Optimal. Leaf size=469 \[ \frac {7 b d^2 n x \sqrt {d+e x^2}}{192 e^2}-\frac {5 b d n x^3 \sqrt {d+e x^2}}{288 e}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}+\frac {5 b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{192 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}} \]
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Rubi [A]
time = 0.41, antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2386, 285,
327, 221, 2392, 12, 14, 201, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {b d^{5/2} n \sqrt {d+e x^2} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{32 e^{5/2} \sqrt {\frac {e x^2}{d}+1}}+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {\frac {e x^2}{d}+1}}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{32 e^{5/2} \sqrt {\frac {e x^2}{d}+1}}+\frac {5 b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{192 e^{5/2} \sqrt {\frac {e x^2}{d}+1}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \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 )}{16 e^{5/2} \sqrt {\frac {e x^2}{d}+1}}+\frac {7 b d^2 n x \sqrt {d+e x^2}}{192 e^2}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}-\frac {5 b d n x^3 \sqrt {d+e x^2}}{288 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 201
Rule 221
Rule 285
Rule 327
Rule 2221
Rule 2317
Rule 2386
Rule 2392
Rule 2438
Rule 3797
Rule 5775
Rubi steps
\begin {align*} \int x^4 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\sqrt {d+e x^2} \int x^4 \sqrt {1+\frac {e x^2}{d}} \left (a+b \log \left (c x^n\right )\right ) \, dx}{\sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \frac {\sqrt {e} x \sqrt {1+\frac {e x^2}{d}} \left (-3 d^2+2 d e x^2+8 e^2 x^4\right )+3 d^{5/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{48 e^{5/2} x} \, dx}{\sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \frac {\sqrt {e} x \sqrt {1+\frac {e x^2}{d}} \left (-3 d^2+2 d e x^2+8 e^2 x^4\right )+3 d^{5/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{48 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \left (-3 d^2 \sqrt {e} \sqrt {1+\frac {e x^2}{d}}+2 d e^{3/2} x^2 \sqrt {1+\frac {e x^2}{d}}+8 e^{5/2} x^4 \sqrt {1+\frac {e x^2}{d}}+\frac {3 d^{5/2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x}\right ) \, dx}{48 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}\\ &=-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int x^4 \sqrt {1+\frac {e x^2}{d}} \, dx}{6 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d^{5/2} n \sqrt {d+e x^2}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^2 n \sqrt {d+e x^2}\right ) \int \sqrt {1+\frac {e x^2}{d}} \, dx}{16 e^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d n \sqrt {d+e x^2}\right ) \int x^2 \sqrt {1+\frac {e x^2}{d}} \, dx}{24 e \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {b d^2 n x \sqrt {d+e x^2}}{32 e^2}-\frac {b d n x^3 \sqrt {d+e x^2}}{96 e}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b n \sqrt {d+e x^2}\right ) \int \frac {x^4}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{36 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d^{5/2} n \sqrt {d+e x^2}\right ) \text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^2 n \sqrt {d+e x^2}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{32 e^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d n \sqrt {d+e x^2}\right ) \int \frac {x^2}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{96 e \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {5 b d^2 n x \sqrt {d+e x^2}}{192 e^2}-\frac {5 b d n x^3 \sqrt {d+e x^2}}{288 e}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^{5/2} n \sqrt {d+e x^2}\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 )}{8 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^2 n \sqrt {d+e x^2}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{192 e^2 \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d n \sqrt {d+e x^2}\right ) \int \frac {x^2}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{48 e \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {7 b d^2 n x \sqrt {d+e x^2}}{192 e^2}-\frac {5 b d n x^3 \sqrt {d+e x^2}}{288 e}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}+\frac {7 b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{192 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \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 )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^{5/2} n \sqrt {d+e x^2}\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 )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b d^2 n \sqrt {d+e x^2}\right ) \int \frac {1}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{96 e^2 \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {7 b d^2 n x \sqrt {d+e x^2}}{192 e^2}-\frac {5 b d n x^3 \sqrt {d+e x^2}}{288 e}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}+\frac {5 b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{192 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \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 )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b d^{5/2} n \sqrt {d+e x^2}\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 )}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {7 b d^2 n x \sqrt {d+e x^2}}{192 e^2}-\frac {5 b d n x^3 \sqrt {d+e x^2}}{288 e}-\frac {1}{36} b n x^5 \sqrt {d+e x^2}+\frac {5 b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{192 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \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 )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {d^2 x \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {d x^3 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{24 e}+\frac {1}{6} x^5 \sqrt {d+e x^2} \left (a+b \log \left (c x^n\right )\right )+\frac {d^{5/2} \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{16 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b d^{5/2} n \sqrt {d+e x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{32 e^{5/2} \sqrt {1+\frac {e x^2}{d}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.39, size = 276, normalized size = 0.59 \begin {gather*} \frac {-48 b e^{5/2} n x^5 \sqrt {d+e x^2} \, _3F_2\left (-\frac {1}{2},\frac {5}{2},\frac {5}{2};\frac {7}{2},\frac {7}{2};-\frac {e x^2}{d}\right )+75 b d^{5/2} n \sqrt {d+e x^2} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)+25 \sqrt {1+\frac {e x^2}{d}} \left (a \sqrt {e} x \sqrt {d+e x^2} \left (-3 d^2+2 d e x^2+8 e^2 x^4\right )+3 d^3 (a-b n \log (x)) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )+b \log \left (c x^n\right ) \left (\sqrt {e} x \sqrt {d+e x^2} \left (-3 d^2+2 d e x^2+8 e^2 x^4\right )+3 d^3 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )\right )\right )}{1200 e^{5/2} \sqrt {1+\frac {e x^2}{d}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{4} \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e \,x^{2}+d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\sqrt {e\,x^2+d}\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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