Optimal. Leaf size=46 \[ \frac {2}{3} a A b x^3+\frac {1}{6} A b^2 x^6+\frac {B \left (a+b x^3\right )^3}{9 b}+a^2 A \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {457, 81, 45}
\begin {gather*} a^2 A \log (x)+\frac {2}{3} a A b x^3+\frac {B \left (a+b x^3\right )^3}{9 b}+\frac {1}{6} A b^2 x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 81
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2 \left (A+B x^3\right )}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^2 (A+B x)}{x} \, dx,x,x^3\right )\\ &=\frac {B \left (a+b x^3\right )^3}{9 b}+\frac {1}{3} A \text {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^3\right )\\ &=\frac {B \left (a+b x^3\right )^3}{9 b}+\frac {1}{3} A \text {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^3\right )\\ &=\frac {2}{3} a A b x^3+\frac {1}{6} A b^2 x^6+\frac {B \left (a+b x^3\right )^3}{9 b}+a^2 A \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.11 \begin {gather*} \frac {1}{3} a (2 A b+a B) x^3+\frac {1}{6} b (A b+2 a B) x^6+\frac {1}{9} b^2 B x^9+a^2 A \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 52, normalized size = 1.13
method | result | size |
norman | \(\left (\frac {1}{6} b^{2} A +\frac {1}{3} a b B \right ) x^{6}+\left (\frac {2}{3} a b A +\frac {1}{3} a^{2} B \right ) x^{3}+\frac {b^{2} B \,x^{9}}{9}+a^{2} A \ln \left (x \right )\) | \(50\) |
default | \(\frac {b^{2} B \,x^{9}}{9}+\frac {A \,b^{2} x^{6}}{6}+\frac {B a b \,x^{6}}{3}+\frac {2 a A b \,x^{3}}{3}+\frac {a^{2} B \,x^{3}}{3}+a^{2} A \ln \left (x \right )\) | \(52\) |
risch | \(\frac {b^{2} B \,x^{9}}{9}+\frac {A \,b^{2} x^{6}}{6}+\frac {B a b \,x^{6}}{3}+\frac {2 a A b \,x^{3}}{3}+\frac {a^{2} B \,x^{3}}{3}+a^{2} A \ln \left (x \right )\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 52, normalized size = 1.13 \begin {gather*} \frac {1}{9} \, B b^{2} x^{9} + \frac {1}{6} \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac {1}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} + \frac {1}{3} \, A a^{2} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.23, size = 49, normalized size = 1.07 \begin {gather*} \frac {1}{9} \, B b^{2} x^{9} + \frac {1}{6} \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac {1}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 53, normalized size = 1.15 \begin {gather*} A a^{2} \log {\left (x \right )} + \frac {B b^{2} x^{9}}{9} + x^{6} \left (\frac {A b^{2}}{6} + \frac {B a b}{3}\right ) + x^{3} \cdot \left (\frac {2 A a b}{3} + \frac {B a^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 52, normalized size = 1.13 \begin {gather*} \frac {1}{9} \, B b^{2} x^{9} + \frac {1}{3} \, B a b x^{6} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{3} \, B a^{2} x^{3} + \frac {2}{3} \, A a b x^{3} + A a^{2} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 49, normalized size = 1.07 \begin {gather*} x^3\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+x^6\,\left (\frac {A\,b^2}{6}+\frac {B\,a\,b}{3}\right )+\frac {B\,b^2\,x^9}{9}+A\,a^2\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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