Optimal. Leaf size=62 \[ -\frac {(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac {b (b+c x)^4 (b B-A c)}{4 c^3}+\frac {B (b+c x)^6}{6 c^3} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ -\frac {(b+c x)^5 (2 b B-A c)}{5 c^3}+\frac {b (b+c x)^4 (b B-A c)}{4 c^3}+\frac {B (b+c x)^6}{6 c^3} \]
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^2} \, dx &=\int \left (\frac {b (b B-A c) (b+c x)^3}{c^2}+\frac {(-2 b B+A c) (b+c x)^4}{c^2}+\frac {B (b+c x)^5}{c^2}\right ) \, dx\\ &=\frac {b (b B-A c) (b+c x)^4}{4 c^3}-\frac {(2 b B-A c) (b+c x)^5}{5 c^3}+\frac {B (b+c x)^6}{6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 67, normalized size = 1.08 \[ \frac {1}{60} x^2 \left (30 A b^3+20 b^2 x (3 A c+b B)+12 c^2 x^3 (A c+3 b B)+45 b c x^2 (A c+b B)+10 B c^3 x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 73, normalized size = 1.18 \[ \frac {1}{6} \, B c^{3} x^{6} + \frac {1}{2} \, A b^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 76, normalized size = 1.23 \[ \frac {1}{6} \, B c^{3} x^{6} + \frac {3}{5} \, B b c^{2} x^{5} + \frac {1}{5} \, A c^{3} x^{5} + \frac {3}{4} \, B b^{2} c x^{4} + \frac {3}{4} \, A b c^{2} x^{4} + \frac {1}{3} \, B b^{3} x^{3} + A b^{2} c x^{3} + \frac {1}{2} \, A b^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 1.23 \[ \frac {B \,c^{3} x^{6}}{6}+\frac {A \,b^{3} x^{2}}{2}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{5}}{5}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{4}}{4}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 73, normalized size = 1.18 \[ \frac {1}{6} \, B c^{3} x^{6} + \frac {1}{2} \, A b^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 68, normalized size = 1.10 \[ x^3\,\left (\frac {B\,b^3}{3}+A\,c\,b^2\right )+x^5\,\left (\frac {A\,c^3}{5}+\frac {3\,B\,b\,c^2}{5}\right )+\frac {A\,b^3\,x^2}{2}+\frac {B\,c^3\,x^6}{6}+\frac {3\,b\,c\,x^4\,\left (A\,c+B\,b\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 80, normalized size = 1.29 \[ \frac {A b^{3} x^{2}}{2} + \frac {B c^{3} x^{6}}{6} + x^{5} \left (\frac {A c^{3}}{5} + \frac {3 B b c^{2}}{5}\right ) + x^{4} \left (\frac {3 A b c^{2}}{4} + \frac {3 B b^{2} c}{4}\right ) + x^{3} \left (A b^{2} c + \frac {B b^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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