Optimal. Leaf size=120 \[ \frac {2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 a A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
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Rubi [A] time = 0.04, 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} \[ \frac {2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 a A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int x^{3/2} (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^{3/2} \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^{3/2}+b (A b+a B) x^{5/2}+b^2 B x^{7/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 a A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 (A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.42 \[ \frac {2 x^{5/2} \sqrt {(a+b x)^2} (9 a (7 A+5 B x)+5 b x (9 A+7 B x))}{315 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 32, normalized size = 0.27 \[ \frac {2}{315} \, {\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \, {\left (B a + A b\right )} x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 53, normalized size = 0.44 \[ \frac {2}{9} \, B b x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, B a x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, A a x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) \]
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 \[ \frac {2 \left (35 B b \,x^{2}+45 A b x +45 B a x +63 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {5}{2}}}{315 \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 35, normalized size = 0.29 \[ \frac {2}{63} \, {\left (7 \, b x^{2} + 9 \, a x\right )} B x^{\frac {5}{2}} + \frac {2}{35} \, {\left (5 \, b x^{2} + 7 \, a x\right )} A x^{\frac {3}{2}} \]
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 \[ \int x^{3/2}\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right ) \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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