Optimal. Leaf size=80 \[ -\frac{2 a^2 p x}{5 b^2}+\frac{2 a^{5/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{5 b^{5/2}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac{2 a p x^3}{15 b}-\frac{2 p x^5}{25} \]
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Rubi [A] time = 0.0465048, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2455, 302, 205} \[ -\frac{2 a^2 p x}{5 b^2}+\frac{2 a^{5/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{5 b^{5/2}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac{2 a p x^3}{15 b}-\frac{2 p x^5}{25} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 302
Rule 205
Rubi steps
\begin{align*} \int x^4 \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac{1}{5} x^5 \log \left (c \left (a+b x^2\right )^p\right )-\frac{1}{5} (2 b p) \int \frac{x^6}{a+b x^2} \, dx\\ &=\frac{1}{5} x^5 \log \left (c \left (a+b x^2\right )^p\right )-\frac{1}{5} (2 b p) \int \left (\frac{a^2}{b^3}-\frac{a x^2}{b^2}+\frac{x^4}{b}-\frac{a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{2 a^2 p x}{5 b^2}+\frac{2 a p x^3}{15 b}-\frac{2 p x^5}{25}+\frac{1}{5} x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac{\left (2 a^3 p\right ) \int \frac{1}{a+b x^2} \, dx}{5 b^2}\\ &=-\frac{2 a^2 p x}{5 b^2}+\frac{2 a p x^3}{15 b}-\frac{2 p x^5}{25}+\frac{2 a^{5/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{5 b^{5/2}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^2\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0444517, size = 74, normalized size = 0.92 \[ \frac{1}{75} \left (-\frac{30 a^2 p x}{b^2}+\frac{30 a^{5/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+15 x^5 \log \left (c \left (a+b x^2\right )^p\right )+\frac{10 a p x^3}{b}-6 p x^5\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.499, size = 229, normalized size = 2.9 \begin{align*}{\frac{{x}^{5}\ln \left ( \left ( b{x}^{2}+a \right ) ^{p} \right ) }{5}}-{\frac{i}{10}}\pi \,{x}^{5} \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{10}}\pi \,{x}^{5} \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{10}}\pi \,{x}^{5}{\it csgn} \left ( i \left ( b{x}^{2}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}-{\frac{i}{10}}\pi \,{x}^{5}{\it csgn} \left ( i \left ( b{x}^{2}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{2}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ){x}^{5}}{5}}-{\frac{2\,p{x}^{5}}{25}}+{\frac{2\,ap{x}^{3}}{15\,b}}+{\frac{{a}^{2}p}{5\,{b}^{3}}\sqrt{-ab}\ln \left ( -\sqrt{-ab}x+a \right ) }-{\frac{{a}^{2}p}{5\,{b}^{3}}\sqrt{-ab}\ln \left ( \sqrt{-ab}x+a \right ) }-{\frac{2\,{a}^{2}px}{5\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10261, size = 436, normalized size = 5.45 \begin{align*} \left [\frac{15 \, b^{2} p x^{5} \log \left (b x^{2} + a\right ) - 6 \, b^{2} p x^{5} + 15 \, b^{2} x^{5} \log \left (c\right ) + 10 \, a b p x^{3} + 15 \, a^{2} p \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 30 \, a^{2} p x}{75 \, b^{2}}, \frac{15 \, b^{2} p x^{5} \log \left (b x^{2} + a\right ) - 6 \, b^{2} p x^{5} + 15 \, b^{2} x^{5} \log \left (c\right ) + 10 \, a b p x^{3} + 30 \, a^{2} p \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 30 \, a^{2} p x}{75 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21173, size = 96, normalized size = 1.2 \begin{align*} \frac{1}{5} \, p x^{5} \log \left (b x^{2} + a\right ) - \frac{1}{25} \,{\left (2 \, p - 5 \, \log \left (c\right )\right )} x^{5} + \frac{2 \, a p x^{3}}{15 \, b} + \frac{2 \, a^{3} p \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{5 \, \sqrt{a b} b^{2}} - \frac{2 \, a^{2} p x}{5 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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