Optimal. Leaf size=52 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 52, 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 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x) \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 664
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 42, normalized size = 0.81 \begin {gather*} \frac {2 (a e+c d x)}{\left (c d^2-a e^2\right ) \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.72, size = 65, normalized size = 1.25
method | result | size |
trager | \(-\frac {2 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{\left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )}\) | \(50\) |
gosper | \(-\frac {2 \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(51\) |
default | \(-\frac {2 \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{e \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.49, size = 57, normalized size = 1.10 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{c d^{2} x e + c d^{3} - a x e^{3} - a d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.69, size = 50, normalized size = 0.96 \begin {gather*} -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\left (a\,e^2-c\,d^2\right )\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________