Optimal. Leaf size=32 \[ -\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]
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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} \[ -\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]
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 = 0.01, size = 32, normalized size = 1.00 \[ -\frac {2 (c+d x)^{3/2}}{3 (a+b x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 65, normalized size = 2.03 \[ -\frac {2 \, \sqrt {b x + a} {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, {\left (a^{2} b c - a^{3} d + {\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.43, size = 152, normalized size = 4.75 \[ -\frac {4 \, {\left (\sqrt {b d} b^{4} c^{2} d - 2 \, \sqrt {b d} a b^{3} c d^{2} + \sqrt {b d} a^{2} b^{2} d^{3} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} d\right )} {\left | b \right |}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.84 \[ \frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 \left (b x +a \right )^{\frac {3}{2}} \left (a d -b c \right )} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \frac {2\,{\left (c+d\,x\right )}^{3/2}}{\left (3\,a\,d-3\,b\,c\right )\,{\left (a+b\,x\right )}^{3/2}} \]
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 \[ \int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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