Optimal. Leaf size=220 \[ -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b^5 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
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Rubi [A]
time = 0.04, antiderivative size = 220, 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 {b^5 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {5 a^2 b^3 x^2 \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}}{x (a+b x)}+\frac {5 a^4 b \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {10 a^3 b^2 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 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (10 a^3 b^7+\frac {a^5 b^5}{x^2}+\frac {5 a^4 b^6}{x}+10 a^2 b^8 x+5 a b^9 x^2+b^{10} x^3\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}}{x (a+b x)}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a^2 b^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b^5 x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {5 a^4 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 = 79, normalized size = 0.36 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-12 a^5+120 a^3 b^2 x^2+60 a^2 b^3 x^3+20 a b^4 x^4+3 b^5 x^5+60 a^4 b x \log (x)\right )}{12 x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 76, normalized size = 0.35
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (3 b^{5} x^{5}+20 a \,b^{4} x^{4}+60 a^{2} b^{3} x^{3}+60 a^{4} b \ln \left (x \right ) x +120 a^{3} x^{2} b^{2}-12 a^{5}\right )}{12 x \left (b x +a \right )^{5}}\) | \(76\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \left (\frac {1}{4} b^{3} x^{4}+\frac {5}{3} a \,b^{2} x^{3}+5 a^{2} b \,x^{2}+10 a^{3} x \right )}{b x +a}-\frac {a^{5} \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {5 a^{4} b \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 191, normalized size = 0.87 \begin {gather*} 5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a^{4} b \log \left (2 \, b^{2} x + 2 \, a b\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a^{4} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {5}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} b^{2} x + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} b + \frac {5}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2} x + \frac {35}{12} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a b - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.75, size = 59, normalized size = 0.27 \begin {gather*} \frac {3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, 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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.91, size = 91, normalized size = 0.41 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a b^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} x \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {a^{5} \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 = -0.00 \begin {gather*} \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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