Optimal. Leaf size=138 \[ \frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{9/2}}-\frac {35 b^2 \sqrt {a x^2+b x^3}}{8 a^4 x^2}+\frac {35 b \sqrt {a x^2+b x^3}}{12 a^3 x^3}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}+\frac {2}{a x^2 \sqrt {a x^2+b x^3}} \]
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Rubi [A] time = 0.19, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2023, 2025, 2008, 206} \begin {gather*} -\frac {35 b^2 \sqrt {a x^2+b x^3}}{8 a^4 x^2}+\frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{9/2}}+\frac {35 b \sqrt {a x^2+b x^3}}{12 a^3 x^3}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}+\frac {2}{a x^2 \sqrt {a x^2+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2023
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac {2}{a x^2 \sqrt {a x^2+b x^3}}+\frac {7 \int \frac {1}{x^3 \sqrt {a x^2+b x^3}} \, dx}{a}\\ &=\frac {2}{a x^2 \sqrt {a x^2+b x^3}}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}-\frac {(35 b) \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx}{6 a^2}\\ &=\frac {2}{a x^2 \sqrt {a x^2+b x^3}}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}+\frac {35 b \sqrt {a x^2+b x^3}}{12 a^3 x^3}+\frac {\left (35 b^2\right ) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{8 a^3}\\ &=\frac {2}{a x^2 \sqrt {a x^2+b x^3}}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}+\frac {35 b \sqrt {a x^2+b x^3}}{12 a^3 x^3}-\frac {35 b^2 \sqrt {a x^2+b x^3}}{8 a^4 x^2}-\frac {\left (35 b^3\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{16 a^4}\\ &=\frac {2}{a x^2 \sqrt {a x^2+b x^3}}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}+\frac {35 b \sqrt {a x^2+b x^3}}{12 a^3 x^3}-\frac {35 b^2 \sqrt {a x^2+b x^3}}{8 a^4 x^2}+\frac {\left (35 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{8 a^4}\\ &=\frac {2}{a x^2 \sqrt {a x^2+b x^3}}-\frac {7 \sqrt {a x^2+b x^3}}{3 a^2 x^4}+\frac {35 b \sqrt {a x^2+b x^3}}{12 a^3 x^3}-\frac {35 b^2 \sqrt {a x^2+b x^3}}{8 a^4 x^2}+\frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{8 a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.28 \begin {gather*} -\frac {2 b^3 x \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {b x}{a}+1\right )}{a^4 \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.04, size = 107, normalized size = 0.78 \begin {gather*} \frac {x \sqrt {a+b x} \left (\frac {35 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {48 a^3-231 a^2 (a+b x)+280 a (a+b x)^2-105 (a+b x)^3}{24 a^4 x^3 \sqrt {a+b x}}\right )}{\sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 241, normalized size = 1.75 \begin {gather*} \left [\frac {105 \, {\left (b^{4} x^{5} + a b^{3} x^{4}\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 (105 \, a b^{3} x^{3} + 35 \, a^{2} b^{2} x^{2} - 14 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{48 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}, -\frac {105 \, {\left (b^{4} x^{5} + a b^{3} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (105 \, a b^{3} x^{3} + 35 \, a^{2} b^{2} x^{2} - 14 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{24 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}\right ] \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}{{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 86, normalized size = 0.62 \begin {gather*} -\frac {\left (b x +a \right ) \left (-105 \sqrt {b x +a}\, b^{3} x^{3} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+105 \sqrt {a}\, b^{3} x^{3}+35 a^{\frac {3}{2}} b^{2} x^{2}-14 a^{\frac {5}{2}} b x +8 a^{\frac {7}{2}}\right )}{24 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {9}{2}}} \end {gather*}
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}{{\left (b x^{3} + a x^{2}\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\,{\left (b\,x^3+a\,x^2\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^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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