Optimal. Leaf size=219 \[ \frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)} \]
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Rubi [A] time = 0.05, antiderivative size = 219, 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} \[ -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x} \]
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^5} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^5} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^{10}+\frac {a^5 b^5}{x^5}+\frac {5 a^4 b^6}{x^4}+\frac {10 a^3 b^7}{x^3}+\frac {10 a^2 b^8}{x^2}+\frac {5 a b^9}{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}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \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 \[ -\frac {\sqrt {(a+b x)^2} \left (3 a^5+20 a^4 b x+60 a^3 b^2 x^2+120 a^2 b^3 x^3-60 a b^4 x^4 \log (x)-12 b^5 x^5\right )}{12 x^4 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 59, normalized size = 0.27 \[ \frac {12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \relax (x) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 91, normalized size = 0.42 \[ b^{5} x \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {120 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 60 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 76, normalized size = 0.35 \[ \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 a \,b^{4} x^{4} \ln \relax (x )+12 b^{5} x^{5}-120 a^{2} b^{3} x^{3}-60 a^{3} b^{2} x^{2}-20 a^{4} b x -3 a^{5}\right )}{12 \left (b x +a \right )^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.56, size = 311, normalized size = 1.42 \[ 5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a b^{4} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5} x}{2 \, a} + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5} x}{4 \, a^{3}} + \frac {35 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{12 \, a^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{4}}{3 \, a^{4}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{3 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{3 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{4 \, a^{2} x^{4}} \]
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 \[ \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^5} \,d x \]
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 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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