Optimal. Leaf size=95 \[ -\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{3/2}}+\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (a+b x)^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{3/2}}+\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (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 \sqrt {x} (a+b x)^{3/2} \, dx &=\frac {1}{3} x^{3/2} (a+b x)^{3/2}+\frac {1}{2} a \int \sqrt {x} \sqrt {a+b x} \, dx\\ &=\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (a+b x)^{3/2}+\frac {1}{8} a^2 \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx\\ &=\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (a+b x)^{3/2}-\frac {a^3 \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b}\\ &=\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (a+b x)^{3/2}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b}\\ &=\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (a+b x)^{3/2}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{8 b}\\ &=\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a+b x}+\frac {1}{3} x^{3/2} (a+b x)^{3/2}-\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 85, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \left (3 a^2+14 a b x+8 b^2 x^2\right )-\frac {3 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}\right )}{24 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 82, normalized size = 0.86 \begin {gather*} \frac {a^3 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{8 b^{3/2}}+\frac {\sqrt {a+b x} \left (3 a^2 \sqrt {x}+14 a b x^{3/2}+8 b^2 x^{5/2}\right )}{24 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 140, normalized size = 1.47 \begin {gather*} \left [\frac {3 \, a^{3} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (8 \, b^{3} x^{2} + 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{2}}, \frac {3 \, a^{3} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} + 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, 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 = 0.01, size = 96, normalized size = 1.01 \begin {gather*} \frac {\sqrt {b x +a}\, a \,x^{\frac {3}{2}}}{4}-\frac {\sqrt {\left (b x +a \right ) x}\, a^{3} \ln \left (\frac {b x +\frac {a}{2}}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{16 \sqrt {b x +a}\, b^{\frac {3}{2}} \sqrt {x}}+\frac {\sqrt {b x +a}\, a^{2} \sqrt {x}}{8 b}+\frac {\left (b x +a \right )^{\frac {3}{2}} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.02, size = 144, normalized size = 1.52 \begin {gather*} \frac {a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{16 \, b^{\frac {3}{2}}} + \frac {\frac {3 \, \sqrt {b x + a} a^{3} b^{2}}{\sqrt {x}} - \frac {8 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{4} - \frac {3 \, {\left (b x + a\right )} b^{3}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x + a\right )}^{3} b}{x^{3}}\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 \sqrt {x}\,{\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.59, size = 124, normalized size = 1.31 \begin {gather*} \frac {a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {1 + \frac {b x}{a}}} + \frac {17 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {1 + \frac {b x}{a}}} + \frac {11 \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {1 + \frac {b x}{a}}} - \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + \frac {b^{2} x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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