Optimal. Leaf size=86 \[ \frac{a^3 x}{b^4 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{2 a x^2}{b^3 \sqrt{c x^2}}+\frac{x^3}{2 b^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.02566, antiderivative size = 86, 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} \[ \frac{a^3 x}{b^4 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{2 a x^2}{b^3 \sqrt{c x^2}}+\frac{x^3}{2 b^2 \sqrt{c x^2}} \]
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)^2} \, dx &=\frac{x \int \frac{x^3}{(a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{2 a x^2}{b^3 \sqrt{c x^2}}+\frac{x^3}{2 b^2 \sqrt{c x^2}}+\frac{a^3 x}{b^4 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x \log (a+b x)}{b^4 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0129685, size = 69, normalized size = 0.8 \[ \frac{x \left (-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)+2 a^3-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 74, normalized size = 0.9 \begin{align*}{\frac{x \left ({b}^{3}{x}^{3}+6\,\ln \left ( bx+a \right ) x{a}^{2}b-3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -4\,{a}^{2}bx+2\,{a}^{3} \right ) }{2\,{b}^{4} \left ( bx+a \right ) }{\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 = 1.16849, size = 159, normalized size = 1.85 \begin{align*} \frac{{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \,{\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (b^{5} c x^{2} + a b^{4} c x\right )}} \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 )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{c x^{2}}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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