Optimal. Leaf size=30 \[ \frac {2 (a+b x)^2}{5 b c \sqrt {\frac {c}{a+b x}}} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac {2 (a+b x)^2}{5 b c \sqrt {\frac {c}{a+b x}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {c}{a+b x}\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (\frac {c}{x}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x^{3/2} \, dx,x,a+b x\right )}{b c \sqrt {\frac {c}{a+b x}} \sqrt {a+b x}}\\ &=\frac {2 (a+b x)^2}{5 b c \sqrt {\frac {c}{a+b x}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 0.70 \[ \frac {2 c}{5 b \left (\frac {c}{a+b x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 46, normalized size = 1.53 \[ \frac {2 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt {\frac {c}{b x + a}}}{5 \, b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 26, normalized size = 0.87 \[ \frac {2 \, {\left (b x + a\right )}^{2}}{5 \, b c \sqrt {\frac {c}{b x + a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.73 \[ \frac {\frac {2 b x}{5}+\frac {2 a}{5}}{\left (\frac {c}{b x +a}\right )^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 17, normalized size = 0.57 \[ \frac {2 \, c}{5 \, b \left (\frac {c}{b x + a}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 54, normalized size = 1.80 \[ \sqrt {\frac {c}{a+b\,x}}\,\left (\frac {6\,a^2\,x}{5\,c^2}+\frac {2\,a^3}{5\,b\,c^2}+\frac {2\,b^2\,x^3}{5\,c^2}+\frac {6\,a\,b\,x^2}{5\,c^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.69, size = 49, normalized size = 1.63 \[ \begin {cases} \frac {2 a}{5 b c^{\frac {3}{2}} \left (\frac {1}{a + b x}\right )^{\frac {3}{2}}} + \frac {2 x}{5 c^{\frac {3}{2}} \left (\frac {1}{a + b x}\right )^{\frac {3}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\frac {c}{a}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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