Optimal. Leaf size=32 \[ -\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \]
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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 (c+d x)^{3/2}}{3 (a+b 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 {c+d x}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 (c+d x)^{3/2}}{3 (b c-a d) (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs.
\(2(26)=52\).
time = 0.16, size = 88, normalized size = 2.75
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 \left (b x +a \right )^{\frac {3}{2}} \left (a d -b c \right )}\) | \(27\) |
default | \(-\frac {\sqrt {d x +c}}{b \left (b x +a \right )^{\frac {3}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{2 b}\) | \(88\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (26) = 52\).
time = 0.74, size = 65, normalized size = 2.03 \begin {gather*} -\frac {2 \, \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} b c - a^{3} d + {\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x\right )}} \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 {c + d x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs.
\(2 (26) = 52\).
time = 1.09, size = 152, normalized size = 4.75 \begin {gather*} -\frac {4 \, {\left (\sqrt {b d} b^{4} c^{2} d - 2 \, \sqrt {b d} a b^{3} c d^{2} + \sqrt {b d} a^{2} b^{2} d^{3} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} d\right )} {\left | b \right |}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 27, normalized size = 0.84 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{3/2}}{\left (3\,a\,d-3\,b\,c\right )\,{\left (a+b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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