Optimal. Leaf size=182 \[ \frac {x^4 (a+b x)}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^3 (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 182, 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} \[ -\frac {a^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^3 (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^4 (a+b x)}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {x^4}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (-\frac {a^3}{b^5}+\frac {a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{b^2}+\frac {a^4}{b^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a^3 x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a x^3 (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x^4 (a+b x)}{4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (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 = 0.02, size = 68, normalized size = 0.37 \[ \frac {(a+b x) \left (12 a^4 \log (a+b x)+b x \left (-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right )\right )}{12 b^5 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 52, normalized size = 0.29 \[ \frac {3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 83, normalized size = 0.46 \[ \frac {a^{4} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{3} x \mathrm {sgn}\left (b x + a\right )}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 67, normalized size = 0.37 \[ \frac {\left (b x +a \right ) \left (3 b^{4} x^{4}-4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-12 a^{3} b x \right )}{12 \sqrt {\left (b x +a \right )^{2}}\, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 115, normalized size = 0.63 \[ \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{3}}{4 \, b^{2}} + \frac {13 \, a^{2} x^{2}}{12 \, b^{3}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x^{2}}{12 \, b^{3}} - \frac {13 \, a^{3} x}{6 \, b^{4}} + \frac {a^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{6 \, b^{5}} \]
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 \[ \int \frac {x^4}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 49, normalized size = 0.27 \[ \frac {a^{4} \log {\left (a + b x \right )}}{b^{5}} - \frac {a^{3} x}{b^{4}} + \frac {a^{2} x^{2}}{2 b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {x^{4}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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