Optimal. Leaf size=126 \[ \frac {a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{7/2}}-\frac {a \sqrt {x} \sqrt {a+b x} (6 A b-5 a B)}{8 b^3}+\frac {x^{3/2} \sqrt {a+b x} (6 A b-5 a B)}{12 b^2}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b} \]
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Rubi [A] time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \[ \frac {a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{7/2}}+\frac {x^{3/2} \sqrt {a+b x} (6 A b-5 a B)}{12 b^2}-\frac {a \sqrt {x} \sqrt {a+b x} (6 A b-5 a B)}{8 b^3}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\sqrt {a+b x}} \, dx &=\frac {B x^{5/2} \sqrt {a+b x}}{3 b}+\frac {\left (3 A b-\frac {5 a B}{2}\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{3 b}\\ &=\frac {(6 A b-5 a B) x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b}-\frac {(a (6 A b-5 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b^2}\\ &=-\frac {a (6 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{8 b^3}+\frac {(6 A b-5 a B) x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b}+\frac {\left (a^2 (6 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^3}\\ &=-\frac {a (6 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{8 b^3}+\frac {(6 A b-5 a B) x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b}+\frac {\left (a^2 (6 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^3}\\ &=-\frac {a (6 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{8 b^3}+\frac {(6 A b-5 a B) x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b}+\frac {\left (a^2 (6 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^3}\\ &=-\frac {a (6 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{8 b^3}+\frac {(6 A b-5 a B) x^{3/2} \sqrt {a+b x}}{12 b^2}+\frac {B x^{5/2} \sqrt {a+b x}}{3 b}+\frac {a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 113, normalized size = 0.90 \[ \frac {\sqrt {b} \sqrt {x} (a+b x) \left (15 a^2 B-2 a b (9 A+5 B x)+4 b^2 x (3 A+2 B x)\right )-3 a^{5/2} \sqrt {\frac {b x}{a}+1} (5 a B-6 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{24 b^{7/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 200, normalized size = 1.59 \[ \left [-\frac {3 \, {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{3} x^{2} + 15 \, B a^{2} b - 18 \, A a b^{2} - 2 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{4}}, \frac {3 \, {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, B b^{3} x^{2} + 15 \, B a^{2} b - 18 \, A a b^{2} - 2 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.02, size = 176, normalized size = 1.40 \[ \frac {\sqrt {b x +a}\, \left (16 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {5}{2}} x^{2}+18 A \,a^{2} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-15 B \,a^{3} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+24 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {5}{2}} x -20 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {3}{2}} x -36 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {3}{2}}+30 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} \sqrt {b}\right ) \sqrt {x}}{48 \sqrt {\left (b x +a \right ) x}\, b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 160, normalized size = 1.27 \[ \frac {\sqrt {b x^{2} + a x} B x^{2}}{3 \, b} - \frac {5 \, \sqrt {b x^{2} + a x} B a x}{12 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} A x}{2 \, b} - \frac {5 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {5}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{8 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a x} A a}{4 \, b^{2}} \]
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 \[ \int \frac {x^{3/2}\,\left (A+B\,x\right )}{\sqrt {a+b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 36.74, size = 245, normalized size = 1.94 \[ - \frac {3 A a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A \sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {1 + \frac {b x}{a}}} + \frac {3 A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {A x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} + \frac {5 B a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 B a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B \sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 + \frac {b x}{a}}} - \frac {5 B a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {B x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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