Optimal. Leaf size=25 \[ -\frac{x}{b c \sqrt{c x^2} (a+b x)} \]
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Rubi [A] time = 0.0038696, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ -\frac{x}{b c \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac{x \int \frac{1}{(a+b x)^2} \, dx}{c \sqrt{c x^2}}\\ &=-\frac{x}{b c \sqrt{c x^2} (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0059425, size = 24, normalized size = 0.96 \[ -\frac{x^3}{b \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3}}{ \left ( bx+a \right ) b} \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 [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 = 1.33424, size = 54, normalized size = 2.16 \begin{align*} -\frac{\sqrt{c x^{2}}}{b^{2} c^{2} x^{2} + a b c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.77616, size = 90, normalized size = 3.6 \begin{align*} \begin{cases} \frac{\tilde{\infty } x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x^{2}}{b^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: b = - \frac{a}{x} \\\frac{x^{4}}{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}} + a b c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \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^{3}}{\left (c x^{2}\right )^{\frac{3}{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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