Optimal. Leaf size=145 \[ -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Rubi [A]
time = 0.03, antiderivative size = 145, 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^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^3 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (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^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^4} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 b^3}{x^4}+\frac {3 a^2 b^4}{x^3}+\frac {3 a b^5}{x^2}+\frac {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}}{3 x^3 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^3 \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 = 57, normalized size = 0.39 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a \left (2 a^2+9 a b x+18 b^2 x^2\right )-6 b^3 x^3 \log (x)\right )}{6 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 54, normalized size = 0.37
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} \left (6 b^{3} \ln \left (x \right ) x^{3}-18 a \,b^{2} x^{2}-9 a^{2} b x -2 a^{3}\right )}{6 x^{3} \left (b x +a \right )^{3}}\) | \(54\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-3 a \,b^{2} x^{2}-\frac {3}{2} a^{2} b x -\frac {1}{3} a^{3}\right )}{\left (b x +a \right ) x^{3}}+\frac {b^{3} \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs.
\(2 (97) = 194\).
time = 0.29, size = 229, normalized size = 1.58 \begin {gather*} \left (-1\right )^{2 \, b^{2} x + 2 \, a b} b^{3} \log \left (2 \, b^{2} x + 2 \, a b\right ) - \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4} x}{2 \, a^{2}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}}{2 \, a} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{3}}{6 \, a^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}}{2 \, a^{2} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{6 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 37, normalized size = 0.26 \begin {gather*} \frac {6 \, b^{3} x^{3} \log \left (x\right ) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \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^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.03, size = 59, normalized size = 0.41 \begin {gather*} b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {18 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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