Optimal. Leaf size=120 \[ \frac {2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \]
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Rubi [A] time = 0.05, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} \frac {2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int x^{7/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^{7/2} \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b x^{7/2}+b (A b+a B) x^{9/2}+b^2 B x^{11/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 a A x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.42 \begin {gather*} \frac {2 x^{9/2} \sqrt {(a+b x)^2} (13 a (11 A+9 B x)+9 b x (13 A+11 B x))}{1287 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 9.97, size = 59, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (143 a A x^{9/2}+117 a B x^{11/2}+117 A b x^{11/2}+99 b B x^{13/2}\right )}{1287 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 32, normalized size = 0.27 \begin {gather*} \frac {2}{1287} \, {\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \, {\left (B a + A b\right )} x^{5}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 53, normalized size = 0.44 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, B a x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, A b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A a x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.37 \begin {gather*} \frac {2 \left (99 B b \,x^{2}+117 A b x +117 B a x +143 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {9}{2}}}{1287 \left (b x +a \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 35, normalized size = 0.29 \begin {gather*} \frac {2}{143} \, {\left (11 \, b x^{2} + 13 \, a x\right )} B x^{\frac {9}{2}} + \frac {2}{99} \, {\left (9 \, b x^{2} + 11 \, a x\right )} A x^{\frac {7}{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 = -0.01 \begin {gather*} \int x^{7/2}\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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