Optimal. Leaf size=94 \[ \frac{\left (a e^2+c d^2\right ) (B d-A e)}{2 e^4 (d+e x)^2}-\frac{a B e^2-2 A c d e+3 B c d^2}{e^4 (d+e x)}-\frac{c (3 B d-A e) \log (d+e x)}{e^4}+\frac{B c x}{e^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.174751, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{\left (a e^2+c d^2\right ) (B d-A e)}{2 e^4 (d+e x)^2}-\frac{a B e^2-2 A c d e+3 B c d^2}{e^4 (d+e x)}-\frac{c (3 B d-A e) \log (d+e x)}{e^4}+\frac{B c x}{e^3} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2))/(d + e*x)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c \int B\, dx}{e^{3}} + \frac{c \left (A e - 3 B d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{- 2 A c d e + B a e^{2} + 3 B c d^{2}}{e^{4} \left (d + e x\right )} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{2 e^{4} \left (d + e x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.139257, size = 88, normalized size = 0.94 \[ \frac{\frac{\left (a e^2+c d^2\right ) (B d-A e)}{(d+e x)^2}-\frac{2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{d+e x}+2 \log (d+e x) (A c e-3 B c d)+2 B c e x}{2 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 144, normalized size = 1.5 \[{\frac{Bcx}{{e}^{3}}}-{\frac{aA}{2\,e \left ( ex+d \right ) ^{2}}}-{\frac{Ac{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bad}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Bc{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ) Ac}{{e}^{3}}}-3\,{\frac{\ln \left ( ex+d \right ) Bcd}{{e}^{4}}}+2\,{\frac{Acd}{{e}^{3} \left ( ex+d \right ) }}-{\frac{Ba}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{Bc{d}^{2}}{{e}^{4} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)/(e*x+d)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.704341, size = 150, normalized size = 1.6 \[ -\frac{5 \, B c d^{3} - 3 \, A c d^{2} e + B a d e^{2} + A a e^{3} + 2 \,{\left (3 \, B c d^{2} e - 2 \, A c d e^{2} + B a e^{3}\right )} x}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac{B c x}{e^{3}} - \frac{{\left (3 \, B c d - A c e\right )} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.264413, size = 227, normalized size = 2.41 \[ \frac{2 \, B c e^{3} x^{3} + 4 \, B c d e^{2} x^{2} - 5 \, B c d^{3} + 3 \, A c d^{2} e - B a d e^{2} - A a e^{3} - 2 \,{\left (2 \, B c d^{2} e - 2 \, A c d e^{2} + B a e^{3}\right )} x - 2 \,{\left (3 \, B c d^{3} - A c d^{2} e +{\left (3 \, B c d e^{2} - A c e^{3}\right )} x^{2} + 2 \,{\left (3 \, B c d^{2} e - A c d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 6.0879, size = 117, normalized size = 1.24 \[ \frac{B c x}{e^{3}} - \frac{c \left (- A e + 3 B d\right ) \log{\left (d + e x \right )}}{e^{4}} - \frac{A a e^{3} - 3 A c d^{2} e + B a d e^{2} + 5 B c d^{3} + x \left (- 4 A c d e^{2} + 2 B a e^{3} + 6 B c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)/(e*x+d)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.312372, size = 130, normalized size = 1.38 \[ B c x e^{\left (-3\right )} -{\left (3 \, B c d - A c e\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (5 \, B c d^{3} - 3 \, A c d^{2} e + B a d e^{2} + A a e^{3} + 2 \,{\left (3 \, B c d^{2} e - 2 \, A c d e^{2} + B a e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^3,x, algorithm="giac")
[Out]