Optimal. Leaf size=112 \[ -\frac{b^2 c \sqrt{c x^2}}{a^3 x^2}-\frac{b^3 c \sqrt{c x^2} \log (x)}{a^4 x}+\frac{b^3 c \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{b c \sqrt{c x^2}}{2 a^2 x^3}-\frac{c \sqrt{c x^2}}{3 a x^4} \]
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Rubi [A] time = 0.0310682, antiderivative size = 112, 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, 44} \[ -\frac{b^2 c \sqrt{c x^2}}{a^3 x^2}-\frac{b^3 c \sqrt{c x^2} \log (x)}{a^4 x}+\frac{b^3 c \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{b c \sqrt{c x^2}}{2 a^2 x^3}-\frac{c \sqrt{c x^2}}{3 a x^4} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2}}{x^7 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{x^4 (a+b x)} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{c \sqrt{c x^2}}{3 a x^4}+\frac{b c \sqrt{c x^2}}{2 a^2 x^3}-\frac{b^2 c \sqrt{c x^2}}{a^3 x^2}-\frac{b^3 c \sqrt{c x^2} \log (x)}{a^4 x}+\frac{b^3 c \sqrt{c x^2} \log (a+b x)}{a^4 x}\\ \end{align*}
Mathematica [A] time = 0.0251771, size = 65, normalized size = 0.58 \[ -\frac{\left (c x^2\right )^{3/2} \left (a \left (2 a^2-3 a b x+6 b^2 x^2\right )-6 b^3 x^3 \log (a+b x)+6 b^3 x^3 \log (x)\right )}{6 a^4 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 62, normalized size = 0.6 \begin{align*} -{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,a{b}^{2}{x}^{2}-3\,{a}^{2}bx+2\,{a}^{3}}{6\,{x}^{6}{a}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04894, size = 89, normalized size = 0.79 \begin{align*} \frac{b^{3} c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{4}} - \frac{b^{3} c^{\frac{3}{2}} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} c^{\frac{3}{2}} x^{2} - 3 \, a b c^{\frac{3}{2}} x + 2 \, a^{2} c^{\frac{3}{2}}}{6 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66753, size = 132, normalized size = 1.18 \begin{align*} \frac{{\left (6 \, b^{3} c x^{3} \log \left (\frac{b x + a}{x}\right ) - 6 \, a b^{2} c x^{2} + 3 \, a^{2} b c x - 2 \, a^{3} c\right )} \sqrt{c x^{2}}}{6 \, a^{4} x^{4}} \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{3}{2}}}{x^{7} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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