{x \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {d+e x}}dx}{\sqrt {c x^2+b x+a}}+(3 b d-a e) \left (-\frac {\frac {(b d+a e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int \frac {1}{x \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {d+e x}}dx}{\sqrt {c x^2+b x+a}}-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

\(\Big \downarrow \) 187

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {4 (c d+b e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int -\frac {1}{e x \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}d\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}+(3 b d-a e) \left (-\frac {-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )-\frac {2 (b d+a e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \int -\frac {1}{e x \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}d\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {4 (c d+b e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \int -\frac {1}{e x \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}+(3 b d-a e) \left (-\frac {-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )-\frac {2 (b d+a e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \int -\frac {1}{e x \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {4 (c d+b e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \int -\frac {1}{e x \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}+(3 b d-a e) \left (-\frac {-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )-\frac {2 (b d+a e) \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \int -\frac {1}{e x \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}d\sqrt {d+e x}}{\sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} (c d+b e) \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \operatorname {EllipticPi}\left (\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {c} d \sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}+(3 b d-a e) \left (-\frac {-c e \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} d \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )-\frac {\sqrt {2} (b d+a e) \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {b+2 c x-\sqrt {b^2-4 a c}} \sqrt {b+2 c x+\sqrt {b^2-4 a c}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \operatorname {EllipticPi}\left (\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {c} d \sqrt {c x^2+b x+a} \sqrt {b+\frac {2 c (d+e x)}{e}-\sqrt {b^2-4 a c}-\frac {2 c d}{e}} \sqrt {b+\frac {2 c (d+e x)}{e}+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}}}{2 a d}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{a d x}\right )}{4 a}-\frac {\sqrt {d+e x} \sqrt {c x^2+b x+a}}{2 a x^2}\)

Input:

Int[Sqrt[d + e*x]/(x^3*Sqrt[a + b*x + c*x^2]),x]
 

Output:

-1/2*(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(a*x^2) - ((2*Sqrt[2]*Sqrt[b^2 
- 4*a*c]*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-( 
(c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 
- 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2 
*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) 
- (2*Sqrt[2]*(c*d + b*e)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[b - 
Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[1 - (2 
*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x) 
)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*c*d - b*e + Sqrt[b^2 
- 4*a*c]*e)/(2*c*d), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - ( 
b - Sqrt[b^2 - 4*a*c])*e]], (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*d - ( 
b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[c]*d*Sqrt[a + b*x + c*x^2]*Sqrt[b - Sqrt 
[b^2 - 4*a*c] - (2*c*d)/e + (2*c*(d + e*x))/e]*Sqrt[b + Sqrt[b^2 - 4*a*c] 
- (2*c*d)/e + (2*c*(d + e*x))/e]) + (3*b*d - a*e)*(-((Sqrt[d + e*x]*Sqrt[a 
 + b*x + c*x^2])/(a*d*x)) - (-(c*e*((Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e* 
x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + 
 Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4* 
a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e*x))/(2*c 
*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[ 
b^2 - 4*a*c]*d*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*...
 

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 187

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] &&  !SimplerQ[e + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

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

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

rule 1282

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[e^2*(d + e*x)^(m + 1)*Sqrt[f + g*x 
]*(Sqrt[a + b*x + c*x^2]/((m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/(2*(m + 1)*(e*f - d*g)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x)^ 
(m + 1)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[2*d*(c*e*f - c*d*g + b* 
e*g)*(m + 1) - e^2*(b*f + a*g)*(2*m + 3) + 2*e*(c*d*g*(m + 1) - e*(c*f + b* 
g)*(m + 2))*x - c*e^2*g*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, 
f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
 

rule 1285

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.) 
*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*Sqrt[f + g*x]* 
(Sqrt[a + b*x + c*x^2]/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*( 
m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqr 
t[a + b*x + c*x^2]))*Simp[2*c*d*f*(m + 1) - e*(a*g + b*f*(2*m + 3)) - 2*(b* 
e*g*(2 + m) - c*(d*g*(m + 1) - e*f*(m + 2)))*x - c*e*g*(2*m + 5)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x 
] && LtQ[m, 0] &&  !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 6.62 (sec) , antiderivative size = 1205, normalized size of antiderivative = 1.68

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1205\)
risch	\(\text {Expression too large to display}\)	\(1577\)
default	\(\text {Expression too large to display}\)	\(5158\)

Input:

int((e*x+d)^(1/2)/x^3/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-1/2/a/x^ 
2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)-1/4/a^2/d*(a*e-3*b*d)*(c 
*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/x-1/2*c/a*e*(d/e-1/2*(b+(-4* 
a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x- 
1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2) 
*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1 
/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/ 
e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c 
)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/4*c*e*(a*e-3*b*d)/a^2/d*( 
d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2)) 
/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2) 
^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2 
)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e- 
1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2) 
^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4 
*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e) 
/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2 
))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))+1/4*(a^2*e^2+2*a*b*d*e 
+4*a*c*d^2-3*b^2*d^2)/a^2/d^2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/ 
(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1...
 

Fricas [F]

\[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} x^{3}} \,d x } \] Input:

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

Output:

integral(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(c*x^5 + b*x^4 + a*x^3), x)
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d + e x}}{x^{3} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((e*x+d)**(1/2)/x**3/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral(sqrt(d + e*x)/(x**3*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} x^{3}} \,d x } \] Input:

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

Output:

integrate(sqrt(e*x + d)/(sqrt(c*x^2 + b*x + a)*x^3), x)
 

Giac [F]

\[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} x^{3}} \,d x } \] Input:

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

Output:

integrate(sqrt(e*x + d)/(sqrt(c*x^2 + b*x + a)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{x^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((d + e*x)^(1/2)/(x^3*(a + b*x + c*x^2)^(1/2)),x)
 

Output:

int((d + e*x)^(1/2)/(x^3*(a + b*x + c*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {d+e x}}{x^3 \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {e x +d}}{x^{3} \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((e*x+d)^(1/2)/x^3/(c*x^2+b*x+a)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((e*x+d)^(1/2)/x^3/(c*x^2+b*x+a)^(1/2),x)