Optimal. Leaf size=155 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {266, 43, 2350, 12, 1251, 897, 1261, 206} \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 206
Rule 266
Rule 897
Rule 1251
Rule 1261
Rule 2350
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-(b n) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{3 e^3 x \left (d+e x^2\right )^{3/2}} \, dx\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{x \left (d+e x^2\right )^{3/2}} \, dx}{3 e^3}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \operatorname {Subst}\left (\int \frac {8 d^2+12 d e x+3 e^2 x^2}{x (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^3}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \operatorname {Subst}\left (\int \frac {-d^2+6 d x^2+3 x^4}{x^2 \left (-\frac {d}{e}+\frac {x^2}{e}\right )} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^4}\\ &=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {(b n) \operatorname {Subst}\left (\int \left (3 e+\frac {d e}{x^2}-\frac {8 d e}{d-x^2}\right ) \, dx,x,\sqrt {d+e x^2}\right )}{3 e^4}\\ &=\frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {(8 b d n) \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d+e x^2}\right )}{3 e^3}\\ &=\frac {b d n}{3 e^3 \sqrt {d+e x^2}}-\frac {b n \sqrt {d+e x^2}}{e^3}+\frac {8 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 205, normalized size = 1.32 \[ \sqrt {d+e x^2} \left (-\frac {d^2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{3 e^3 \left (d+e x^2\right )^2}+\frac {d \left (6 a+6 b \left (\log \left (c x^n\right )-n \log (x)\right )+b n\right )}{3 e^3 \left (d+e x^2\right )}+\frac {a+b \left (\log \left (c x^n\right )-n \log (x)\right )-b n}{e^3}\right )+\frac {b n \log (x) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {8 b \sqrt {d} n \log \left (\sqrt {d} \sqrt {d+e x^2}+d\right )}{3 e^3}-\frac {8 b \sqrt {d} n \log (x)}{3 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 401, normalized size = 2.59 \[ \left [\frac {4 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {d} \log \left (-\frac {e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - {\left (3 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + 2 \, b d^{2} n - 8 \, a d^{2} + {\left (5 \, b d e n - 12 \, a d e\right )} x^{2} - {\left (3 \, b e^{2} x^{4} + 12 \, b d e x^{2} + 8 \, b d^{2}\right )} \log \relax (c) - {\left (3 \, b e^{2} n x^{4} + 12 \, b d e n x^{2} + 8 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (e^{5} x^{4} + 2 \, d e^{4} x^{2} + d^{2} e^{3}\right )}}, -\frac {8 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (3 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + 2 \, b d^{2} n - 8 \, a d^{2} + {\left (5 \, b d e n - 12 \, a d e\right )} x^{2} - {\left (3 \, b e^{2} x^{4} + 12 \, b d e x^{2} + 8 \, b d^{2}\right )} \log \relax (c) - {\left (3 \, b e^{2} n x^{4} + 12 \, b d e n x^{2} + 8 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x^{2} + d}}{3 \, {\left (e^{5} x^{4} + 2 \, d e^{4} x^{2} + d^{2} e^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{5}}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 193, normalized size = 1.25 \[ -\frac {1}{3} \, b n {\left (\frac {4 \, \sqrt {d} \log \left (\frac {\sqrt {e x^{2} + d} - \sqrt {d}}{\sqrt {e x^{2} + d} + \sqrt {d}}\right )}{e^{3}} + \frac {3 \, \sqrt {e x^{2} + d}}{e^{3}} - \frac {d}{\sqrt {e x^{2} + d} e^{3}}\right )} + \frac {1}{3} \, {\left (\frac {3 \, x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {12 \, d x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}} + \frac {8 \, d^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{3}}\right )} b \log \left (c x^{n}\right ) + \frac {1}{3} \, {\left (\frac {3 \, x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {12 \, d x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}} + \frac {8 \, d^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________