Optimal. Leaf size=110 \[ \frac {4 \left (32 a^3 x+39 a b\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{105 b^3 x^2}-\frac {4 \left (15 b-32 a^2 x\right ) \sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{105 b^3 x^3} \]
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Rubi [F] time = 2.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a^2 x}\right ) \int \frac {1}{\sqrt {x} \sqrt {-b+a^2 x} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx}{\sqrt {-b x+a^2 x^2}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a^2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a^2 x^2} \left (a x^4+x^2 \sqrt {-b x^2+a^2 x^4}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a^2 x^2}}\\ \end {align*}
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Mathematica [F] time = 1.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.31, size = 110, normalized size = 1.00 \begin {gather*} \frac {4 \left (39 a b+32 a^3 x\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{105 b^3 x^2}-\frac {4 \left (15 b-32 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{105 b^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 71, normalized size = 0.65 \begin {gather*} \frac {4 \, {\left (32 \, a^{3} x^{2} + 39 \, a b x + \sqrt {a^{2} x^{2} - b x} {\left (32 \, a^{2} x - 15 \, b\right )}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{105 \, b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]
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 {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{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}{\sqrt {a^2\,x^2-b\,x}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,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}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \sqrt {x \left (a^{2} x - b\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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