Optimal. Leaf size=171 \[ -\frac {3 a^2}{b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 a}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4}{4 b^5 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 a^3}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 171, 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} \[ -\frac {a^4}{4 b^5 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 a^3}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a^2}{b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 a}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {x^4}{\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^4}{\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 {a^4}{b^9 (a+b x)^5}-\frac {4 a^3}{b^9 (a+b x)^4}+\frac {6 a^2}{b^9 (a+b x)^3}-\frac {4 a}{b^9 (a+b x)^2}+\frac {1}{b^9 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {4 a}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^4}{4 b^5 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {4 a^3}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a^2}{b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 0.43 \[ \frac {a \left (25 a^3+88 a^2 b x+108 a b^2 x^2+48 b^3 x^3\right )+12 (a+b x)^4 \log (a+b x)}{12 b^5 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 127, normalized size = 0.74 \[ \frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4} + 12 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} \]
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 = 123, normalized size = 0.72 \[ \frac {\left (12 b^{4} x^{4} \ln \left (b x +a \right )+48 a \,b^{3} x^{3} \ln \left (b x +a \right )+72 a^{2} b^{2} x^{2} \ln \left (b x +a \right )+48 a \,b^{3} x^{3}+48 a^{3} b x \ln \left (b x +a \right )+108 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )+88 a^{3} b x +25 a^{4}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 92, normalized size = 0.54 \[ \frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}} + \frac {\log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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