Optimal. Leaf size=57 \[ \frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3 b}{a^2 \sqrt {a+b x}}-\frac {1}{a x \sqrt {a+b x}} \]
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Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 208} \begin {gather*} -\frac {3 \sqrt {a+b x}}{a^2 x}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{a x \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{3/2}} \, dx &=\frac {2}{a x \sqrt {a+b x}}+\frac {3 \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{a}\\ &=\frac {2}{a x \sqrt {a+b x}}-\frac {3 \sqrt {a+b x}}{a^2 x}-\frac {(3 b) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a^2}\\ &=\frac {2}{a x \sqrt {a+b x}}-\frac {3 \sqrt {a+b x}}{a^2 x}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^2}\\ &=\frac {2}{a x \sqrt {a+b x}}-\frac {3 \sqrt {a+b x}}{a^2 x}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.54 \begin {gather*} -\frac {2 b \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x}{a}+1\right )}{a^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 52, normalized size = 0.91 \begin {gather*} \frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 a-3 (a+b x)}{a^2 x \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 151, normalized size = 2.65 \begin {gather*} \left [\frac {3 \, {\left (b^{2} x^{2} + a b x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (3 \, a b x + a^{2}\right )} \sqrt {b x + a}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {3 \, {\left (b^{2} x^{2} + a b x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x + a^{2}\right )} \sqrt {b x + a}}{a^{3} b x^{2} + a^{4} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 64, normalized size = 1.12 \begin {gather*} -\frac {3 \, b \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {3 \, {\left (b x + a\right )} b - 2 \, a b}{{\left ({\left (b x + a\right )}^{\frac {3}{2}} - \sqrt {b x + a} a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.96 \begin {gather*} 2 \left (-\frac {1}{\sqrt {b x +a}\, a^{2}}-\frac {-\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {\sqrt {b x +a}}{2 b x}}{a^{2}}\right ) b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 76, normalized size = 1.33 \begin {gather*} -\frac {3 \, {\left (b x + a\right )} b - 2 \, a b}{{\left (b x + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {b x + a} a^{3}} - \frac {3 \, b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2 \, a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 60, normalized size = 1.05 \begin {gather*} \frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,b}{a}-\frac {3\,b\,\left (a+b\,x\right )}{a^2}}{a\,\sqrt {a+b\,x}-{\left (a+b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.41, size = 73, normalized size = 1.28 \begin {gather*} - \frac {1}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 \sqrt {b}}{a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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