Optimal. Leaf size=75 \[ \frac{-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac{b B}{e^3 (d+e x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.122256, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac{b B}{e^3 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(A + B*x))/(d + e*x)^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.5867, size = 66, normalized size = 0.88 \[ - \frac{B b}{e^{3} \left (d + e x\right )} - \frac{A b e + B a e - 2 B b d}{2 e^{3} \left (d + e x\right )^{2}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )}{3 e^{3} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(B*x+A)/(e*x+d)**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0524363, size = 63, normalized size = 0.84 \[ -\frac{a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{6 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(A + B*x))/(d + e*x)^4,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 79, normalized size = 1.1 \[ -{\frac{aA{e}^{2}-Abde-Bade+bB{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Bb}{{e}^{3} \left ( ex+d \right ) }}-{\frac{Abe+Bae-2\,Bbd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(B*x+A)/(e*x+d)^4,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.33904, size = 126, normalized size = 1.68 \[ -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(e*x + d)^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.203348, size = 126, normalized size = 1.68 \[ -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(e*x + d)^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 7.24132, size = 107, normalized size = 1.43 \[ - \frac{2 A a e^{2} + A b d e + B a d e + 2 B b d^{2} + 6 B b e^{2} x^{2} + x \left (3 A b e^{2} + 3 B a e^{2} + 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(B*x+A)/(e*x+d)**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.216404, size = 93, normalized size = 1.24 \[ -\frac{{\left (6 \, B b x^{2} e^{2} + 6 \, B b d x e + 2 \, B b d^{2} + 3 \, B a x e^{2} + 3 \, A b x e^{2} + B a d e + A b d e + 2 \, A a e^{2}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(e*x + d)^4,x, algorithm="giac")
[Out]