Optimal. Leaf size=71 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}}+\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2} \]
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Rubi [A] time = 0.02, antiderivative size = 71, 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 = {50, 63, 217, 206} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}}+\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx &=\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{4} (3 a) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{8} \left (3 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{4} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {1}{4} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {3}{4} a \sqrt {x} \sqrt {a+b x}+\frac {1}{2} \sqrt {x} (a+b x)^{3/2}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 69, normalized size = 0.97 \begin {gather*} \frac {1}{4} \sqrt {a+b x} \left (\frac {3 a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\frac {b x}{a}+1}}+\sqrt {x} (5 a+2 b x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 66, normalized size = 0.93 \begin {gather*} \frac {1}{4} \sqrt {a+b x} \left (5 a \sqrt {x}+2 b x^{3/2}\right )-\frac {3 a^2 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 119, normalized size = 1.68 \begin {gather*} \left [\frac {3 \, 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 + 5 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x + 5 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b}\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 = 0.01, size = 78, normalized size = 1.10 \begin {gather*} \frac {3 \sqrt {\left (b x +a \right ) x}\, a^{2} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 \sqrt {b x +a}\, \sqrt {b}\, \sqrt {x}}+\frac {3 \sqrt {b x +a}\, a \sqrt {x}}{4}+\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.99, size = 107, normalized size = 1.51 \begin {gather*} -\frac {3 \, 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 \, \sqrt {b}} - \frac {\frac {3 \, \sqrt {b x + a} a^{2} b}{\sqrt {x}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{2} - \frac {2 \, {\left (b x + a\right )} b}{x} + \frac {{\left (b x + a\right )}^{2}}{x^{2}}\right )}} \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 )}^{3/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.17, size = 75, normalized size = 1.06 \begin {gather*} \frac {5 a^{\frac {3}{2}} \sqrt {x} \sqrt {1 + \frac {b x}{a}}}{4} + \frac {\sqrt {a} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x}{a}}}{2} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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