Optimal. Leaf size=27 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {629} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int (a+b 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}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.93 \begin {gather*} \frac {(a+b x)^6 \sqrt {(a+b x)^2}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 18, normalized size = 0.67 \begin {gather*} \frac {\left ((a+b x)^2\right )^{7/2}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 64, normalized size = 2.37 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 120, normalized size = 4.44 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} \mathrm {sgn}\left (b x + a\right ) + a b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} x \mathrm {sgn}\left (b x + a\right ) + \frac {a^{7} \mathrm {sgn}\left (b x + a\right )}{7 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 82, normalized size = 3.04 \begin {gather*} \frac {\left (b^{6} x^{6}+7 a \,b^{5} x^{5}+21 a^{2} b^{4} x^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} b^{2} x^{2}+21 a^{5} b x +7 a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{7 \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 = 0.68, size = 23, normalized size = 0.85 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.24, size = 14, normalized size = 0.52 \begin {gather*} \frac {{\left ({\left (a+b\,x\right )}^2\right )}^{7/2}}{7\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.52, size = 226, normalized size = 8.37 \begin {gather*} \begin {cases} \frac {a^{6} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7 b} + \frac {6 a^{5} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {15 a^{4} b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {20 a^{3} b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {15 a^{2} b^{3} x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {6 a b^{4} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} + \frac {b^{5} x^{6} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}{7} & \text {for}\: b \neq 0 \\a x \left (a^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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