Optimal. Leaf size=61 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}-\frac {a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {640, 609} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}-\frac {a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 640
Rubi steps
\begin {align*} \int x \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}-\frac {a \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx}{b}\\ &=-\frac {a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 77, normalized size = 1.26 \begin {gather*} \frac {x^2 \sqrt {(a+b x)^2} \left (21 a^5+70 a^4 b x+105 a^3 b^2 x^2+84 a^2 b^3 x^3+35 a b^4 x^4+6 b^5 x^5\right )}{42 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int x \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 = 0.39, size = 57, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, b^{5} x^{7} + \frac {5}{6} \, a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{5} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{3} \, a^{4} b x^{3} + \frac {1}{2} \, a^{5} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 107, normalized size = 1.75 \begin {gather*} \frac {1}{7} \, b^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a b^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{4} b x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{7} \mathrm {sgn}\left (b x + a\right )}{42 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 74, normalized size = 1.21 \begin {gather*} \frac {\left (6 b^{5} x^{5}+35 a \,b^{4} x^{4}+84 a^{2} b^{3} x^{3}+105 a^{3} b^{2} x^{2}+70 a^{4} b x +21 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{2}}{42 \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 = 1.34, size = 75, normalized size = 1.23 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b^{2}} \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 = -0.02 \begin {gather*} \int x\,{\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 \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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