Optimal. Leaf size=48 \[ -\frac {2 (d+e x)^{3/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {662}
\begin {gather*} -\frac {2 (d+e x)^{3/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^{3/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.77 \begin {gather*} -\frac {2 (d+e x)^{3/2}}{3 c d ((a e+c d x) (d+e x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 42, normalized size = 0.88
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} c d}\) | \(42\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c d \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 30, normalized size = 0.62 \begin {gather*} -\frac {2}{3 \, {\left (c^{2} d^{2} x + a c d e\right )} \sqrt {c d x + a e}} \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. 107 vs.
\(2 (44) = 88\).
time = 3.51, size = 107, normalized size = 2.23 \begin {gather*} -\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3 \, {\left (c^{3} d^{4} x^{2} + a^{2} c d x e^{3} + {\left (2 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2} + {\left (c^{3} d^{3} x^{3} + 2 \, a c^{2} d^{3} x\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 88, normalized size = 1.83 \begin {gather*} -\frac {2 \, e^{2}}{3 \, {\left (\sqrt {-c d^{2} e + a e^{3}} c^{2} d^{3} - \sqrt {-c d^{2} e + a e^{3}} a c d e^{2}\right )}} - \frac {2 \, e^{3}}{3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.32, size = 110, normalized size = 2.29 \begin {gather*} -\frac {2\,\sqrt {d+e\,x}\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,\left (a^2\,c\,d^2\,e^2+a^2\,c\,d\,e^3\,x+2\,a\,c^2\,d^3\,e\,x+2\,a\,c^2\,d^2\,e^2\,x^2+c^3\,d^4\,x^2+c^3\,d^3\,e\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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