Optimal. Leaf size=63 \[ 3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 63, 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 {2 (a+b x)^{3/2}}{\sqrt {x}}+3 b \sqrt {x} \sqrt {a+b x}+3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \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 {(a+b x)^{3/2}}{x^{3/2}} \, dx &=-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+(3 b) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+\frac {1}{2} (3 a b) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+(3 a b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+(3 a b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=3 b \sqrt {x} \sqrt {a+b x}-\frac {2 (a+b x)^{3/2}}{\sqrt {x}}+3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 54, normalized size = 0.86 \begin {gather*} \frac {(-2 a+b x) \sqrt {a+b x}}{\sqrt {x}}-3 a \sqrt {b} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 3.44, size = 84, normalized size = 1.33 \begin {gather*} \frac {-2 a \left (a+b x\right )+3 a^{\frac {3}{2}} \sqrt {b} \sqrt {x} \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}-b x \left (a+b x\right )+\frac {b^2 x^2 \left (a+b x\right )}{a}}{\sqrt {a} \sqrt {x} \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 71, normalized size = 1.13
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-b x +2 a \right )}{\sqrt {x}}+\frac {3 a \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right ) \sqrt {x \left (b x +a \right )}}{2 \sqrt {x}\, \sqrt {b x +a}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 84, normalized size = 1.33 \begin {gather*} -\frac {3}{2} \, a \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + a} a}{\sqrt {x}} - \frac {\sqrt {b x + a} a b}{{\left (b - \frac {b x + a}{x}\right )} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 109, normalized size = 1.73 \begin {gather*} \left [\frac {3 \, a \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )} \sqrt {x}}{2 \, x}, -\frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - \sqrt {b x + a} {\left (b x - 2 \, a\right )} \sqrt {x}}{x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.52, size = 92, normalized size = 1.46 \begin {gather*} - \frac {2 a^{\frac {3}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} - \frac {\sqrt {a} b \sqrt {x}}{\sqrt {1 + \frac {b x}{a}}} + 3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {b^{2} x^{\frac {3}{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 = 10.28, size = 126, normalized size = 2.00 \begin {gather*} \frac {b b^{2} \left (\frac {2 \left (\frac {1}{2} \sqrt {a+b x} \sqrt {a+b x}-\frac {3}{2} a\right ) \sqrt {a+b x} \sqrt {-a b+b \left (a+b x\right )}}{-a b+b \left (a+b x\right )}-\frac {6 a \ln \left |\sqrt {-a b+b \left (a+b x\right )}-\sqrt {b} \sqrt {a+b x}\right |}{2 \sqrt {b}}\right )}{\left |b\right | b} \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 = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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