Optimal. Leaf size=126 \[ -\frac {a^2 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{5/2}}+\frac {a \sqrt {x} \sqrt {a+b x} (2 A b-a B)}{8 b^2}+\frac {x^{3/2} \sqrt {a+b x} (2 A b-a B)}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b} \]
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Rubi [A] time = 0.06, 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} \[ -\frac {a^2 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{5/2}}+\frac {a \sqrt {x} \sqrt {a+b x} (2 A b-a B)}{8 b^2}+\frac {x^{3/2} \sqrt {a+b x} (2 A b-a B)}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {a+b x} (A+B x) \, dx &=\frac {B x^{3/2} (a+b x)^{3/2}}{3 b}+\frac {\left (3 A b-\frac {3 a B}{2}\right ) \int \sqrt {x} \sqrt {a+b x} \, dx}{3 b}\\ &=\frac {(2 A b-a B) x^{3/2} \sqrt {a+b x}}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b}+\frac {(a (2 A b-a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b}\\ &=\frac {a (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {(2 A b-a B) x^{3/2} \sqrt {a+b x}}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b}-\frac {\left (a^2 (2 A b-a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^2}\\ &=\frac {a (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {(2 A b-a B) x^{3/2} \sqrt {a+b x}}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b}-\frac {\left (a^2 (2 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^2}\\ &=\frac {a (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {(2 A b-a B) x^{3/2} \sqrt {a+b x}}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b}-\frac {\left (a^2 (2 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 )}{8 b^2}\\ &=\frac {a (2 A b-a B) \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {(2 A b-a B) x^{3/2} \sqrt {a+b x}}{4 b}+\frac {B x^{3/2} (a+b x)^{3/2}}{3 b}-\frac {a^2 (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 106, normalized size = 0.84 \[ \frac {\sqrt {a+b x} \left (\frac {3 a^{3/2} (a B-2 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}+\sqrt {b} \sqrt {x} \left (-3 a^2 B+2 a b (3 A+B x)+4 b^2 x (3 A+2 B x)\right )\right )}{24 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 197, normalized size = 1.56 \[ \left [-\frac {3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{3} x^{2} - 3 \, B a^{2} b + 6 \, A a b^{2} + 2 \, {\left (B a b^{2} + 6 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{3}}, -\frac {3 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (8 \, B b^{3} x^{2} - 3 \, B a^{2} b + 6 \, A a b^{2} + 2 \, {\left (B a b^{2} + 6 \, A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{3}}\right ] \]
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 = 176, normalized size = 1.40 \[ -\frac {\sqrt {b x +a}\, \left (-16 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {5}{2}} x^{2}+6 A \,a^{2} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-3 B \,a^{3} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-24 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {5}{2}} x -4 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {3}{2}} x -12 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {3}{2}}+6 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} \sqrt {b}\right ) \sqrt {x}}{48 \sqrt {\left (b x +a \right ) x}\, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 154, normalized size = 1.22 \[ \frac {1}{2} \, \sqrt {b x^{2} + a x} A x - \frac {\sqrt {b x^{2} + a x} B a x}{4 \, b} + \frac {B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} - \frac {A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a x} B a^{2}}{8 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{3 \, b} + \frac {\sqrt {b x^{2} + a x} A a}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.63, size = 399, normalized size = 3.17 \[ \frac {\frac {x^{11/2}\,\left (\frac {A\,a^2\,b^4}{2}-\frac {B\,a^3\,b^3}{4}\right )}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}}+\frac {x^{9/2}\,\left (\frac {17\,B\,a^3\,b^2}{12}+\frac {5\,A\,a^2\,b^3}{2}\right )}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9}-\frac {x^{7/2}\,\left (3\,A\,a^2\,b^2-\frac {19\,B\,a^3\,b}{2}\right )}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7}+\frac {x^{5/2}\,\left (\frac {19\,B\,a^3}{2}-3\,A\,a^2\,b\right )}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}-\frac {\sqrt {x}\,\left (B\,a^3-2\,A\,a^2\,b\right )}{4\,b^2\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}+\frac {x^{3/2}\,\left (17\,B\,a^3+30\,A\,b\,a^2\right )}{12\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}}{\frac {15\,b^2\,x^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}-\frac {20\,b^3\,x^3}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}+\frac {15\,b^4\,x^4}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}-\frac {6\,b^5\,x^5}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}+\frac {b^6\,x^6}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}-\frac {6\,b\,x}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}+1}-\frac {a^2\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )\,\left (2\,A\,b-B\,a\right )}{4\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.11, size = 673, normalized size = 5.34 \[ \frac {A a^{\frac {3}{2}} \sqrt {x}}{4 b \sqrt {1 + \frac {b x}{a}}} + \frac {3 A \sqrt {a} x^{\frac {3}{2}}}{4 \sqrt {1 + \frac {b x}{a}}} - \frac {A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {3}{2}}} + \frac {A b x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} - \frac {2 B a \left (\begin {cases} \frac {a^{\frac {3}{2}} \sqrt {a + b x}}{8 \sqrt {b} \sqrt {\frac {b x}{a}}} - \frac {3 \sqrt {a} \left (a + b x\right )^{\frac {3}{2}}}{8 \sqrt {b} \sqrt {\frac {b x}{a}}} - \frac {a^{2} \operatorname {acosh}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {\left (a + b x\right )^{\frac {5}{2}}}{4 \sqrt {a} \sqrt {b} \sqrt {\frac {b x}{a}}} & \text {for}\: \left |{1 + \frac {b x}{a}}\right | > 1 \\- \frac {i a^{\frac {3}{2}} \sqrt {a + b x}}{8 \sqrt {b} \sqrt {- \frac {b x}{a}}} + \frac {3 i \sqrt {a} \left (a + b x\right )^{\frac {3}{2}}}{8 \sqrt {b} \sqrt {- \frac {b x}{a}}} + \frac {i a^{2} \operatorname {asin}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{8 \sqrt {b}} - \frac {i \left (a + b x\right )^{\frac {5}{2}}}{4 \sqrt {a} \sqrt {b} \sqrt {- \frac {b x}{a}}} & \text {otherwise} \end {cases}\right )}{b^{2}} + \frac {2 B \left (\begin {cases} \frac {a^{\frac {5}{2}} \sqrt {a + b x}}{16 \sqrt {b} \sqrt {\frac {b x}{a}}} - \frac {a^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}}{48 \sqrt {b} \sqrt {\frac {b x}{a}}} - \frac {5 \sqrt {a} \left (a + b x\right )^{\frac {5}{2}}}{24 \sqrt {b} \sqrt {\frac {b x}{a}}} - \frac {a^{3} \operatorname {acosh}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {\left (a + b x\right )^{\frac {7}{2}}}{6 \sqrt {a} \sqrt {b} \sqrt {\frac {b x}{a}}} & \text {for}\: \left |{1 + \frac {b x}{a}}\right | > 1 \\- \frac {i a^{\frac {5}{2}} \sqrt {a + b x}}{16 \sqrt {b} \sqrt {- \frac {b x}{a}}} + \frac {i a^{\frac {3}{2}} \left (a + b x\right )^{\frac {3}{2}}}{48 \sqrt {b} \sqrt {- \frac {b x}{a}}} + \frac {5 i \sqrt {a} \left (a + b x\right )^{\frac {5}{2}}}{24 \sqrt {b} \sqrt {- \frac {b x}{a}}} + \frac {i a^{3} \operatorname {asin}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{16 \sqrt {b}} - \frac {i \left (a + b x\right )^{\frac {7}{2}}}{6 \sqrt {a} \sqrt {b} \sqrt {- \frac {b x}{a}}} & \text {otherwise} \end {cases}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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