Optimal. Leaf size=51 \[ -\frac{a e^2+c d^2}{2 e^3 (d+e x)^2}+\frac{2 c d}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0840314, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{a e^2+c d^2}{2 e^3 (d+e x)^2}+\frac{2 c d}{e^3 (d+e x)}+\frac{c \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(d + e*x)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.7032, size = 46, normalized size = 0.9 \[ \frac{2 c d}{e^{3} \left (d + e x\right )} + \frac{c \log{\left (d + e x \right )}}{e^{3}} - \frac{a e^{2} + c d^{2}}{2 e^{3} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0311481, size = 48, normalized size = 0.94 \[ \frac{-a e^2+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(d + e*x)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 56, normalized size = 1.1 \[ -{\frac{a}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{c{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{c\ln \left ( ex+d \right ) }{{e}^{3}}}+2\,{\frac{cd}{{e}^{3} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.72556, size = 77, normalized size = 1.51 \[ \frac{4 \, c d e x + 3 \, c d^{2} - a e^{2}}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{c \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.20205, size = 99, normalized size = 1.94 \[ \frac{4 \, c d e x + 3 \, c d^{2} - a e^{2} + 2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.94483, size = 56, normalized size = 1.1 \[ \frac{c \log{\left (d + e x \right )}}{e^{3}} + \frac{- a e^{2} + 3 c d^{2} + 4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.213046, size = 62, normalized size = 1.22 \[ c e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (4 \, c d x +{\left (3 \, c d^{2} - a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d)^3,x, algorithm="giac")
[Out]