Optimal. Leaf size=32 \[ \frac {2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \begin {gather*} \frac {2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{(c+d x)^{5/2}} \, dx &=\frac {2 (a+b x)^{3/2}}{3 (b c-a d) (c+d x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 (a+b x)^{3/2}}{3 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.60, size = 65, normalized size = 2.03 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}}{3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 51, normalized size = 1.59 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{4} d}{3 \, {\left (b c d {\left | b \right |} - a d^{2} {\left | b \right |}\right )} {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.84 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a d -b c \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 130, normalized size = 4.06 \begin {gather*} -\frac {\left (\frac {2\,a\,\sqrt {a+b\,x}}{3\,a\,d^3-3\,b\,c\,d^2}+\frac {2\,b\,x\,\sqrt {a+b\,x}}{3\,a\,d^3-3\,b\,c\,d^2}\right )\,\sqrt {c+d\,x}}{x^2-\frac {3\,b\,c^3-3\,a\,c^2\,d}{3\,a\,d^3-3\,b\,c\,d^2}+\frac {6\,c\,d\,x\,\left (a\,d-b\,c\right )}{3\,a\,d^3-3\,b\,c\,d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b x}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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