\(\int \frac {1}{(c+d x)^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx\) [64]

Optimal result

Integrand size = 33, antiderivative size = 475 \[ \int \frac {1}{(c+d x)^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=-\frac {(c-d x)^3 \sqrt {e+f x}}{5 c (d e-c f) \left (c^2-d^2 x^2\right )^{5/2}}-\frac {2 (d e-3 c f) (c-d x)^2 \sqrt {e+f x}}{15 c^2 (d e-c f)^2 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {\left (4 d^2 e^2-15 c d e f+27 c^2 f^2\right ) (c-d x) \sqrt {e+f x}}{30 c^3 (d e-c f)^3 \sqrt {c^2-d^2 x^2}}-\frac {\left (4 d^2 e^2-15 c d e f+27 c^2 f^2\right ) \sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right )|-\frac {2 c f}{d e-c f}\right )}{30 c^{5/2} (d e-c f)^3 \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {c^2-d^2 x^2}}+\frac {\left (4 d^2 e^2-11 c d e f+15 c^2 f^2\right ) \sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{30 c^{5/2} d (d e-c f)^2 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \] Output:

-1/5*(-d*x+c)^3*(f*x+e)^(1/2)/c/(-c*f+d*e)/(-d^2*x^2+c^2)^(5/2)-2/15*(-3*c 
*f+d*e)*(-d*x+c)^2*(f*x+e)^(1/2)/c^2/(-c*f+d*e)^2/(-d^2*x^2+c^2)^(3/2)-1/3 
0*(27*c^2*f^2-15*c*d*e*f+4*d^2*e^2)*(-d*x+c)*(f*x+e)^(1/2)/c^3/(-c*f+d*e)^ 
3/(-d^2*x^2+c^2)^(1/2)-1/30*(27*c^2*f^2-15*c*d*e*f+4*d^2*e^2)*(d*x+c)^(1/2 
)*(1-d*x/c)^(1/2)*(f*x+e)^(1/2)*EllipticE(1/2*(d*x+c)^(1/2)*2^(1/2)/c^(1/2 
),(-2*c*f/(-c*f+d*e))^(1/2))/c^(5/2)/(-c*f+d*e)^3/(d*(f*x+e)/(-c*f+d*e))^( 
1/2)/(-d^2*x^2+c^2)^(1/2)+1/30*(15*c^2*f^2-11*c*d*e*f+4*d^2*e^2)*(d*x+c)^( 
1/2)*(1-d*x/c)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*EllipticF(1/2*(d*x+c)^(1 
/2)*2^(1/2)/c^(1/2),(-2*c*f/(-c*f+d*e))^(1/2))/c^(5/2)/d/(-c*f+d*e)^2/(f*x 
+e)^(1/2)/(-d^2*x^2+c^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.92 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(c+d x)^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (f \left (4 d e^2-15 c e f+\frac {27 c^2 f^2}{d}\right )-\frac {(e+f x) \left (6 c^2 (d e-c f)^2+4 c (d e-3 c f) (d e-c f) (c+d x)+\left (4 d^2 e^2-15 c d e f+27 c^2 f^2\right ) (c+d x)^2\right )}{(c+d x)^3}+\frac {i \left (4 d^3 e^3-11 c d^2 e^2 f+12 c^2 d e f^2+27 c^3 f^3\right ) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right )|\frac {d e-c f}{d e+c f}\right )}{f \sqrt {-\frac {d e+c f}{d}} \left (c^2-d^2 x^2\right )}-\frac {2 i c \left (2 d^2 e^2-7 c d e f+21 c^2 f^2\right ) \sqrt {\frac {f (-c+d x)}{d (e+f x)}} \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {d e+c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d e-c f}{d e+c f}\right )}{\sqrt {-\frac {d e+c f}{d}} \left (c^2-d^2 x^2\right )}\right )}{30 c^3 (d e-c f)^3 \sqrt {e+f x}} \] Input:

