Optimal. Leaf size=159 \[ \frac {5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{3/2}}+\frac {5 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-a B)}{64 b}+\frac {\sqrt {x} (a+b x)^{5/2} (8 A b-a B)}{24 b}+\frac {5 a \sqrt {x} (a+b x)^{3/2} (8 A b-a B)}{96 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b} \]
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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} \begin {gather*} \frac {5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{3/2}}+\frac {5 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-a B)}{64 b}+\frac {\sqrt {x} (a+b x)^{5/2} (8 A b-a B)}{24 b}+\frac {5 a \sqrt {x} (a+b x)^{3/2} (8 A b-a B)}{96 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{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 {(a+b x)^{5/2} (A+B x)}{\sqrt {x}} \, dx &=\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {\left (4 A b-\frac {a B}{2}\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {x}} \, dx}{4 b}\\ &=\frac {(8 A b-a B) \sqrt {x} (a+b x)^{5/2}}{24 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {(5 a (8 A b-a B)) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx}{48 b}\\ &=\frac {5 a (8 A b-a B) \sqrt {x} (a+b x)^{3/2}}{96 b}+\frac {(8 A b-a B) \sqrt {x} (a+b x)^{5/2}}{24 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {\left (5 a^2 (8 A b-a B)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx}{64 b}\\ &=\frac {5 a^2 (8 A b-a B) \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5 a (8 A b-a B) \sqrt {x} (a+b x)^{3/2}}{96 b}+\frac {(8 A b-a B) \sqrt {x} (a+b x)^{5/2}}{24 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {\left (5 a^3 (8 A b-a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b}\\ &=\frac {5 a^2 (8 A b-a B) \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5 a (8 A b-a B) \sqrt {x} (a+b x)^{3/2}}{96 b}+\frac {(8 A b-a B) \sqrt {x} (a+b x)^{5/2}}{24 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {\left (5 a^3 (8 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b}\\ &=\frac {5 a^2 (8 A b-a B) \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5 a (8 A b-a B) \sqrt {x} (a+b x)^{3/2}}{96 b}+\frac {(8 A b-a B) \sqrt {x} (a+b x)^{5/2}}{24 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {\left (5 a^3 (8 A b-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}\\ &=\frac {5 a^2 (8 A b-a B) \sqrt {x} \sqrt {a+b x}}{64 b}+\frac {5 a (8 A b-a B) \sqrt {x} (a+b x)^{3/2}}{96 b}+\frac {(8 A b-a B) \sqrt {x} (a+b x)^{5/2}}{24 b}+\frac {B \sqrt {x} (a+b x)^{7/2}}{4 b}+\frac {5 a^3 (8 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 126, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \left (15 a^3 B+2 a^2 b (132 A+59 B x)+8 a b^2 x (26 A+17 B x)+16 b^3 x^2 (4 A+3 B x)\right )-\frac {15 a^{5/2} (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^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 144, normalized size = 0.91 \begin {gather*} \frac {5 \left (a^4 B-8 a^3 A b\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{64 b^{3/2}}+\frac {\sqrt {a+b x} \left (15 a^3 B \sqrt {x}+264 a^2 A b \sqrt {x}+118 a^2 b B x^{3/2}+208 a A b^2 x^{3/2}+136 a b^2 B x^{5/2}+64 A b^3 x^{5/2}+48 b^3 B x^{7/2}\right )}{192 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.55, size = 246, normalized size = 1.55 \begin {gather*} \left [-\frac {15 \, {\left (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 + 264 \, A a^{2} b^{2} + 8 \, {\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \, {\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{2}}, \frac {15 \, {\left (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 + 264 \, A a^{2} b^{2} + 8 \, {\left (17 \, B a b^{3} + 8 \, A b^{4}\right )} x^{2} + 2 \, {\left (59 \, B a^{2} b^{2} + 104 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{2}}\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}+272 \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 )-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 )+416 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {5}{2}} x +236 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {3}{2}} x +528 \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 {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.93, size = 454, normalized size = 2.86 \begin {gather*} \frac {1}{4} \, \sqrt {b x^{2} + a x} B b^{2} x^{3} - \frac {7}{24} \, \sqrt {b x^{2} + a x} B a b x^{2} + \frac {35}{96} \, \sqrt {b x^{2} + a x} B a^{2} x + \frac {35 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {3}{2}}} + \frac {A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}} - \frac {35 \, \sqrt {b x^{2} + a x} B a^{3}}{64 \, b} + \frac {{\left (3 \, B a b^{2} + A b^{3}\right )} \sqrt {b x^{2} + a x} x^{2}}{3 \, b} - \frac {5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \sqrt {b x^{2} + a x} a x}{12 \, b^{2}} + \frac {3 \, {\left (B a^{2} b + A a b^{2}\right )} \sqrt {b x^{2} + a x} x}{2 \, b} - \frac {5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {7}{2}}} + \frac {9 \, {\left (B a^{2} b + A a b^{2}\right )} a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {5}{2}}} - \frac {{\left (B a^{3} + 3 \, A a^{2} b\right )} a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, b^{\frac {3}{2}}} + \frac {5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \sqrt {b x^{2} + a x} a^{2}}{8 \, b^{3}} - \frac {9 \, {\left (B a^{2} b + A a b^{2}\right )} \sqrt {b x^{2} + a x} a}{4 \, b^{2}} + \frac {{\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {b x^{2} + a x}}{b} \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 {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 87.99, size = 262, normalized size = 1.65 \begin {gather*} A \left (\frac {11 a^{\frac {5}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{8} + \frac {13 a^{\frac {3}{2}} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{12} + \frac {\sqrt {a} b^{2} x^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}}}{3} + \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 \sqrt {b}}\right ) + B \left (\frac {5 a^{\frac {7}{2}} \sqrt {x}}{64 b \sqrt {1 + \frac {b x}{a}}} + \frac {133 a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 \sqrt {1 + \frac {b x}{a}}} + \frac {127 a^{\frac {3}{2}} b x^{\frac {5}{2}}}{96 \sqrt {1 + \frac {b x}{a}}} + \frac {23 \sqrt {a} b^{2} x^{\frac {7}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} - \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} + \frac {b^{3} x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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