Optimal. Leaf size=190 \[ \frac {2 x^{3/2} (a+b x) (A b-a B)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a \sqrt {x} (a+b x) (A b-a B)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \begin {gather*} \frac {2 x^{3/2} (a+b x) (A b-a B)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a \sqrt {x} (a+b x) (A b-a B)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {x^{3/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {5 A b^2}{2}-\frac {5 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{a b+b^2 x} \, dx}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 a \left (\frac {5 A b^2}{2}-\frac {5 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a (A b-a B) \sqrt {x} (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 a^2 \left (\frac {5 A b^2}{2}-\frac {5 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{5 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a (A b-a B) \sqrt {x} (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (4 a^2 \left (\frac {5 A b^2}{2}-\frac {5 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{5 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a (A b-a B) \sqrt {x} (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{5/2} (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^{3/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 0.53 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {x} \left (15 a^2 B-5 a b (3 A+B x)+b^2 x (5 A+3 B x)\right )-15 a^{3/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{15 b^{7/2} \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 10.36, size = 105, normalized size = 0.55 \begin {gather*} \frac {(a+b x) \left (\frac {2 \sqrt {x} \left (15 a^2 B-15 a A b-5 a b B x+5 A b^2 x+3 b^2 B x^2\right )}{15 b^3}-\frac {2 \left (a^{5/2} B-a^{3/2} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 180, normalized size = 0.95 \begin {gather*} \left [-\frac {15 \, {\left (B a^{2} - A a b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \, {\left (B a b - A b^{2}\right )} x\right )} \sqrt {x}}{15 \, b^{3}}, -\frac {2 \, {\left (15 \, {\left (B a^{2} - A a b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \, {\left (B a b - A b^{2}\right )} x\right )} \sqrt {x}\right )}}{15 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 133, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (B a^{3} \mathrm {sgn}\left (b x + a\right ) - A a^{2} b \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (3 \, B b^{4} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) - 5 \, B a b^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 5 \, A b^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 15 \, B a^{2} b^{2} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - 15 \, A a b^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right )\right )}}{15 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 129, normalized size = 0.68 \begin {gather*} \frac {2 \left (b x +a \right ) \left (3 \sqrt {a b}\, B \,b^{2} x^{\frac {5}{2}}+15 A \,a^{2} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-15 B \,a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+5 \sqrt {a b}\, A \,b^{2} x^{\frac {3}{2}}-5 \sqrt {a b}\, B a b \,x^{\frac {3}{2}}-15 \sqrt {a b}\, A a b \sqrt {x}+15 \sqrt {a b}\, B \,a^{2} \sqrt {x}\right )}{15 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {a b}\, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.59, size = 147, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} x^{2} + 5 \, B a b x\right )} x^{\frac {3}{2}} + {\left (3 \, {\left (7 \, B a b - 5 \, A b^{2}\right )} x^{2} + 5 \, {\left (5 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {x}}{15 \, {\left (b^{3} x + a b^{2}\right )}} - \frac {2 \, {\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} - \frac {{\left (7 \, B a b - 5 \, A b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (B a^{2} - A a b\right )} \sqrt {x}}{3 \, b^{3}} \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 \frac {x^{3/2}\,\left (A+B\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,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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