Optimal. Leaf size=66 \[ \frac {a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}-\frac {A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac {B \log \left (a+b x^3\right )}{3 b^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78}
\begin {gather*} -\frac {A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac {a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}+\frac {B \log \left (a+b x^3\right )}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x (A+B x)}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^3}+\frac {A b-2 a B}{b^2 (a+b x)^2}+\frac {B}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {a (A b-a B)}{6 b^3 \left (a+b x^3\right )^2}-\frac {A b-2 a B}{3 b^3 \left (a+b x^3\right )}+\frac {B \log \left (a+b x^3\right )}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 64, normalized size = 0.97 \begin {gather*} \frac {3 a^2 B-2 A b^2 x^3-a b \left (A-4 B x^3\right )+2 B \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 b^3 \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 61, normalized size = 0.92
method | result | size |
norman | \(\frac {-\frac {a \left (A b -3 B a \right )}{6 b^{3}}-\frac {\left (A b -2 B a \right ) x^{3}}{3 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {B \ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(57\) |
risch | \(\frac {-\frac {a \left (A b -3 B a \right )}{6 b^{3}}-\frac {\left (A b -2 B a \right ) x^{3}}{3 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {B \ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(57\) |
default | \(-\frac {A b -2 B a}{3 b^{3} \left (b \,x^{3}+a \right )}+\frac {a \left (A b -B a \right )}{6 b^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {B \ln \left (b \,x^{3}+a \right )}{3 b^{3}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 72, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (2 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b}{6 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {B \log \left (b x^{3} + a\right )}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.98, size = 89, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (2 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b + 2 \, {\left (B b^{2} x^{6} + 2 \, B a b x^{3} + B a^{2}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.07, size = 70, normalized size = 1.06 \begin {gather*} \frac {B \log {\left (a + b x^{3} \right )}}{3 b^{3}} + \frac {- A a b + 3 B a^{2} + x^{3} \left (- 2 A b^{2} + 4 B a b\right )}{6 a^{2} b^{3} + 12 a b^{4} x^{3} + 6 b^{5} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 61, normalized size = 0.92 \begin {gather*} \frac {B \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac {2 \, {\left (2 \, B a - A b\right )} x^{3} + \frac {3 \, B a^{2} - A a b}{b}}{6 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.38, size = 70, normalized size = 1.06 \begin {gather*} \frac {\frac {3\,B\,a^2-A\,a\,b}{6\,b^3}-\frac {x^3\,\left (A\,b-2\,B\,a\right )}{3\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {B\,\ln \left (b\,x^3+a\right )}{3\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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