Optimal. Leaf size=132 \[ \frac {4 \left (256 a^5 x^2+32 a^3 b x+245 a b^2\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{1155 b^4 x^3}-\frac {4 \sqrt {a^2 x^2-b x} \left (-256 a^4 x^2-160 a^2 b x+105 b^2\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{1155 b^4 x^4} \]
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Rubi [F] time = 4.78, 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}{x \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}{x \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}{x^{3/2} \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}{x^2 \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 = 1.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \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 = 5.26, size = 132, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {-b x+a^2 x^2} \left (105 b^2-160 a^2 b x-256 a^4 x^2\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{1155 b^4 x^4}+\frac {4 \left (245 a b^2+32 a^3 b x+256 a^5 x^2\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{1155 b^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 93, normalized size = 0.70 \begin {gather*} \frac {4 \, {\left (256 \, a^{5} x^{3} + 32 \, a^{3} b x^{2} + 245 \, a b^{2} x + {\left (256 \, a^{4} x^{2} + 160 \, a^{2} b x - 105 \, b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{1155 \, b^{4} x^{4}} \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}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]
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}} x}\,{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\,\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}{x \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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