Optimal. Leaf size=93 \[ \frac {(4 A b-a B) \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b}+\frac {a (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {81, 52, 65, 223,
212} \begin {gather*} \frac {a (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{3/2}}+\frac {\sqrt {x} \sqrt {a+b x} (4 A b-a B)}{4 b}+\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{\sqrt {x}} \, dx &=\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b}+\frac {\left (2 A b-\frac {a B}{2}\right ) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx}{2 b}\\ &=\frac {(4 A b-a B) \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b}+\frac {(a (4 A b-a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b}\\ &=\frac {(4 A b-a B) \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b}+\frac {(a (4 A b-a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b}\\ &=\frac {(4 A b-a B) \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b}+\frac {(a (4 A b-a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b}\\ &=\frac {(4 A b-a B) \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {B \sqrt {x} (a+b x)^{3/2}}{2 b}+\frac {a (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 76, normalized size = 0.82 \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x} (4 A b+a B+2 b B x)}{4 b}+\frac {a (-4 A b+a B) \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{4 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 136, normalized size = 1.46
method | result | size |
risch | \(\frac {\left (2 b B x +4 A b +B a \right ) \sqrt {b x +a}\, \sqrt {x}}{4 b}+\frac {\left (\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) A}{2 \sqrt {b}}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) B}{8 b^{\frac {3}{2}}}\right ) \sqrt {\left (b x +a \right ) x}}{\sqrt {b x +a}\, \sqrt {x}}\) | \(115\) |
default | \(\frac {\sqrt {b x +a}\, \sqrt {x}\, \left (4 B \,b^{\frac {3}{2}} x \sqrt {\left (b x +a \right ) x}+4 A \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a b +8 A \,b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}-B \ln \left (\frac {2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2}+2 B a \sqrt {b}\, \sqrt {\left (b x +a \right ) x}\right )}{8 b^{\frac {3}{2}} \sqrt {\left (b x +a \right ) x}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 109, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2} + a x} B x - \frac {B a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {3}{2}}} + \frac {A a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, \sqrt {b}} + \sqrt {b x^{2} + a x} A + \frac {\sqrt {b x^{2} + a x} B a}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.69, size = 146, normalized size = 1.57 \begin {gather*} \left [-\frac {{\left (B a^{2} - 4 \, A a b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (2 \, B b^{2} x + B a b + 4 \, A b^{2}\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b^{2}}, \frac {{\left (B a^{2} - 4 \, A a b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (2 \, B b^{2} x + B a b + 4 \, A b^{2}\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.48, size = 568, normalized size = 6.11 \begin {gather*} \frac {2 A \left (\begin {cases} \frac {\sqrt {a} \sqrt {b} \sqrt {\frac {b x}{a}} \sqrt {a + b x}}{2} + \frac {a \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{2} & \text {for}\: \left |{1 + \frac {b x}{a}}\right | > 1 \\\frac {i \sqrt {a} \sqrt {b} \sqrt {a + b x}}{2 \sqrt {- \frac {b x}{a}}} - \frac {i a \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{2} - \frac {i \sqrt {b} \left (a + b x\right )^{\frac {3}{2}}}{2 \sqrt {a} \sqrt {- \frac {b x}{a}}} & \text {otherwise} \end {cases}\right )}{b} - \frac {2 B a \left (\begin {cases} \frac {\sqrt {a} \sqrt {b} \sqrt {\frac {b x}{a}} \sqrt {a + b x}}{2} + \frac {a \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{2} & \text {for}\: \left |{1 + \frac {b x}{a}}\right | > 1 \\\frac {i \sqrt {a} \sqrt {b} \sqrt {a + b x}}{2 \sqrt {- \frac {b x}{a}}} - \frac {i a \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{2} - \frac {i \sqrt {b} \left (a + b x\right )^{\frac {3}{2}}}{2 \sqrt {a} \sqrt {- \frac {b x}{a}}} & \text {otherwise} \end {cases}\right )}{b^{2}} + \frac {2 B \left (\begin {cases} - \frac {3 a^{\frac {3}{2}} \sqrt {b} \sqrt {a + b x}}{8 \sqrt {\frac {b x}{a}}} + \frac {\sqrt {a} \sqrt {b} \left (a + b x\right )^{\frac {3}{2}}}{8 \sqrt {\frac {b x}{a}}} + \frac {3 a^{2} \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{8} + \frac {\sqrt {b} \left (a + b x\right )^{\frac {5}{2}}}{4 \sqrt {a} \sqrt {\frac {b x}{a}}} & \text {for}\: \left |{1 + \frac {b x}{a}}\right | > 1 \\\frac {3 i a^{\frac {3}{2}} \sqrt {b} \sqrt {a + b x}}{8 \sqrt {- \frac {b x}{a}}} - \frac {i \sqrt {a} \sqrt {b} \left (a + b x\right )^{\frac {3}{2}}}{8 \sqrt {- \frac {b x}{a}}} - \frac {3 i a^{2} \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {a + b x}}{\sqrt {a}} \right )}}{8} - \frac {i \sqrt {b} \left (a + b x\right )^{\frac {5}{2}}}{4 \sqrt {a} \sqrt {- \frac {b x}{a}}} & \text {otherwise} \end {cases}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 96, normalized size = 1.03 \begin {gather*} A\,\sqrt {x}\,\sqrt {a+b\,x}+\frac {2\,A\,a\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a+b\,x}-\sqrt {a}}\right )}{\sqrt {b}}+B\,\sqrt {x}\,\left (\frac {x}{2}+\frac {a}{4\,b}\right )\,\sqrt {a+b\,x}-\frac {B\,a^2\,\ln \left (a+2\,b\,x+2\,\sqrt {b}\,\sqrt {x}\,\sqrt {a+b\,x}\right )}{8\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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