Integrate[1/((c + d*x)^3*Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*(f*(4*d*e^2 - 15*c*e*f + (27*c^2*f^2)/d) - ((e + f*x) 
*(6*c^2*(d*e - c*f)^2 + 4*c*(d*e - 3*c*f)*(d*e - c*f)*(c + d*x) + (4*d^2*e 
^2 - 15*c*d*e*f + 27*c^2*f^2)*(c + d*x)^2))/(c + d*x)^3 + (I*(4*d^3*e^3 - 
11*c*d^2*e^2*f + 12*c^2*d*e*f^2 + 27*c^3*f^3)*Sqrt[(f*(-c + d*x))/(d*(e + 
f*x))]*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*EllipticE[I*ArcSi 
nh[Sqrt[-((d*e + c*f)/d)]/Sqrt[e + f*x]], (d*e - c*f)/(d*e + c*f)])/(f*Sqr 
t[-((d*e + c*f)/d)]*(c^2 - d^2*x^2)) - ((2*I)*c*(2*d^2*e^2 - 7*c*d*e*f + 2 
1*c^2*f^2)*Sqrt[(f*(-c + d*x))/(d*(e + f*x))]*Sqrt[(f*(c + d*x))/(d*(e + f 
*x))]*(e + f*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-((d*e + c*f)/d)]/Sqrt[e + 
f*x]], (d*e - c*f)/(d*e + c*f)])/(Sqrt[-((d*e + c*f)/d)]*(c^2 - d^2*x^2))) 
)/(30*c^3*(d*e - c*f)^3*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {718, 115, 27, 169, 25, 27, 169, 27, 176, 124, 27, 123, 131, 27, 131, 129}

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.

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

\(\Big \downarrow \) 718

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \int \frac {1}{\sqrt {c-d x} (c+d x)^{7/2} \sqrt {e+f x}}dx}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 115

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (-\frac {\int -\frac {d (4 d e-9 c f+3 d f x)}{2 \sqrt {c-d x} (c+d x)^{5/2} \sqrt {e+f x}}dx}{5 c d (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\int \frac {4 d e-9 c f+3 d f x}{\sqrt {c-d x} (c+d x)^{5/2} \sqrt {e+f x}}dx}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {-\frac {\int -\frac {d \left (4 d^2 e^2-13 c d f e+21 c^2 f^2+2 d f (d e-3 c f) x\right )}{\sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}dx}{3 c d (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {\int \frac {d \left (4 d^2 e^2-13 c d f e+21 c^2 f^2+2 d f (d e-3 c f) x\right )}{\sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}dx}{3 c d (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {\int \frac {4 d^2 e^2-13 c d f e+21 c^2 f^2+2 d f (d e-3 c f) x}{\sqrt {c-d x} (c+d x)^{3/2} \sqrt {e+f x}}dx}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {\int \frac {d f \left (f (d e+15 c f) c^2+d \left (4 d^2 e^2-15 c d f e+27 c^2 f^2\right ) x\right )}{2 \sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{c d (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \int \frac {f (d e+15 c f) c^2+d \left (4 d^2 e^2-15 c d f e+27 c^2 f^2\right ) x}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \int \frac {\sqrt {c-d x}}{\sqrt {c+d x} \sqrt {e+f x}}dx\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\frac {\sqrt {2} \sqrt {c-d x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {2} \sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\frac {\sqrt {c-d x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (2 c \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c-d x} \sqrt {c+d x} \sqrt {e+f x}}dx-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (\frac {\sqrt {2} c \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {\sqrt {2}}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {c-d x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (\frac {2 c \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {c-d x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (\frac {2 c \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {c-d x} \sqrt {e+f x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {\sqrt {c-d x} \sqrt {c+d x} \left (\frac {\frac {-\frac {f \left (\frac {4 c^{3/2} \sqrt {\frac {c-d x}{c}} \left (21 c^2 f^2-7 c d e f+2 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{d \sqrt {c-d x} \sqrt {e+f x}}-\frac {2 \sqrt {2} \sqrt {c-d x} \sqrt {c f-d e} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {1}{2} \left (1-\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {\frac {c-d x}{c}} \sqrt {e+f x}}\right )}{2 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x} \left (27 c^2 f^2-15 c d e f+4 d^2 e^2\right )}{c \sqrt {c+d x} (d e-c f)}}{3 c (d e-c f)}-\frac {4 \sqrt {c-d x} \sqrt {e+f x} (d e-3 c f)}{3 c (c+d x)^{3/2} (d e-c f)}}{10 c (d e-c f)}-\frac {\sqrt {c-d x} \sqrt {e+f x}}{5 c (c+d x)^{5/2} (d e-c f)}\right )}{\sqrt {c^2-d^2 x^2}}\)

