Optimal. Leaf size=116 \[ \frac {2 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 b B x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 116, 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 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 b B x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{3/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{x^{3/2}} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{x^{3/2}}+\frac {b (A b+a B)}{\sqrt {x}}+b^2 B \sqrt {x}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 (A b+a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b B x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.41 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.04, size = 48, normalized size = 0.41 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-3 a A+3 a B x+3 A b x+b B x^2\right )}{3 \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.22 \begin {gather*} \frac {2 \, {\left (B b x^{2} - 3 \, A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, \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 = 0.46 \begin {gather*} \frac {2}{3} \, B b x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 2 \, A b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, A a \mathrm {sgn}\left (b x + a\right )}{\sqrt {x}} \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 = 0.38 \begin {gather*} -\frac {2 \left (-B b \,x^{2}-3 A b x -3 B a x +3 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (b x +a \right ) \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 33, normalized size = 0.28 \begin {gather*} \frac {2 \, {\left (b x^{2} + 3 \, a x\right )} B}{3 \, \sqrt {x}} + \frac {2 \, {\left (b x^{2} - a x\right )} A}{x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 53, normalized size = 0.46 \begin {gather*} \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^2}{3}-\frac {2\,A\,a}{b}+\frac {x\,\left (6\,A\,b+6\,B\,a\right )}{3\,b}\right )}{x^{3/2}+\frac {a\,\sqrt {x}}{b}} \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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