Optimal. Leaf size=17 \[ \frac{(a+b x)^3}{3 b c^3} \]
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Rubi [A] time = 0.0040241, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {21, 32} \[ \frac{(a+b x)^3}{3 b c^3} \]
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^3} \, dx &=\frac{\int (a+b x)^2 \, dx}{c^3}\\ &=\frac{(a+b x)^3}{3 b c^3}\\ \end{align*}
Mathematica [A] time = 0.0009627, size = 17, normalized size = 1. \[ \frac{(a+b x)^3}{3 b c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 16, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{3}}{3\,b{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01137, size = 35, normalized size = 2.06 \begin{align*} \frac{b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45946, size = 55, normalized size = 3.24 \begin{align*} \frac{b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.094063, size = 29, normalized size = 1.71 \begin{align*} \frac{a^{2} x}{c^{3}} + \frac{a b x^{2}}{c^{3}} + \frac{b^{2} x^{3}}{3 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05813, size = 35, normalized size = 2.06 \begin{align*} \frac{b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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