Optimal. Leaf size=142 \[ \frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]
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Rubi [A] time = 0.03, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \begin {gather*} -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (3 a b^5+\frac {a^3 b^3}{x^2}+\frac {3 a^2 b^4}{x}+b^6 x\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 0.39 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-2 a^3+6 a^2 b x \log (x)+6 a b^2 x^2+b^3 x^3\right )}{2 x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.66, size = 292, normalized size = 2.06 \begin {gather*} -\frac {3}{2} a^2 \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-\frac {3}{2} a^2 \sqrt {b^2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )+3 a^2 b \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 a b x+b^2 x^2}}{a}\right )+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b-21 a^2 b^2 x+24 a b^3 x^2+4 b^4 x^3\right )+\sqrt {b^2} \left (8 a^4+29 a^3 b x-3 a^2 b^2 x^2-28 a b^3 x^3-4 b^4 x^4\right )}{8 x \left (a b+b^2 x\right )-8 \sqrt {b^2} x \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 36, normalized size = 0.25 \begin {gather*} \frac {b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \relax (x) - 2 \, a^{3}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 57, normalized size = 0.40 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{2} x \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {a^{3} \mathrm {sgn}\left (b x + a\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 53, normalized size = 0.37 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} x^{3}+6 a^{2} b x \ln \relax (x )+6 a \,b^{2} x^{2}-2 a^{3}\right )}{2 \left (b x +a \right )^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 140, normalized size = 0.99 \begin {gather*} 3 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a^{2} b \log \left (2 \, b^{2} x + 2 \, a b\right ) - 3 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a^{2} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2} x + \frac {9}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a b - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{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 {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x^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 {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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