Input:

Int[1/((c + d*x)^3*Sqrt[e + f*x]*Sqrt[c^2 - d^2*x^2]),x]
 

Output:

(Sqrt[c - d*x]*Sqrt[c + d*x]*(-1/5*(Sqrt[c - d*x]*Sqrt[e + f*x])/(c*(d*e - 
 c*f)*(c + d*x)^(5/2)) + ((-4*(d*e - 3*c*f)*Sqrt[c - d*x]*Sqrt[e + f*x])/( 
3*c*(d*e - c*f)*(c + d*x)^(3/2)) + (-(((4*d^2*e^2 - 15*c*d*e*f + 27*c^2*f^ 
2)*Sqrt[c - d*x]*Sqrt[e + f*x])/(c*(d*e - c*f)*Sqrt[c + d*x])) - (f*((-2*S 
qrt[2]*Sqrt[-(d*e) + c*f]*(4*d^2*e^2 - 15*c*d*e*f + 27*c^2*f^2)*Sqrt[c - d 
*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x 
])/Sqrt[-(d*e) + c*f]], (1 - (d*e)/(c*f))/2])/(d*Sqrt[f]*Sqrt[(c - d*x)/c] 
*Sqrt[e + f*x]) + (4*c^(3/2)*(2*d^2*e^2 - 7*c*d*e*f + 21*c^2*f^2)*Sqrt[(c 
- d*x)/c]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[ArcSin[Sqrt[c + d*x]/( 
Sqrt[2]*Sqrt[c])], (-2*c*f)/(d*e - c*f)])/(d*Sqrt[c - d*x]*Sqrt[e + f*x])) 
)/(2*c*(d*e - c*f)))/(3*c*(d*e - c*f)))/(10*c*(d*e - c*f))))/Sqrt[c^2 - d^ 
2*x^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 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

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

rule 129

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

rule 131

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

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 176

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

rule 718

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]* 
(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/ 
e)*x)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 
 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(861\) vs. \(2(421)=842\).

Time = 10.82 (sec) , antiderivative size = 862, normalized size of antiderivative = 1.81

method	result	size
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\frac {\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}{5 c \left (c f -d e \right ) d^{3} \left (x +\frac {c}{d}\right )^{3}}+\frac {2 \left (3 c f -d e \right ) \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}{15 c^{2} \left (c f -d e \right )^{2} d^{2} \left (x +\frac {c}{d}\right )^{2}}+\frac {\left (-d^{2} f \,x^{2}+c d f x -d^{2} e x +d e c \right ) \left (27 c^{2} f^{2}-15 c d e f +4 d^{2} e^{2}\right )}{30 c^{3} \left (c f -d e \right )^{3} d \sqrt {\left (x +\frac {c}{d}\right ) \left (-d^{2} f \,x^{2}+c d f x -d^{2} e x +d e c \right )}}+\frac {2 \left (-\frac {f \left (3 c f -d e \right )}{15 c^{2} \left (c f -d e \right )^{2}}+\frac {\left (2 c f -d e \right ) \left (27 c^{2} f^{2}-15 c d e f +4 d^{2} e^{2}\right )}{60 c^{3} \left (c f -d e \right )^{3}}-\frac {\left (d f c -d^{2} e \right ) \left (27 c^{2} f^{2}-15 c d e f +4 d^{2} e^{2}\right )}{60 c^{3} \left (c f -d e \right )^{3} d}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {d f \left (27 c^{2} f^{2}-15 c d e f +4 d^{2} e^{2}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{d}\right )}{30 c^{3} \left (c f -d e \right )^{3} \sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\right )}{\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}\)	\(862\)
default	\(\text {Expression too large to display}\)	\(3366\)

