Optimal. Leaf size=96 \[ -\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 a^{5/2} \sqrt {b}}+\frac {a B-3 A b}{3 a^2 b x^{3/2}}+\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.01, 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, 325, 329, 275, 205} \[ -\frac {3 A b-a B}{3 a^2 b x^{3/2}}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 a^{5/2} \sqrt {b}}+\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 325
Rule 329
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^{5/2} \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )}+\frac {\left (\frac {9 A b}{2}-\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {3 A b-a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )}-\frac {(3 A b-a B) \int \frac {\sqrt {x}}{a+b x^3} \, dx}{2 a^2}\\ &=-\frac {3 A b-a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )}-\frac {(3 A b-a B) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^6} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {3 A b-a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )}-\frac {(3 A b-a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^{3/2}\right )}{3 a^2}\\ &=-\frac {3 A b-a B}{3 a^2 b x^{3/2}}+\frac {A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 79, normalized size = 0.82 \[ \frac {\frac {(a B-3 A b) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} \left (-2 a A+a B x^3-3 A b x^3\right )}{x^{3/2} \left (a+b x^3\right )}}{3 a^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 232, normalized size = 2.42 \[ \left [\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{5} + {\left (B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{3} + 2 \, \sqrt {-a b} x^{\frac {3}{2}} - a}{b x^{3} + a}\right ) - 2 \, {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{3}\right )} \sqrt {x}}{6 \, {\left (a^{3} b^{2} x^{5} + a^{4} b x^{2}\right )}}, \frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{5} + {\left (B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{\frac {3}{2}}}{a}\right ) - {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{3}\right )} \sqrt {x}}{3 \, {\left (a^{3} b^{2} x^{5} + a^{4} b x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 0.69 \[ \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a^{2}} + \frac {B a x^{3} - 3 \, A b x^{3} - 2 \, A a}{3 \, {\left (b x^{\frac {9}{2}} + a x^{\frac {3}{2}}\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 93, normalized size = 0.97 \[ -\frac {A b \,x^{\frac {3}{2}}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {B \,x^{\frac {3}{2}}}{3 \left (b \,x^{3}+a \right ) a}-\frac {A b \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {B \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \sqrt {a b}\, a}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 67, normalized size = 0.70 \[ \frac {{\left (B a - 3 \, A b\right )} x^{3} - 2 \, A a}{3 \, {\left (a^{2} b x^{\frac {9}{2}} + a^{3} x^{\frac {3}{2}}\right )}} + \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 139, normalized size = 1.45 \[ -\frac {2\,A\,a^{3/2}\,\sqrt {b}-B\,a^2\,x^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )+3\,A\,b^2\,x^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )+3\,A\,\sqrt {a}\,b^{3/2}\,x^3-B\,a^{3/2}\,\sqrt {b}\,x^3+3\,A\,a\,b\,x^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )-B\,a\,b\,x^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )}{3\,a^{7/2}\,\sqrt {b}\,x^{3/2}+3\,a^{5/2}\,b^{3/2}\,x^{9/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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