Optimal. Leaf size=110 \[ -\frac {15 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{7/2}}+\frac {15 b \sqrt {a x^2+b x^3}}{4 a^3 x^2}-\frac {5 \sqrt {a x^2+b x^3}}{2 a^2 x^3}+\frac {2}{a x \sqrt {a x^2+b x^3}} \]
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Rubi [A] time = 0.11, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2006, 2025, 2008, 206} \[ -\frac {15 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{7/2}}+\frac {15 b \sqrt {a x^2+b x^3}}{4 a^3 x^2}-\frac {5 \sqrt {a x^2+b x^3}}{2 a^2 x^3}+\frac {2}{a x \sqrt {a x^2+b x^3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2006
Rule 2008
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac {2}{a x \sqrt {a x^2+b x^3}}+\frac {5 \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx}{a}\\ &=\frac {2}{a x \sqrt {a x^2+b x^3}}-\frac {5 \sqrt {a x^2+b x^3}}{2 a^2 x^3}-\frac {(15 b) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{4 a^2}\\ &=\frac {2}{a x \sqrt {a x^2+b x^3}}-\frac {5 \sqrt {a x^2+b x^3}}{2 a^2 x^3}+\frac {15 b \sqrt {a x^2+b x^3}}{4 a^3 x^2}+\frac {\left (15 b^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{8 a^3}\\ &=\frac {2}{a x \sqrt {a x^2+b x^3}}-\frac {5 \sqrt {a x^2+b x^3}}{2 a^2 x^3}+\frac {15 b \sqrt {a x^2+b x^3}}{4 a^3 x^2}-\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{4 a^3}\\ &=\frac {2}{a x \sqrt {a x^2+b x^3}}-\frac {5 \sqrt {a x^2+b x^3}}{2 a^2 x^3}+\frac {15 b \sqrt {a x^2+b x^3}}{4 a^3 x^2}-\frac {15 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 38, normalized size = 0.35 \[ \frac {2 b^2 x \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {b x}{a}+1\right )}{a^3 \sqrt {x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 219, normalized size = 1.99 \[ \left [\frac {15 \, {\left (b^{3} x^{4} + a b^{2} x^{3}\right )} \sqrt {a} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{8 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}, \frac {15 \, {\left (b^{3} x^{4} + a b^{2} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{4 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}\right ] \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 76, normalized size = 0.69 \[ -\frac {\left (b x +a \right ) \left (15 \sqrt {b x +a}\, b^{2} x^{2} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-15 \sqrt {a}\, b^{2} x^{2}-5 a^{\frac {3}{2}} b x +2 a^{\frac {5}{2}}\right ) x}{4 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {7}{2}}} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.43, size = 42, normalized size = 0.38 \[ -\frac {2\,x\,{\left (\frac {a}{b\,x}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {7}{2};\ \frac {9}{2};\ -\frac {a}{b\,x}\right )}{7\,{\left (b\,x^3+a\,x^2\right )}^{3/2}} \]
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 \[ \int \frac {1}{\left (a x^{2} + b x^{3}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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