Input:

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

Output:

((f*x+e)*(-d^2*x^2+c^2))^(1/2)/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2)*(1/5/c/( 
c*f-d*e)/d^3*(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)/(x+c/d)^3+2/15*(3* 
c*f-d*e)/c^2/(c*f-d*e)^2/d^2*(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)/(x 
+c/d)^2+1/30*(-d^2*f*x^2+c*d*f*x-d^2*e*x+c*d*e)/c^3/(c*f-d*e)^3*(27*c^2*f^ 
2-15*c*d*e*f+4*d^2*e^2)/d/((x+c/d)*(-d^2*f*x^2+c*d*f*x-d^2*e*x+c*d*e))^(1/ 
2)+2*(-1/15*f*(3*c*f-d*e)/c^2/(c*f-d*e)^2+1/60*(2*c*f-d*e)*(27*c^2*f^2-15* 
c*d*e*f+4*d^2*e^2)/c^3/(c*f-d*e)^3-1/60*(c*d*f-d^2*e)/c^3/(c*f-d*e)^3*(27* 
c^2*f^2-15*c*d*e*f+4*d^2*e^2)/d)*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x-c 
/d)/(-e/f-c/d))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(-d^2*f*x^3-d^2*e*x^2+c^2 
*f*x+c^2*e)^(1/2)*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f-c/ 
d))^(1/2))+1/30*d*f*(27*c^2*f^2-15*c*d*e*f+4*d^2*e^2)/c^3/(c*f-d*e)^3*(e/f 
-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x-c/d)/(-e/f-c/d))^(1/2)*((x+c/d)/(-e/f+ 
c/d))^(1/2)/(-d^2*f*x^3-d^2*e*x^2+c^2*f*x+c^2*e)^(1/2)*((-e/f-c/d)*Ellipti 
cE(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f-c/d))^(1/2))+c/d*EllipticF( 
((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f-c/d))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (432) = 864\).

Time = 0.13 (sec) , antiderivative size = 868, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(c+d x)^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="fri 
cas")
 

Output:

