Optimal. Leaf size=159 \[ -\frac {5 a^3 (8 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{9/2}}+\frac {5 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-7 a B)}{64 b^4}-\frac {5 a x^{3/2} \sqrt {a+b x} (8 A b-7 a B)}{96 b^3}+\frac {x^{5/2} \sqrt {a+b x} (8 A b-7 a B)}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b} \]
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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} \begin {gather*} \frac {5 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-7 a B)}{64 b^4}-\frac {5 a^3 (8 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{9/2}}+\frac {x^{5/2} \sqrt {a+b x} (8 A b-7 a B)}{24 b^2}-\frac {5 a x^{3/2} \sqrt {a+b x} (8 A b-7 a B)}{96 b^3}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b} \end {gather*}
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^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx &=\frac {B x^{7/2} \sqrt {a+b x}}{4 b}+\frac {\left (4 A b-\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{4 b}\\ &=\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}-\frac {(5 a (8 A b-7 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^2}\\ &=-\frac {5 a (8 A b-7 a B) x^{3/2} \sqrt {a+b x}}{96 b^3}+\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}+\frac {\left (5 a^2 (8 A b-7 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b^3}\\ &=\frac {5 a^2 (8 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{64 b^4}-\frac {5 a (8 A b-7 a B) x^{3/2} \sqrt {a+b x}}{96 b^3}+\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}-\frac {\left (5 a^3 (8 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^4}\\ &=\frac {5 a^2 (8 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{64 b^4}-\frac {5 a (8 A b-7 a B) x^{3/2} \sqrt {a+b x}}{96 b^3}+\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}-\frac {\left (5 a^3 (8 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^4}\\ &=\frac {5 a^2 (8 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{64 b^4}-\frac {5 a (8 A b-7 a B) x^{3/2} \sqrt {a+b x}}{96 b^3}+\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}-\frac {\left (5 a^3 (8 A b-7 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^4}\\ &=\frac {5 a^2 (8 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{64 b^4}-\frac {5 a (8 A b-7 a B) x^{3/2} \sqrt {a+b x}}{96 b^3}+\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}-\frac {5 a^3 (8 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 122, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {(8 A b-7 a B) \left (b x \sqrt {\frac {b x}{a}+1} \left (15 a^2-10 a b x+8 b^2 x^2\right )-15 a^{5/2} \sqrt {b} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{\sqrt {\frac {b x}{a}+1}}+48 b^4 B x^4\right )}{192 b^5 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 145, normalized size = 0.91 \begin {gather*} \frac {\sqrt {a+b x} \left (-105 a^3 B \sqrt {x}+120 a^2 A b \sqrt {x}+70 a^2 b B x^{3/2}-80 a A b^2 x^{3/2}-56 a b^2 B x^{5/2}+64 A b^3 x^{5/2}+48 b^3 B x^{7/2}\right )}{192 b^4}-\frac {5 \left (7 a^4 B-8 a^3 A b\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{64 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.75, size = 249, normalized size = 1.57 \begin {gather*} \left [-\frac {15 \, {\left (7 \, 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} - 105 \, B a^{3} b + 120 \, A a^{2} b^{2} - 8 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{5}}, -\frac {15 \, {\left (7 \, 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} - 105 \, B a^{3} b + 120 \, A a^{2} b^{2} - 8 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{5}}\right ] \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 218, normalized size = 1.37 \begin {gather*} -\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}+112 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {5}{2}} x^{2}+120 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 )-105 B \,a^{4} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+160 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {5}{2}} x -140 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {3}{2}} x -240 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {3}{2}}+210 \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 {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 206, normalized size = 1.30 \begin {gather*} \frac {\sqrt {b x^{2} + a x} B x^{3}}{4 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} B a x^{2}}{24 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} A x^{2}}{3 \, b} + \frac {35 \, \sqrt {b x^{2} + a x} B a^{2} x}{96 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a x} A a x}{12 \, b^{2}} + \frac {35 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {9}{2}}} - \frac {5 \, A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {7}{2}}} - \frac {35 \, \sqrt {b x^{2} + a x} B a^{3}}{64 \, b^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{8 \, 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^{5/2}\,\left (A+B\,x\right )}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 60.58, size = 303, normalized size = 1.91 \begin {gather*} \frac {5 A a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 A a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A \sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 + \frac {b x}{a}}} - \frac {5 A a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {A x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} - \frac {35 B a^{\frac {7}{2}} \sqrt {x}}{64 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {35 B a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {7 B a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B \sqrt {a} x^{\frac {7}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {35 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {9}{2}}} + \frac {B x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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