Optimal. Leaf size=95 \[ \frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{5/2}}-\frac {x^{3/2} (A b-3 a B)}{3 a b^2}+\frac {x^{9/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 321, 329, 275, 205} \begin {gather*} -\frac {x^{3/2} (A b-3 a B)}{3 a b^2}+\frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{5/2}}+\frac {x^{9/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 321
Rule 329
Rule 457
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x^{9/2}}{3 a b \left (a+b x^3\right )}+\frac {\left (-\frac {3 A b}{2}+\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{a+b x^3} \, dx}{3 a b}\\ &=-\frac {(A b-3 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{9/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-3 a B) \int \frac {\sqrt {x}}{a+b x^3} \, dx}{2 b^2}\\ &=-\frac {(A b-3 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{9/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-3 a B) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^6} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {(A b-3 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{9/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-3 a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^{3/2}\right )}{3 b^2}\\ &=-\frac {(A b-3 a B) x^{3/2}}{3 a b^2}+\frac {(A b-a B) x^{9/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 77, normalized size = 0.81 \begin {gather*} \frac {\frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\sqrt {b} x^{3/2} \left (3 a B-A b+2 b B x^3\right )}{a+b x^3}}{3 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 77, normalized size = 0.81 \begin {gather*} \frac {(A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{5/2}}+\frac {x^{3/2} \left (3 a B-A b+2 b B x^3\right )}{3 b^2 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 222, normalized size = 2.34 \begin {gather*} \left [\frac {{\left ({\left (3 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b\right )} \sqrt {-a b} \log \left (\frac {b x^{3} - 2 \, \sqrt {-a b} x^{\frac {3}{2}} - a}{b x^{3} + a}\right ) + 2 \, {\left (2 \, B a b^{2} x^{4} + {\left (3 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}}{6 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}}, -\frac {{\left ({\left (3 \, B a b - A b^{2}\right )} x^{3} + 3 \, B a^{2} - A a b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{\frac {3}{2}}}{a}\right ) - {\left (2 \, B a b^{2} x^{4} + {\left (3 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}}{3 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 68, normalized size = 0.72 \begin {gather*} \frac {2 \, B x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {{\left (3 \, B a - A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b^{2}} + \frac {B a x^{\frac {3}{2}} - A b x^{\frac {3}{2}}}{3 \, {\left (b x^{3} + a\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 93, normalized size = 0.98 \begin {gather*} -\frac {A \,x^{\frac {3}{2}}}{3 \left (b \,x^{3}+a \right ) b}+\frac {B a \,x^{\frac {3}{2}}}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {A \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \sqrt {a b}\, b}-\frac {B a \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {2 B \,x^{\frac {3}{2}}}{3 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 68, normalized size = 0.72 \begin {gather*} \frac {{\left (B a - A b\right )} x^{\frac {3}{2}}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} + \frac {2 \, B x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {{\left (3 \, B a - A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 116, normalized size = 1.22 \begin {gather*} \frac {2\,B\,x^{3/2}}{3\,b^2}-\frac {x^{3/2}\,\left (\frac {A\,b}{3}-\frac {B\,a}{3}\right )}{b^3\,x^3+a\,b^2}+\frac {\mathrm {atan}\left (\frac {36\,\sqrt {a}\,b^{3/2}\,x^{3/2}\,\left (A^2\,b^2-6\,A\,B\,a\,b+9\,B^2\,a^2\right )}{\left (A\,b-3\,B\,a\right )\,\left (36\,A\,a\,b^2-108\,B\,a^2\,b\right )}\right )\,\left (A\,b-3\,B\,a\right )}{3\,\sqrt {a}\,b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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