Optimal. Leaf size=83 \[ \frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}} \]
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Rubi [A] time = 0.0240749, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ \frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{c x^2} (a+b x)} \, dx &=\frac{x \int \frac{x^3}{a+b x} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}}-\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0103318, size = 51, normalized size = 0.61 \[ \frac{x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 50, normalized size = 0.6 \begin{align*} -{\frac{x \left ( -2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx \right ) }{6\,{b}^{4}}{\frac{1}{\sqrt{c{x}^{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 = 1.82468, size = 116, normalized size = 1.4 \begin{align*} \frac{{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{c x^{2}} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10552, size = 108, normalized size = 1.3 \begin{align*} \frac{1}{6} \, \sqrt{c x^{2}}{\left (x{\left (\frac{2 \, x}{b c} - \frac{3 \, a}{b^{2} c}\right )} + \frac{6 \, a^{2}}{b^{3} c}\right )} + \frac{a^{3} \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{4} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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