Optimal. Leaf size=98 \[ \frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{5/2}}-\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x} \]
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Rubi [A] time = 0.03, antiderivative size = 98, 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^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{5/2}}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {a+b x} \, dx &=\frac {1}{3} x^{5/2} \sqrt {a+b x}+\frac {1}{6} a \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx\\ &=\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x}-\frac {a^2 \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x}+\frac {a^3 \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^2}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^2}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x}+\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^2}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a+b x}}{8 b^2}+\frac {a x^{3/2} \sqrt {a+b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a+b x}+\frac {a^3 \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 = 0.11, size = 85, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {3 a^{5/2} \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+2 a b x+8 b^2 x^2\right )\right )}{24 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 82, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+b x} \left (-3 a^2 \sqrt {x}+2 a b x^{3/2}+8 b^2 x^{5/2}\right )}{24 b^2}-\frac {a^3 \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{8 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 141, normalized size = 1.44 \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} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{48 \, b^{3}}, -\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} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {b x + a} \sqrt {x}}{24 \, b^{3}}\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 = 0.00, size = 102, normalized size = 1.04 \begin {gather*} \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 {5}{2}} \sqrt {x}}+\frac {\sqrt {b x +a}\, a^{2} \sqrt {x}}{8 b^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}} x^{\frac {3}{2}}}{3 b}-\frac {\left (b x +a \right )^{\frac {3}{2}} a \sqrt {x}}{4 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.04, size = 146, normalized size = 1.49 \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 {5}{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^{5} - \frac {3 \, {\left (b x + a\right )} b^{4}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b^{3}}{x^{2}} - \frac {{\left (b x + a\right )}^{3} b^{2}}{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 x^{3/2}\,\sqrt {a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.38, size = 122, normalized size = 1.24 \begin {gather*} - \frac {a^{\frac {5}{2}} \sqrt {x}}{8 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {5 \sqrt {a} 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 {5}{2}}} + \frac {b 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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