-1/90*((4*c^3*d^3*e^3 - 15*c^4*d^2*e^2*f + 24*c^5*d*e*f^2 - 45*c^6*f^3 + ( 
4*d^6*e^3 - 15*c*d^5*e^2*f + 24*c^2*d^4*e*f^2 - 45*c^3*d^3*f^3)*x^3 + 3*(4 
*c*d^5*e^3 - 15*c^2*d^4*e^2*f + 24*c^3*d^3*e*f^2 - 45*c^4*d^2*f^3)*x^2 + 3 
*(4*c^2*d^4*e^3 - 15*c^3*d^3*e^2*f + 24*c^4*d^2*e*f^2 - 45*c^5*d*f^3)*x)*s 
qrt(-d^2*f)*weierstrassPInverse(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f^2), -8/27 
*(d^2*e^3 - 9*c^2*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f) + 3*(4*c^3*d^3*e^2* 
f - 15*c^4*d^2*e*f^2 + 27*c^5*d*f^3 + (4*d^6*e^2*f - 15*c*d^5*e*f^2 + 27*c 
^2*d^4*f^3)*x^3 + 3*(4*c*d^5*e^2*f - 15*c^2*d^4*e*f^2 + 27*c^3*d^3*f^3)*x^ 
2 + 3*(4*c^2*d^4*e^2*f - 15*c^3*d^3*e*f^2 + 27*c^4*d^2*f^3)*x)*sqrt(-d^2*f 
)*weierstrassZeta(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f^2), -8/27*(d^2*e^3 - 9* 
c^2*e*f^2)/(d^2*f^3), weierstrassPInverse(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f 
^2), -8/27*(d^2*e^3 - 9*c^2*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f)) + 3*(14* 
c^2*d^4*e^2*f - 43*c^3*d^3*e*f^2 + 45*c^4*d^2*f^3 + (4*d^6*e^2*f - 15*c*d^ 
5*e*f^2 + 27*c^2*d^4*f^3)*x^2 + 2*(6*c*d^5*e^2*f - 23*c^2*d^4*e*f^2 + 33*c 
^3*d^3*f^3)*x)*sqrt(-d^2*x^2 + c^2)*sqrt(f*x + e))/(c^6*d^5*e^3*f - 3*c^7* 
d^4*e^2*f^2 + 3*c^8*d^3*e*f^3 - c^9*d^2*f^4 + (c^3*d^8*e^3*f - 3*c^4*d^7*e 
^2*f^2 + 3*c^5*d^6*e*f^3 - c^6*d^5*f^4)*x^3 + 3*(c^4*d^7*e^3*f - 3*c^5*d^6 
*e^2*f^2 + 3*c^6*d^5*e*f^3 - c^7*d^4*f^4)*x^2 + 3*(c^5*d^6*e^3*f - 3*c^6*d 
^5*e^2*f^2 + 3*c^7*d^4*e*f^3 - c^8*d^3*f^4)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x)^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\text {Timed out} \] Input:

integrate(1/(d*x+c)**3/(f*x+e)**(1/2)/(-d**2*x**2+c**2)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/(d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="max 
ima")
 

Output:

integrate(1/(sqrt(-d^2*x^2 + c^2)*(d*x + c)^3*sqrt(f*x + e)), x)
 

Giac [F]

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

integrate(1/(d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x, algorithm="gia 
c")
 

Output:

integrate(1/(sqrt(-d^2*x^2 + c^2)*(d*x + c)^3*sqrt(f*x + e)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(1/((e + f*x)^(1/2)*(c^2 - d^2*x^2)^(1/2)*(c + d*x)^3),x)
 

Output:

int(1/((e + f*x)^(1/2)*(c^2 - d^2*x^2)^(1/2)*(c + d*x)^3), x)
 

Reduce [F]

\[ \int \frac {1}{(c+d x)^3 \sqrt {e+f x} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {\sqrt {f x +e}\, \sqrt {-d^{2} x^{2}+c^{2}}}{-d^{5} f \,x^{6}-3 c \,d^{4} f \,x^{5}-d^{5} e \,x^{5}-2 c^{2} d^{3} f \,x^{4}-3 c \,d^{4} e \,x^{4}+2 c^{3} d^{2} f \,x^{3}-2 c^{2} d^{3} e \,x^{3}+3 c^{4} d f \,x^{2}+2 c^{3} d^{2} e \,x^{2}+c^{5} f x +3 c^{4} d e x +c^{5} e}d x \] Input:

int(1/(d*x+c)^3/(f*x+e)^(1/2)/(-d^2*x^2+c^2)^(1/2),x)
 

Output:

int((sqrt(e + f*x)*sqrt(c**2 - d**2*x**2))/(c**5*e + c**5*f*x + 3*c**4*d*e 
*x + 3*c**4*d*f*x**2 + 2*c**3*d**2*e*x**2 + 2*c**3*d**2*f*x**3 - 2*c**2*d* 
*3*e*x**3 - 2*c**2*d**3*f*x**4 - 3*c*d**4*e*x**4 - 3*c*d**4*f*x**5 - d**5* 
e*x**5 - d**5*f*x**6),x)