Optimal. Leaf size=74 \[ \frac {a \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a+b x}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {52, 65, 223,
212} \begin {gather*} -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 b^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {a+b x}+\frac {a \sqrt {x} \sqrt {a+b x}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {a+b x} \, dx &=\frac {1}{2} x^{3/2} \sqrt {a+b x}+\frac {1}{4} a \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx\\ &=\frac {a \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a+b x}-\frac {a^2 \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{8 b}\\ &=\frac {a \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a+b x}-\frac {a^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{4 b}\\ &=\frac {a \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a+b x}-\frac {a^2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{4 b}\\ &=\frac {a \sqrt {x} \sqrt {a+b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a+b x}-\frac {a^2 \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 = 0.05, size = 63, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x} (a+2 b x)}{4 b}+\frac {a^2 \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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Mathics [A]
time = 3.99, size = 90, normalized size = 1.22 \begin {gather*} \frac {-a^{\frac {5}{2}} b \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}+a b^{\frac {3}{2}} \sqrt {x} \left (a+b x\right )+3 b^{\frac {5}{2}} x^{\frac {3}{2}} \left (a+b x\right )+\frac {2 b^{\frac {7}{2}} x^{\frac {5}{2}} \left (a+b x\right )}{a}}{4 \sqrt {a} b^{\frac {5}{2}} \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 84, normalized size = 1.14
method | result | size |
risch | \(\frac {\left (2 b x +a \right ) \sqrt {x}\, \sqrt {b x +a}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right ) \sqrt {x \left (b x +a \right )}}{8 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(74\) |
default | \(\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{2}}}{2 b}-\frac {a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4 b}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (52) = 104\).
time = 0.35, size = 108, normalized size = 1.46 \begin {gather*} \frac {a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {\frac {\sqrt {b x + a} a^{2} b}{\sqrt {x}} + \frac {{\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{3} - \frac {2 \, {\left (b x + a\right )} b^{2}}{x} + \frac {{\left (b x + a\right )}^{2} b}{x^{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 114, normalized size = 1.54 \begin {gather*} \left [\frac {a^{2} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x + a b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b^{2}}, \frac {a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (2 \, b^{2} x + a b\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 [A]
time = 2.01, size = 97, normalized size = 1.31 \begin {gather*} \frac {a^{\frac {3}{2}} \sqrt {x}}{4 b \sqrt {1 + \frac {b x}{a}}} + \frac {3 \sqrt {a} x^{\frac {3}{2}}}{4 \sqrt {1 + \frac {b x}{a}}} - \frac {a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {3}{2}}} + \frac {b x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 92, normalized size = 1.24 \begin {gather*} 2 \left (2 \left (\frac {\frac {1}{32}\cdot 4 b^{2} \sqrt {x} \sqrt {x}}{b^{2}}+\frac {\frac {1}{32}\cdot 2 b a}{b^{2}}\right ) \sqrt {x} \sqrt {a+b x}+\frac {2 a^{2} \ln \left |\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right |}{16 b \sqrt {b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 52, normalized size = 0.70 \begin {gather*} \sqrt {x}\,\left (\frac {x}{2}+\frac {a}{4\,b}\right )\,\sqrt {a+b\,x}-\frac {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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