Optimal. Leaf size=107 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3}-\frac {2 a \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3}+\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 107, 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} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3}-\frac {2 a \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3}+\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (a b+b^2 x\right )^5 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^2 \left (a b+b^2 x\right )^5}{b^2}-\frac {2 a \left (a b+b^2 x\right )^6}{b^3}+\frac {\left (a b+b^2 x\right )^7}{b^4}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {a^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}-\frac {2 a (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {(a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 0.72 \begin {gather*} \frac {x^3 \sqrt {(a+b x)^2} \left (56 a^5+210 a^4 b x+336 a^3 b^2 x^2+280 a^2 b^3 x^3+120 a b^4 x^4+21 b^5 x^5\right )}{168 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.71, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 57, normalized size = 0.53 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} + \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{4} \, a^{4} b x^{4} + \frac {1}{3} \, a^{5} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 107, normalized size = 1.00 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, a b^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{4} b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{8} \mathrm {sgn}\left (b x + a\right )}{168 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 0.69 \begin {gather*} \frac {\left (21 b^{5} x^{5}+120 a \,b^{4} x^{4}+280 a^{2} b^{3} x^{3}+336 a^{3} b^{2} x^{2}+210 a^{4} b x +56 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{3}}{168 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 102, normalized size = 0.95 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3}}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x}{8 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} \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 = -0.01 \begin {gather*} \int x^2\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \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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