Optimal. Leaf size=273 \[ \frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a x+b x^3}}-\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {3 \sqrt {b} x \left (a+b x^2\right )}{a^2 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {1}{a \sqrt {a x+b x^3}} \]
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Rubi [A] time = 0.22, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2006, 2025, 2032, 329, 305, 220, 1196} \[ \frac {3 \sqrt {b} x \left (a+b x^2\right )}{a^2 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a x+b x^3}}-\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a x+b x^3}}+\frac {1}{a \sqrt {a x+b x^3}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2006
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{\left (a x+b x^3\right )^{3/2}} \, dx &=\frac {1}{a \sqrt {a x+b x^3}}+\frac {3 \int \frac {1}{x \sqrt {a x+b x^3}} \, dx}{2 a}\\ &=\frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {(3 b) \int \frac {x}{\sqrt {a x+b x^3}} \, dx}{2 a^2}\\ &=\frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {\left (3 b \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{2 a^2 \sqrt {a x+b x^3}}\\ &=\frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {\left (3 b \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a x+b x^3}}\\ &=\frac {1}{a \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}+\frac {\left (3 \sqrt {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 )}{a^{3/2} \sqrt {a x+b x^3}}-\frac {\left (3 \sqrt {b} \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {a x+b x^3}}\\ &=\frac {1}{a \sqrt {a x+b x^3}}+\frac {3 \sqrt {b} x \left (a+b x^2\right )}{a^2 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {3 \sqrt {a x+b x^3}}{a^2 x}-\frac {3 \sqrt [4]{b} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{7/4} \sqrt {a x+b x^3}}+\frac {3 \sqrt [4]{b} \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 )}{2 a^{7/4} \sqrt {a x+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 51, normalized size = 0.19 \[ -\frac {2 \sqrt {\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {b x^2}{a}\right )}{a \sqrt {x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x}}{b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 206, normalized size = 0.75 \[ -\frac {b \,x^{2}}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}\, a^{2}}-\frac {2 \left (b \,x^{2}+a \right )}{\sqrt {\left (b \,x^{2}+a \right ) x}\, a^{2}}+\frac {3 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 \sqrt {b \,x^{3}+a x}\, a^{2}} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 40, normalized size = 0.15 \[ -\frac {2\,x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{2};\ \frac {3}{4};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^3+a\,x\right )}^{3/2}} \]
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 \[ \int \frac {1}{\left (a x + b x^{3}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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