Optimal. Leaf size=142 \[ -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {3 a b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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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 = {660, 45}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
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 = 0.01, size = 56, normalized size = 0.39 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-2 a^3+6 a b^2 x^2+b^3 x^3+6 a^2 b x \log (x)\right )}{2 x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 53, normalized size = 0.37
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (b^{3} x^{3}+6 a^{2} b \ln \left (x \right ) x +6 a \,b^{2} x^{2}-2 a^{3}\right )}{2 x \left (b x +a \right )^{3}}\) | \(53\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (\frac {1}{2} b \,x^{2}+3 a x \right )}{b x +a}-\frac {a^{3} \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {3 a^{2} b \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, 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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Fricas [A]
time = 1.03, 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 \left (x\right ) - 2 \, a^{3}}{2 \, 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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Giac [A]
time = 2.21, 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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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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