Optimal. Leaf size=159 \[ \frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}}-\frac {a^2 \sqrt {x} \sqrt {a+b x} (8 A b-5 a B)}{64 b^3}+\frac {a x^{3/2} \sqrt {a+b x} (8 A b-5 a B)}{96 b^2}+\frac {x^{5/2} \sqrt {a+b x} (8 A b-5 a B)}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b} \]
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Rubi [A] time = 0.07, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {a^2 \sqrt {x} \sqrt {a+b x} (8 A b-5 a B)}{64 b^3}+\frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}}+\frac {a x^{3/2} \sqrt {a+b x} (8 A b-5 a B)}{96 b^2}+\frac {x^{5/2} \sqrt {a+b x} (8 A b-5 a B)}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {a+b x} (A+B x) \, dx &=\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (4 A b-\frac {5 a B}{2}\right ) \int x^{3/2} \sqrt {a+b x} \, dx}{4 b}\\ &=\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {(a (8 A b-5 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{48 b}\\ &=\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}-\frac {\left (a^2 (8 A b-5 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b^2}\\ &=-\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^3}\\ &=-\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^3}\\ &=-\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {\left (a^3 (8 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 )}{64 b^3}\\ &=-\frac {a^2 (8 A b-5 a B) \sqrt {x} \sqrt {a+b x}}{64 b^3}+\frac {a (8 A b-5 a B) x^{3/2} \sqrt {a+b x}}{96 b^2}+\frac {(8 A b-5 a B) x^{5/2} \sqrt {a+b x}}{24 b}+\frac {B x^{5/2} (a+b x)^{3/2}}{4 b}+\frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 126, normalized size = 0.79 \[ \frac {\sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \left (15 a^3 B-2 a^2 b (12 A+5 B x)+8 a b^2 x (2 A+B x)+16 b^3 x^2 (4 A+3 B x)\right )-\frac {3 a^{5/2} (5 a B-8 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}\right )}{192 b^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 246, normalized size = 1.55 \[ \left [-\frac {3 \, {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b - 24 \, A a^{2} b^{2} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{4}}, \frac {3 \, {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b - 24 \, A a^{2} b^{2} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{4}}\right ] \]
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 = 0.01, size = 218, normalized size = 1.37 \[ \frac {\sqrt {b x +a}\, \left (96 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {7}{2}} x^{3}+128 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {7}{2}} x^{2}+16 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {5}{2}} x^{2}+24 A \,a^{3} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-15 B \,a^{4} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+32 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {5}{2}} x -20 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {3}{2}} x -48 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {3}{2}}+30 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} \sqrt {b}\right ) \sqrt {x}}{384 \sqrt {\left (b x +a \right ) x}\, b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 198, normalized size = 1.25 \[ \frac {5 \, \sqrt {b x^{2} + a x} B a^{2} x}{32 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B x}{4 \, b} - \frac {\sqrt {b x^{2} + a x} A a x}{4 \, b} - \frac {5 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} + \frac {A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} B a^{3}}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{24 \, b^{2}} - \frac {\sqrt {b x^{2} + a x} A a^{2}}{8 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, b} \]
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 \[ \int x^{3/2}\,\left (A+B\,x\right )\,\sqrt {a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 25.95, size = 1527, normalized size = 9.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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