Optimal. Leaf size=101 \[ \frac {(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}-\frac {\log (x) (3 A b-a B)}{a^4}-\frac {2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac {A}{3 a^3 x^3}-\frac {A b-a B}{6 a^2 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.10, antiderivative size = 101, 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 = {446, 77} \[ -\frac {2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac {A b-a B}{6 a^2 \left (a+b x^3\right )^2}+\frac {(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}-\frac {\log (x) (3 A b-a B)}{a^4}-\frac {A}{3 a^3 x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^4 \left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A}{a^3 x^2}+\frac {-3 A b+a B}{a^4 x}-\frac {b (-A b+a B)}{a^2 (a+b x)^3}-\frac {b (-2 A b+a B)}{a^3 (a+b x)^2}-\frac {b (-3 A b+a B)}{a^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {A}{3 a^3 x^3}-\frac {A b-a B}{6 a^2 \left (a+b x^3\right )^2}-\frac {2 A b-a B}{3 a^3 \left (a+b x^3\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^3\right )}{3 a^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 87, normalized size = 0.86 \[ \frac {\frac {a^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac {2 a (a B-2 A b)}{a+b x^3}+2 (3 A b-a B) \log \left (a+b x^3\right )+6 \log (x) (a B-3 A b)-\frac {2 a A}{x^3}}{6 a^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 197, normalized size = 1.95 \[ \frac {2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} - 2 \, A a^{3} + 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3} - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{9} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{9} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{6} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 136, normalized size = 1.35 \[ \frac {{\left (B a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} + \frac {3 \, B a b^{2} x^{6} - 9 \, A b^{3} x^{6} + 8 \, B a^{2} b x^{3} - 22 \, A a b^{2} x^{3} + 6 \, B a^{3} - 14 \, A a^{2} b}{6 \, {\left (b x^{3} + a\right )}^{2} a^{4}} - \frac {B a x^{3} - 3 \, A b x^{3} + A a}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 117, normalized size = 1.16 \[ -\frac {A b}{6 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {B}{6 \left (b \,x^{3}+a \right )^{2} a}-\frac {2 A b}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {3 A b \ln \relax (x )}{a^{4}}+\frac {A b \ln \left (b \,x^{3}+a \right )}{a^{4}}+\frac {B}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {B \ln \relax (x )}{a^{3}}-\frac {B \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {A}{3 a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 109, normalized size = 1.08 \[ \frac {2 \, {\left (B a b - 3 \, A b^{2}\right )} x^{6} + 3 \, {\left (B a^{2} - 3 \, A a b\right )} x^{3} - 2 \, A a^{2}}{6 \, {\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.46, size = 107, normalized size = 1.06 \[ \frac {\ln \left (b\,x^3+a\right )\,\left (3\,A\,b-B\,a\right )}{3\,a^4}-\frac {\frac {A}{3\,a}+\frac {x^3\,\left (3\,A\,b-B\,a\right )}{2\,a^2}+\frac {b\,x^6\,\left (3\,A\,b-B\,a\right )}{3\,a^3}}{a^2\,x^3+2\,a\,b\,x^6+b^2\,x^9}-\frac {\ln \relax (x)\,\left (3\,A\,b-B\,a\right )}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.27, size = 107, normalized size = 1.06 \[ \frac {- 2 A a^{2} + x^{6} \left (- 6 A b^{2} + 2 B a b\right ) + x^{3} \left (- 9 A a b + 3 B a^{2}\right )}{6 a^{5} x^{3} + 12 a^{4} b x^{6} + 6 a^{3} b^{2} x^{9}} + \frac {\left (- 3 A b + B a\right ) \log {\relax (x )}}{a^{4}} - \frac {\left (- 3 A b + B a\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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