Optimal. Leaf size=176 \[ -\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {277, 197, 2392,
12, 1279, 462, 223, 212} \begin {gather*} \frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 212
Rule 223
Rule 277
Rule 462
Rule 1279
Rule 2392
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-(b n) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^4 \sqrt {d+e x^2}} \, dx\\ &=-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b n) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{x^4 \sqrt {d+e x^2}} \, dx}{3 d^3}\\ &=-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(b n) \int \frac {-14 d^2 e-24 d e^2 x^2}{x^2 \sqrt {d+e x^2}} \, dx}{9 d^4}\\ &=-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {\left (8 b e^2 n\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{3 d^3}\\ &=-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {\left (8 b e^2 n\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{3 d^3}\\ &=-\frac {b n \sqrt {d+e x^2}}{9 d^2 x^3}+\frac {14 b e n \sqrt {d+e x^2}}{9 d^3 x}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{3 d^3}-\frac {a+b \log \left (c x^n\right )}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \log \left (c x^n\right )\right )}{3 d^3 \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 144, normalized size = 0.82 \begin {gather*} \frac {-3 a d^2-b d^2 n+12 a d e x^2+13 b d e n x^2+24 a e^2 x^4+14 b e^2 n x^4-3 b \left (d^2-4 d e x^2-8 e^2 x^4\right ) \log \left (c x^n\right )-24 b e^{3/2} n x^3 \sqrt {d+e x^2} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{9 d^3 x^3 \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
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 [A]
time = 0.45, size = 186, normalized size = 1.06 \begin {gather*} \frac {12 \, {\left (b n x^{5} e^{2} + b d n x^{3} e\right )} e^{\frac {1}{2}} \log \left (-2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right ) + {\left (2 \, {\left (7 \, b n + 12 \, a\right )} x^{4} e^{2} - b d^{2} n + {\left (13 \, b d n + 12 \, a d\right )} x^{2} e - 3 \, a d^{2} + 3 \, {\left (8 \, b x^{4} e^{2} + 4 \, b d x^{2} e - b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (8 \, b n x^{4} e^{2} + 4 \, b d n x^{2} e - b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x^{2} e + d}}{9 \, {\left (d^{3} x^{5} e + d^{4} x^{3}\right )}} \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 \frac {a + b \log {\left (c x^{n} \right )}}{x^{4} \left (d + e x^{2}\right )^{\frac {3}{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 = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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