Optimal. Leaf size=76 \[ \frac {8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 2014} \begin {gather*} \frac {4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{13/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac {x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {4 \int \frac {x^{11/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 \int \frac {x^{9/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x^{13/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {4 x^{11/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {8 x^{9/2}}{105 a^3 \left (a x+b x^3\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.72 \begin {gather*} \frac {x^{5/2} \sqrt {x \left (a+b x^2\right )} \left (35 a^2+28 a b x^2+8 b^2 x^4\right )}{105 a^3 \left (a+b x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.92, size = 46, normalized size = 0.61 \begin {gather*} \frac {x^{13/2} \left (35 a^2+28 a b x^2+8 b^2 x^4\right )}{105 a^3 \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 87, normalized size = 1.14 \begin {gather*} \frac {{\left (8 \, b^{2} x^{6} + 28 \, a b x^{4} + 35 \, a^{2} x^{2}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{105 \, {\left (a^{3} b^{4} x^{8} + 4 \, a^{4} b^{3} x^{6} + 6 \, a^{5} b^{2} x^{4} + 4 \, a^{6} b x^{2} + a^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 43, normalized size = 0.57 \begin {gather*} \frac {{\left (4 \, x^{2} {\left (\frac {2 \, b^{2} x^{2}}{a^{3}} + \frac {7 \, b}{a^{2}}\right )} + \frac {35}{a}\right )} x^{3}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 0.63 \begin {gather*} \frac {\left (b \,x^{2}+a \right ) \left (8 b^{2} x^{4}+28 a b \,x^{2}+35 a^{2}\right ) x^{\frac {15}{2}}}{105 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {13}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{13/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \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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