Optimal. Leaf size=142 \[ \frac{a^4 c^2 \sqrt{c x^2}}{b^5}-\frac{a^3 c^2 x \sqrt{c x^2}}{2 b^4}+\frac{a^2 c^2 x^2 \sqrt{c x^2}}{3 b^3}-\frac{a^5 c^2 \sqrt{c x^2} \log (a+b x)}{b^6 x}-\frac{a c^2 x^3 \sqrt{c x^2}}{4 b^2}+\frac{c^2 x^4 \sqrt{c x^2}}{5 b} \]
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Rubi [A] time = 0.044857, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac{a^4 c^2 \sqrt{c x^2}}{b^5}-\frac{a^3 c^2 x \sqrt{c x^2}}{2 b^4}+\frac{a^2 c^2 x^2 \sqrt{c x^2}}{3 b^3}-\frac{a^5 c^2 \sqrt{c x^2} \log (a+b x)}{b^6 x}-\frac{a c^2 x^3 \sqrt{c x^2}}{4 b^2}+\frac{c^2 x^4 \sqrt{c x^2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2}}{a+b x} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \frac{x^5}{a+b x} \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (\frac{a^4}{b^5}-\frac{a^3 x}{b^4}+\frac{a^2 x^2}{b^3}-\frac{a x^3}{b^2}+\frac{x^4}{b}-\frac{a^5}{b^5 (a+b x)}\right ) \, dx}{x}\\ &=\frac{a^4 c^2 \sqrt{c x^2}}{b^5}-\frac{a^3 c^2 x \sqrt{c x^2}}{2 b^4}+\frac{a^2 c^2 x^2 \sqrt{c x^2}}{3 b^3}-\frac{a c^2 x^3 \sqrt{c x^2}}{4 b^2}+\frac{c^2 x^4 \sqrt{c x^2}}{5 b}-\frac{a^5 c^2 \sqrt{c x^2} \log (a+b x)}{b^6 x}\\ \end{align*}
Mathematica [A] time = 0.0215217, size = 76, normalized size = 0.54 \[ \frac{c^3 x \left (b x \left (20 a^2 b^2 x^2-30 a^3 b x+60 a^4-15 a b^3 x^3+12 b^4 x^4\right )-60 a^5 \log (a+b x)\right )}{60 b^6 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 74, normalized size = 0.5 \begin{align*} -{\frac{-12\,{b}^{5}{x}^{5}+15\,a{b}^{4}{x}^{4}-20\,{a}^{2}{b}^{3}{x}^{3}+30\,{a}^{3}{b}^{2}{x}^{2}+60\,{a}^{5}\ln \left ( bx+a \right ) -60\,{a}^{4}bx}{60\,{x}^{5}{b}^{6}} \left ( c{x}^{2} \right ) ^{{\frac{5}{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 = 1.59848, size = 198, normalized size = 1.39 \begin{align*} \frac{{\left (12 \, b^{5} c^{2} x^{5} - 15 \, a b^{4} c^{2} x^{4} + 20 \, a^{2} b^{3} c^{2} x^{3} - 30 \, a^{3} b^{2} c^{2} x^{2} + 60 \, a^{4} b c^{2} x - 60 \, a^{5} c^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{60 \, b^{6} 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{\left (c x^{2}\right )^{\frac{5}{2}}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04936, size = 157, normalized size = 1.11 \begin{align*} -\frac{1}{60} \,{\left (\frac{60 \, a^{5} c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{6}} - \frac{60 \, a^{5} c^{2} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{6}} - \frac{12 \, b^{4} c^{2} x^{5} \mathrm{sgn}\left (x\right ) - 15 \, a b^{3} c^{2} x^{4} \mathrm{sgn}\left (x\right ) + 20 \, a^{2} b^{2} c^{2} x^{3} \mathrm{sgn}\left (x\right ) - 30 \, a^{3} b c^{2} x^{2} \mathrm{sgn}\left (x\right ) + 60 \, a^{4} c^{2} x \mathrm{sgn}\left (x\right )}{b^{5}}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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