Optimal. Leaf size=119 \[ -\frac {b^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{3 a x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2025, 2011, 329, 220} \[ -\frac {b^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{3 a x^2} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a x+b x^3}} \, dx &=-\frac {2 \sqrt {a x+b x^3}}{3 a x^2}-\frac {b \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a x+b x^3}}{3 a x^2}-\frac {\left (b \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{3 a \sqrt {a x+b x^3}}\\ &=-\frac {2 \sqrt {a x+b x^3}}{3 a x^2}-\frac {\left (2 b \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{3 a \sqrt {a x+b x^3}}\\ &=-\frac {2 \sqrt {a x+b x^3}}{3 a x^2}-\frac {b^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 53, normalized size = 0.45 \[ -\frac {2 \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};-\frac {b x^2}{a}\right )}{3 x \sqrt {x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x}}{b x^{5} + a x^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {b x^{3} + a x} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 129, normalized size = 1.08 \[ -\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {b \,x^{3}+a x}\, a}-\frac {2 \sqrt {b \,x^{3}+a x}}{3 a \,x^{2}} \]
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 \[ \int \frac {1}{\sqrt {b x^{3} + a x} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\sqrt {b\,x^3+a\,x}} \,d x \]
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 \[ \int \frac {1}{x^{2} \sqrt {x \left (a + b x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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