Optimal. Leaf size=54 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 \left (c d^2-a e^2\right ) (d+e x)^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {664}
\begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 \left (c d^2-a e^2\right ) (d+e x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 43, normalized size = 0.80 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2}}{3 \left (c d^2-a e^2\right ) (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 65, normalized size = 1.20
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (e x +d \right )^{2} \left (e^{2} a -c \,d^{2}\right )}\) | \(58\) |
trager | \(-\frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (e x +d \right )^{2} \left (e^{2} a -c \,d^{2}\right )}\) | \(58\) |
default | \(-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{3} \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}\) | \(65\) |
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 [A]
time = 4.06, size = 88, normalized size = 1.63 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x + a e\right )}}{3 \, {\left (2 \, c d^{3} x e + c d^{4} - a x^{2} e^{4} - 2 \, a d x e^{3} + {\left (c d^{2} x^{2} - a d^{2}\right )} e^{2}\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 {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.08, size = 58, normalized size = 1.07 \begin {gather*} -\frac {2\,\left (a\,e+c\,d\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{3\,\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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