Optimal. Leaf size=68 \[ -\frac {2 x^{3/2}}{b \sqrt {a+b x}}+\frac {3 \sqrt {x} \sqrt {a+b x}}{b^2}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {49, 52, 65, 223,
212} \begin {gather*} -\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {a+b x}}{b^2}-\frac {2 x^{3/2}}{b \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 x^{3/2}}{b \sqrt {a+b x}}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{b}\\ &=-\frac {2 x^{3/2}}{b \sqrt {a+b x}}+\frac {3 \sqrt {x} \sqrt {a+b x}}{b^2}-\frac {(3 a) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{2 b^2}\\ &=-\frac {2 x^{3/2}}{b \sqrt {a+b x}}+\frac {3 \sqrt {x} \sqrt {a+b x}}{b^2}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {2 x^{3/2}}{b \sqrt {a+b x}}+\frac {3 \sqrt {x} \sqrt {a+b x}}{b^2}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{b^2}\\ &=-\frac {2 x^{3/2}}{b \sqrt {a+b x}}+\frac {3 \sqrt {x} \sqrt {a+b x}}{b^2}-\frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 57, normalized size = 0.84 \begin {gather*} \frac {\sqrt {x} (3 a+b x)}{b^2 \sqrt {a+b x}}+\frac {3 a \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.74, size = 77, normalized size = 1.13 \begin {gather*} \frac {\sqrt {a} \left (-3 \sqrt {a} b^3 \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (a+b x\right )+3 a b^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {a+b x}{a}}+b^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {a+b x}{a}}\right )}{b^{\frac {11}{2}} \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(105\) vs.
\(2(52)=104\).
time = 0.13, size = 106, normalized size = 1.56
method | result | size |
risch | \(\frac {\sqrt {x}\, \sqrt {b x +a}}{b^{2}}+\frac {\left (-\frac {3 a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right )}{2 b^{\frac {5}{2}}}+\frac {2 a \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{b^{3} \left (x +\frac {a}{b}\right )}\right ) \sqrt {x \left (b x +a \right )}}{\sqrt {x}\, \sqrt {b x +a}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 92, normalized size = 1.35 \begin {gather*} \frac {2 \, a b - \frac {3 \, {\left (b x + a\right )} a}{x}}{\frac {\sqrt {b x + a} b^{3}}{\sqrt {x}} - \frac {{\left (b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}}} + \frac {3 \, a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{2 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 145, normalized size = 2.13 \begin {gather*} \left [\frac {3 \, {\left (a b x + a^{2}\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{2 \, {\left (b^{4} x + a b^{3}\right )}}, \frac {3 \, {\left (a b x + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{b^{4} x + a b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.99, size = 71, normalized size = 1.04 \begin {gather*} \frac {3 \sqrt {a} \sqrt {x}}{b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {5}{2}}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} b \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 = 97, normalized size = 1.43 \begin {gather*} 2 \left (\frac {2 \left (\frac {\frac {1}{4} b^{2} \sqrt {x} \sqrt {x}}{b^{3}}+\frac {\frac {1}{4}\cdot 3 b a}{b^{3}}\right ) \sqrt {x} \sqrt {a+b x}}{a+b x}+\frac {6 a \ln \left |\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right |}{4 b^{2} \sqrt {b}}\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 {x^{3/2}}{{\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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