Optimal. Leaf size=120 \[ \frac {2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac {2 a A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b B x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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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^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac {2 a A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b B x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \sqrt {x} (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 \sqrt {x} \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 \sqrt {x}+b (A b+a B) x^{3/2}+b^2 B x^{5/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 a A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 (A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b B x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (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^{3/2} \sqrt {(a+b x)^2} (7 a (5 A+3 B x)+3 b x (7 A+5 B x))}{105 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.83, size = 59, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (35 a A x^{3/2}+21 a B x^{5/2}+21 A b x^{5/2}+15 b B x^{7/2}\right )}{105 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 30, normalized size = 0.25 \begin {gather*} \frac {2}{105} \, {\left (15 \, B b x^{3} + 35 \, A a x + 21 \, {\left (B a + A b\right )} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.44 \begin {gather*} \frac {2}{7} \, B b x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, B a x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, A b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A a x^{\frac {3}{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 (15 B b \,x^{2}+21 A b x +21 B a x +35 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {3}{2}}}{105 \left (b x +a \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 35, normalized size = 0.29 \begin {gather*} \frac {2}{35} \, {\left (5 \, b x^{2} + 7 \, a x\right )} B x^{\frac {3}{2}} + \frac {2}{15} \, {\left (3 \, b x^{2} + 5 \, a x\right )} A \sqrt {x} \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 \sqrt {x}\,\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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