Optimal. Leaf size=295 \[ -\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b \sqrt {d} e n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^2+d}}\right )-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-\frac {3}{2} b \sqrt {d} e n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {266, 47, 50, 63, 208, 2350, 12, 14, 5984, 5918, 2402, 2315} \[ -\frac {3}{4} b \sqrt {d} e n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )-\frac {3}{2} b \sqrt {d} e n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rule 2315
Rule 2350
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-\left (d-2 e x^2\right ) \sqrt {d+e x^2}-3 \sqrt {d} e x^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 x^3} \, dx\\ &=\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \frac {-\left (d-2 e x^2\right ) \sqrt {d+e x^2}-3 \sqrt {d} e x^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x^3} \, dx\\ &=\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} (b n) \int \left (-\frac {d \sqrt {d+e x^2}}{x^3}+\frac {2 e \sqrt {d+e x^2}}{x}-\frac {3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x}\right ) \, dx\\ &=\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} (b d n) \int \frac {\sqrt {d+e x^2}}{x^3} \, dx-(b e n) \int \frac {\sqrt {d+e x^2}}{x} \, dx+\frac {1}{2} \left (3 b \sqrt {d} e n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{x} \, dx\\ &=\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (b d n) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2} \, dx,x,x^2\right )-\frac {1}{2} (b e n) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (3 b \sqrt {d} e n\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx,x,x^2\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} \left (3 b \sqrt {d} e n\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x^2}\right )+\frac {1}{8} (b d e n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )-\frac {1}{2} (b d e n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{4} (b d n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )-(b d n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )-\frac {1}{2} (3 b e n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )+\frac {1}{2} (3 b e n) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {1}{2} \left (3 b \sqrt {d} e n\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x^2}}{\sqrt {d}}}\right )\\ &=-b e n \sqrt {d+e x^2}-\frac {b d n \sqrt {d+e x^2}}{4 x^2}+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+\frac {3}{4} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )^2+\frac {3}{2} e \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{2} b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^2}}\right )-\frac {3}{4} b \sqrt {d} e n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x^2}}{\sqrt {d}}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.96, size = 349, normalized size = 1.18 \[ \frac {b e n \sqrt {d+e x^2} \left (-\, _3F_2\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2};\frac {1}{2},\frac {1}{2};-\frac {d}{e x^2}\right )+\log (x) \sqrt {\frac {d}{e x^2}+1}-\frac {\sqrt {d} \log (x) \sinh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right )}{\sqrt {e} x}\right )}{\sqrt {\frac {d}{e x^2}+1}}-\frac {b \sqrt {d} n \sqrt {d+e x^2} \left (2 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-\frac {d}{e x^2}\right )+(2 \log (x)+1) \left (\sqrt {d} \sqrt {\frac {d}{e x^2}+1}+\sqrt {e} x \sinh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {e} x}\right )\right )\right )}{4 x^2 \sqrt {\frac {d}{e x^2}+1}}+\frac {3}{2} \sqrt {d} e \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac {\left (d-2 e x^2\right ) \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 x^2}-\frac {3}{2} \sqrt {d} e \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b e x^{2} + b d\right )} \sqrt {e x^{2} + d} \log \left (c x^{n}\right ) + {\left (a e x^{2} + a d\right )} \sqrt {e x^{2} + d}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (3 \, \sqrt {d} e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right ) - 3 \, \sqrt {e x^{2} + d} e - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d} + \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{2}}\right )} a + b \int \frac {{\left (e x^{2} \log \relax (c) + d \log \relax (c) + {\left (e x^{2} + d\right )} \log \left (x^{n}\right )\right )} \sqrt {e x^{2} + d}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________