Optimal. Leaf size=205 \[ -\frac {10 a^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 a^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 205, 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 {10 a^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 a^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {x^5}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{b^{10}}-\frac {a^5}{b^{10} (a+b x)^5}+\frac {5 a^4}{b^{10} (a+b x)^4}-\frac {10 a^3}{b^{10} (a+b x)^3}+\frac {10 a^2}{b^{10} (a+b x)^2}-\frac {5 a}{b^{10} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {10 a^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 a^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 93, normalized size = 0.45 \begin {gather*} \frac {-77 a^5-248 a^4 b x-252 a^3 b^2 x^2-48 a^2 b^3 x^3+48 a b^4 x^4+12 b^5 x^5-60 a (a+b x)^4 \log (a+b x)}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 145, normalized size = 0.71
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, x}{\left (b x +a \right ) b^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-10 a^{2} b^{2} x^{3}-25 a^{3} b \,x^{2}-\frac {65 a^{4} x}{3}-\frac {77 a^{5}}{12 b}\right )}{\left (b x +a \right )^{5} b^{5}}-\frac {5 \sqrt {\left (b x +a \right )^{2}}\, a \ln \left (b x +a \right )}{\left (b x +a \right ) b^{6}}\) | \(106\) |
default | \(-\frac {\left (60 \ln \left (b x +a \right ) a \,b^{4} x^{4}-12 b^{5} x^{5}+240 \ln \left (b x +a \right ) a^{2} b^{3} x^{3}-48 a \,b^{4} x^{4}+360 \ln \left (b x +a \right ) a^{3} b^{2} x^{2}+48 a^{2} b^{3} x^{3}+240 \ln \left (b x +a \right ) a^{4} b x +252 a^{3} x^{2} b^{2}+60 \ln \left (b x +a \right ) a^{5}+248 a^{4} b x +77 a^{5}\right ) \left (b x +a \right )}{12 b^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 113, normalized size = 0.55 \begin {gather*} \frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} - \frac {5 \, a \log \left (b x + a\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.28, size = 149, normalized size = 0.73 \begin {gather*} \frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5} - 60 \, {\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \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 {x^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 90, normalized size = 0.44 \begin {gather*} \frac {x}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {5 \, a \log \left ({\left | b x + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {120 \, a^{2} b^{3} x^{3} + 300 \, a^{3} b^{2} x^{2} + 260 \, a^{4} b x + 77 \, a^{5}}{12 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} \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 {x^5}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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