Optimal. Leaf size=38 \[ \frac{B (b+c x)^5}{5 c^2}-\frac{(b+c x)^4 (b B-A c)}{4 c^2} \]
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Rubi [A] time = 0.0200986, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{B (b+c x)^5}{5 c^2}-\frac{(b+c x)^4 (b B-A c)}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^3} \, dx &=\int \left (\frac{(-b B+A c) (b+c x)^3}{c}+\frac{B (b+c x)^4}{c}\right ) \, dx\\ &=-\frac{(b B-A c) (b+c x)^4}{4 c^2}+\frac{B (b+c x)^5}{5 c^2}\\ \end{align*}
Mathematica [A] time = 0.0088554, size = 67, normalized size = 1.76 \[ \frac{1}{2} b^2 x^2 (3 A c+b B)+A b^3 x+\frac{1}{4} c^2 x^4 (A c+3 b B)+b c x^3 (A c+b B)+\frac{1}{5} B c^3 x^5 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 73, normalized size = 1.9 \begin{align*}{\frac{B{c}^{3}{x}^{5}}{5}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{2}}{2}}+A{b}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15159, size = 93, normalized size = 2.45 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + A b^{3} x + \frac{1}{4} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} +{\left (B b^{2} c + A b c^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76434, size = 150, normalized size = 3.95 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + A b^{3} x + \frac{1}{4} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} +{\left (B b^{2} c + A b c^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.077004, size = 73, normalized size = 1.92 \begin{align*} A b^{3} x + \frac{B c^{3} x^{5}}{5} + x^{4} \left (\frac{A c^{3}}{4} + \frac{3 B b c^{2}}{4}\right ) + x^{3} \left (A b c^{2} + B b^{2} c\right ) + x^{2} \left (\frac{3 A b^{2} c}{2} + \frac{B b^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21094, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + \frac{3}{4} \, B b c^{2} x^{4} + \frac{1}{4} \, A c^{3} x^{4} + B b^{2} c x^{3} + A b c^{2} x^{3} + \frac{1}{2} \, B b^{3} x^{2} + \frac{3}{2} \, A b^{2} c x^{2} + A b^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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