Optimal. Leaf size=205 \[ -\frac {10 a^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 a^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, 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 = {646, 43} \[ \frac {a^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 a^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 a^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
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 = 0.03, size = 93, normalized size = 0.45 \[ \frac {-77 a^5-248 a^4 b x-252 a^3 b^2 x^2-48 a^2 b^3 x^3+48 a b^4 x^4-60 a (a+b x)^4 \log (a+b x)+12 b^5 x^5}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 149, normalized size = 0.73 \[ \frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5} - 60 \, {\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 145, normalized size = 0.71 \[ -\frac {\left (60 a \,b^{4} x^{4} \ln \left (b x +a \right )-12 b^{5} x^{5}+240 a^{2} b^{3} x^{3} \ln \left (b x +a \right )-48 a \,b^{4} x^{4}+360 a^{3} b^{2} x^{2} \ln \left (b x +a \right )+48 a^{2} b^{3} x^{3}+240 a^{4} b x \ln \left (b x +a \right )+252 a^{3} b^{2} x^{2}+60 a^{5} \ln \left (b x +a \right )+248 a^{4} b x +77 a^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 113, normalized size = 0.55 \[ \frac {12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} - \frac {5 \, a \log \left (b x + a\right )}{b^{6}} \]
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 \[ \int \frac {x^5}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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