Optimal. Leaf size=50 \[ -\frac {2 (d+e x)}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {636} \begin {gather*} -\frac {2 (d+e x)}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 636
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)}{\left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 39, normalized size = 0.78 \begin {gather*} -\frac {2 (d+e x)}{\left (c d^2-a e^2\right ) \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.45, size = 55, normalized size = 1.10 \begin {gather*} -\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\left (c d^2-a e^2\right ) (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 65, normalized size = 1.30 \begin {gather*} -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.52, size = 109, normalized size = 2.18 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (c d^{2} e - a e^{3}\right )} x}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {c d^{3} - a d e^{2}}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}\right )}}{\sqrt {c d x^{2} e + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 58, normalized size = 1.16 \begin {gather*} \frac {2 \left (c d x +a e \right ) \left (e x +d \right )^{2}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 0.76, size = 53, normalized size = 1.06 \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+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )} \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 {d + e x}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________