Optimal. Leaf size=86 \[ -\frac {16 b^2 \sqrt {a x^2+b x^3}}{15 a^3 x^{3/2}}+\frac {8 b \sqrt {a x^2+b x^3}}{15 a^2 x^{5/2}}-\frac {2 \sqrt {a x^2+b x^3}}{5 a x^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \begin {gather*} -\frac {16 b^2 \sqrt {a x^2+b x^3}}{15 a^3 x^{3/2}}+\frac {8 b \sqrt {a x^2+b x^3}}{15 a^2 x^{5/2}}-\frac {2 \sqrt {a x^2+b x^3}}{5 a x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \sqrt {a x^2+b x^3}} \, dx &=-\frac {2 \sqrt {a x^2+b x^3}}{5 a x^{7/2}}-\frac {(4 b) \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^3}} \, dx}{5 a}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{5 a x^{7/2}}+\frac {8 b \sqrt {a x^2+b x^3}}{15 a^2 x^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx}{15 a^2}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{5 a x^{7/2}}+\frac {8 b \sqrt {a x^2+b x^3}}{15 a^2 x^{5/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{15 a^3 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.51 \begin {gather*} -\frac {2 \sqrt {x^2 (a+b x)} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 46, normalized size = 0.53 \begin {gather*} -\frac {2 \left (3 a^2-4 a b x+8 b^2 x^2\right ) \sqrt {a x^2+b x^3}}{15 a^3 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 40, normalized size = 0.47 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )} \sqrt {b x^{3} + a x^{2}}}{15 \, a^{3} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 77, normalized size = 0.90 \begin {gather*} \frac {32 \, {\left (10 \, {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{4} - 5 \, a {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} + a^{2}\right )} b^{\frac {5}{2}}}{15 \, {\left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.53 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}-4 a b x +3 a^{2}\right )}{15 \sqrt {b \,x^{3}+a \,x^{2}}\, a^{3} x^{\frac {3}{2}}} \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 {1}{\sqrt {b x^{3} + a x^{2}} x^{\frac {5}{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 {1}{x^{5/2}\,\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {5}{2}} \sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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