Optimal. Leaf size=114 \[ \frac{4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d+22 e x}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.346936, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d+22 e x}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(x*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 46.6765, size = 97, normalized size = 0.85 \[ \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} \left (d - e x\right )^{3}} + \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{3} \left (d - e x\right )^{2}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{4}} + \frac{22 \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{4} \left (d - e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/x/(-e**2*x**2+d**2)**(7/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.145697, size = 77, normalized size = 0.68 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (32 d^2-51 d e x+22 e^2 x^2\right )}{(d-e x)^3}-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 \log (x)}{15 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(x*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 158, normalized size = 1.4 \[{\frac{4\,d}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{1}{3\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{4\,ex}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{11\,ex}{15\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{22\,ex}{15\,{d}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/x/(-e^2*x^2+d^2)^(7/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.289158, size = 447, normalized size = 3.92 \[ \frac{54 \, e^{5} x^{5} - 145 \, d e^{4} x^{4} - 5 \, d^{2} e^{3} x^{3} + 270 \, d^{3} e^{2} x^{2} - 180 \, d^{4} e x + 15 \,{\left (e^{5} x^{5} - 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} + 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x + 4 \, d^{5} +{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 5 \,{\left (2 \, e^{4} x^{4} + 19 \, d e^{3} x^{3} - 54 \, d^{2} e^{2} x^{2} + 36 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{4} e^{5} x^{5} - 5 \, d^{5} e^{4} x^{4} + 5 \, d^{6} e^{3} x^{3} + 5 \, d^{7} e^{2} x^{2} - 10 \, d^{8} e x + 4 \, d^{9} +{\left (d^{4} e^{4} x^{4} - 7 \, d^{6} e^{2} x^{2} + 10 \, d^{7} e x - 4 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/x/(-e**2*x**2+d**2)**(7/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.293, size = 158, normalized size = 1.39 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{22 \, x e^{5}}{d^{4}} + \frac{15 \, e^{4}}{d^{3}}\right )} - \frac{55 \, e^{3}}{d^{2}}\right )} x - \frac{35 \, e^{2}}{d}\right )} x + 45 \, e\right )} x + 32 \, d\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="giac")
[Out]