Optimal. Leaf size=31 \[ \frac{x (a+b x)^{n+1}}{b c (n+1) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0067218, antiderivative size = 31, 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} \[ \frac{x (a+b x)^{n+1}}{b c (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{x^3 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int (a+b x)^n \, dx}{c \sqrt{c x^2}}\\ &=\frac{x (a+b x)^{1+n}}{b c (1+n) \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0141891, size = 30, normalized size = 0.97 \[ \frac{x^3 (a+b x)^{n+1}}{b (n+1) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{3}}{b \left ( 1+n \right ) } \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 = 1.02099, size = 42, normalized size = 1.35 \begin{align*} \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b c^{2}{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28886, size = 77, normalized size = 2.48 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b c^{2} n + b c^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n} x^{3}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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