Optimal. Leaf size=383 \[ -\frac {b n x}{3 e^2 \sqrt {d+e x^2}}+\frac {4 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{5/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{5/2} \sqrt {d+e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.38, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2386, 294,
221, 2392, 21, 393, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{5/2} \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{5/2} \sqrt {d+e x^2}}+\frac {4 b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {b n x}{3 e^2 \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 221
Rule 294
Rule 393
Rule 2221
Rule 2317
Rule 2386
Rule 2392
Rule 2438
Rule 3797
Rule 5775
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1+\frac {e x^2}{d}} \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{\left (1+\frac {e x^2}{d}\right )^{5/2}} \, dx}{d^2 \sqrt {d+e x^2}}\\ &=-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \left (-\frac {d^3 \left (3 d+4 e x^2\right ) \sqrt {1+\frac {e x^2}{d}}}{3 e^2 \left (d+e x^2\right )^2}+\frac {d^{5/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2} x}\right ) \, dx}{d^2 \sqrt {d+e x^2}}\\ &=-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{e^{5/2} \sqrt {d+e x^2}}+\frac {\left (b d n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\left (3 d+4 e x^2\right ) \sqrt {1+\frac {e x^2}{d}}}{\left (d+e x^2\right )^2} \, dx}{3 e^2 \sqrt {d+e x^2}}\\ &=-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{5/2} \sqrt {d+e x^2}}+\frac {\left (b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {3 d+4 e x^2}{\left (1+\frac {e x^2}{d}\right )^{3/2}} \, dx}{3 d e^2 \sqrt {d+e x^2}}\\ &=-\frac {b n x}{3 e^2 \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{5/2} \sqrt {d+e x^2}}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}+\frac {\left (2 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{5/2} \sqrt {d+e x^2}}+\frac {\left (4 b n \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{3 e^2 \sqrt {d+e x^2}}\\ &=-\frac {b n x}{3 e^2 \sqrt {d+e x^2}}+\frac {4 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{5/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}+\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{e^{5/2} \sqrt {d+e x^2}}\\ &=-\frac {b n x}{3 e^2 \sqrt {d+e x^2}}+\frac {4 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{5/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}+\frac {\left (b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{5/2} \sqrt {d+e x^2}}\\ &=-\frac {b n x}{3 e^2 \sqrt {d+e x^2}}+\frac {4 b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{5/2} \sqrt {d+e x^2}}+\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e \left (d+e x^2\right )^{3/2}}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2} \sqrt {d+e x^2}}-\frac {b \sqrt {d} n \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 e^{5/2} \sqrt {d+e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.57, size = 244, normalized size = 0.64 \begin {gather*} -\frac {b n \sqrt {1+\frac {e x^2}{d}} \left (3 e^{5/2} x^5 \left (d+e x^2\right )^2 \, _3F_2\left (\frac {5}{2},\frac {5}{2},\frac {5}{2};\frac {7}{2},\frac {7}{2};-\frac {e x^2}{d}\right )+25 d^3 \sqrt {e} x \left (3 d+4 e x^2\right ) \sqrt {1+\frac {e x^2}{d}} \log (x)-75 d^{5/2} \left (d+e x^2\right )^2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)\right )}{75 d^2 e^{5/2} \left (d+e x^2\right )^{5/2}}-\frac {x \left (3 d+4 e x^2\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________