Optimal. Leaf size=222 \[ \frac {b^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {5 a b^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {10 a^2 b^3 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]
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Rubi [A] time = 0.05, antiderivative size = 222, 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^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {5 a b^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {10 a^2 b^3 \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 )^{5/2}}{x^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^4} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (5 a b^9+\frac {a^5 b^5}{x^4}+\frac {5 a^4 b^6}{x^3}+\frac {10 a^3 b^7}{x^2}+\frac {10 a^2 b^8}{x}+b^{10} x\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac {10 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {5 a b^4 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {b^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac {10 a^2 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 = 0.02, size = 79, normalized size = 0.36 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-2 a^5-15 a^4 b x-60 a^3 b^2 x^2+60 a^2 b^3 x^3 \log (x)+30 a b^4 x^4+3 b^5 x^5\right )}{6 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.82, size = 338, normalized size = 1.52 \begin {gather*} -5 a^2 \left (b^2\right )^{3/2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )-5 a^2 \left (b^2\right )^{3/2} \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )+10 a^2 b^3 \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^5 b-60 a^4 b^2 x-240 a^3 b^3 x^2-391 a^2 b^4 x^3+120 a b^5 x^4+12 b^6 x^5\right )+\sqrt {b^2} \left (8 a^6+68 a^5 b x+300 a^4 b^2 x^2+631 a^3 b^3 x^3+271 a^2 b^4 x^4-132 a b^5 x^5-12 b^6 x^6\right )}{24 x^3 \left (a b+b^2 x\right )-24 \sqrt {b^2} x^3 \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 = 59, normalized size = 0.27 \begin {gather*} \frac {3 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} \log \relax (x) - 60 \, a^{3} b^{2} x^{2} - 15 \, a^{4} b x - 2 \, a^{5}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 92, normalized size = 0.41 \begin {gather*} \frac {1}{2} \, b^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} x \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {60 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{5} \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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maple [A] time = 0.06, size = 76, normalized size = 0.34 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (3 b^{5} x^{5}+60 a^{2} b^{3} x^{3} \ln \relax (x )+30 a \,b^{4} x^{4}-60 a^{3} b^{2} x^{2}-15 a^{4} b x -2 a^{5}\right )}{6 \left (b x +a \right )^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 284, normalized size = 1.28 \begin {gather*} 10 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a^{2} b^{3} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 10 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a^{2} b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + 5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4} x + 15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a b^{3} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4} x}{2 \, a^{2}} + \frac {35 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{3}}{6 \, a} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{6 \, a^{3}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{6 \, a^{2} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{6 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{3 \, a^{2} 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 = -0.00 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^4} \,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 {5}{2}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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