Optimal. Leaf size=230 \[ \frac{e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac{5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}-\frac{(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac{(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac{5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac{10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}+\frac{B e^5 (a+b x)^4}{4 b^7} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.739837, antiderivative size = 230, 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 \[ \frac{e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac{5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}-\frac{(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac{(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac{5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac{10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}+\frac{B e^5 (a+b x)^4}{4 b^7} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 92.9921, size = 231, normalized size = 1. \[ \frac{B e^{5} \left (a + b x\right )^{4}}{4 b^{7}} + \frac{10 e^{2} x \left (a e - b d\right )^{2} \left (A b e - 2 B a e + B b d\right )}{b^{6}} + \frac{e^{4} \left (a + b x\right )^{3} \left (A b e - 6 B a e + 5 B b d\right )}{3 b^{7}} - \frac{5 e^{3} \left (a + b x\right )^{2} \left (a e - b d\right ) \left (A b e - 3 B a e + 2 B b d\right )}{2 b^{7}} - \frac{5 e \left (a e - b d\right )^{3} \left (2 A b e - 3 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{7}} - \frac{\left (a e - b d\right )^{4} \left (5 A b e - 6 B a e + B b d\right )}{b^{7} \left (a + b x\right )} + \frac{\left (A b - B a\right ) \left (a e - b d\right )^{5}}{2 b^{7} \left (a + b x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**5/(b*x+a)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.259735, size = 254, normalized size = 1.1 \[ \frac{6 b^2 e^3 x^2 \left (6 a^2 B e^2-3 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+12 b e^2 x \left (-10 a^3 B e^3+6 a^2 b e^2 (A e+5 B d)-15 a b^2 d e (A e+2 B d)+10 b^3 d^2 (A e+B d)\right )+4 b^3 e^4 x^3 (-3 a B e+A b e+5 b B d)-\frac{12 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{a+b x}-\frac{6 (A b-a B) (b d-a e)^5}{(a+b x)^2}+60 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)+3 b^4 B e^5 x^4}{12 b^7} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.022, size = 833, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^5/(b*x+a)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.38444, size = 799, normalized size = 3.47 \[ -\frac{{\left (B a b^{5} + A b^{6}\right )} d^{5} - 5 \,{\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \,{\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \,{\left (7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (9 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} d e^{4} -{\left (11 \, B a^{6} - 9 \, A a^{5} b\right )} e^{5} + 2 \,{\left (B b^{6} d^{5} - 5 \,{\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \,{\left (4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \,{\left (5 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} d e^{4} -{\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x}{2 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac{3 \, B b^{3} e^{5} x^{4} + 4 \,{\left (5 \, B b^{3} d e^{4} -{\left (3 \, B a b^{2} - A b^{3}\right )} e^{5}\right )} x^{3} + 6 \,{\left (10 \, B b^{3} d^{2} e^{3} - 5 \,{\left (3 \, B a b^{2} - A b^{3}\right )} d e^{4} + 3 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} e^{5}\right )} x^{2} + 12 \,{\left (10 \, B b^{3} d^{3} e^{2} - 10 \,{\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{3} + 15 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{4} - 2 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{5}\right )} x}{12 \, b^{6}} + \frac{5 \,{\left (B b^{4} d^{4} e - 2 \,{\left (3 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{3} - 2 \,{\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^5/(b*x + a)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.218078, size = 1231, normalized size = 5.35 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^5/(b*x + a)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 45.0004, size = 602, normalized size = 2.62 \[ \frac{B e^{5} x^{4}}{4 b^{3}} + \frac{- 9 A a^{5} b e^{5} + 35 A a^{4} b^{2} d e^{4} - 50 A a^{3} b^{3} d^{2} e^{3} + 30 A a^{2} b^{4} d^{3} e^{2} - 5 A a b^{5} d^{4} e - A b^{6} d^{5} + 11 B a^{6} e^{5} - 45 B a^{5} b d e^{4} + 70 B a^{4} b^{2} d^{2} e^{3} - 50 B a^{3} b^{3} d^{3} e^{2} + 15 B a^{2} b^{4} d^{4} e - B a b^{5} d^{5} + x \left (- 10 A a^{4} b^{2} e^{5} + 40 A a^{3} b^{3} d e^{4} - 60 A a^{2} b^{4} d^{2} e^{3} + 40 A a b^{5} d^{3} e^{2} - 10 A b^{6} d^{4} e + 12 B a^{5} b e^{5} - 50 B a^{4} b^{2} d e^{4} + 80 B a^{3} b^{3} d^{2} e^{3} - 60 B a^{2} b^{4} d^{3} e^{2} + 20 B a b^{5} d^{4} e - 2 B b^{6} d^{5}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} - \frac{x^{3} \left (- A b e^{5} + 3 B a e^{5} - 5 B b d e^{4}\right )}{3 b^{4}} + \frac{x^{2} \left (- 3 A a b e^{5} + 5 A b^{2} d e^{4} + 6 B a^{2} e^{5} - 15 B a b d e^{4} + 10 B b^{2} d^{2} e^{3}\right )}{2 b^{5}} - \frac{x \left (- 6 A a^{2} b e^{5} + 15 A a b^{2} d e^{4} - 10 A b^{3} d^{2} e^{3} + 10 B a^{3} e^{5} - 30 B a^{2} b d e^{4} + 30 B a b^{2} d^{2} e^{3} - 10 B b^{3} d^{3} e^{2}\right )}{b^{6}} + \frac{5 e \left (a e - b d\right )^{3} \left (- 2 A b e + 3 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**5/(b*x+a)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.229095, size = 807, normalized size = 3.51 \[ \frac{5 \,{\left (B b^{4} d^{4} e - 6 \, B a b^{3} d^{3} e^{2} + 2 \, A b^{4} d^{3} e^{2} + 12 \, B a^{2} b^{2} d^{2} e^{3} - 6 \, A a b^{3} d^{2} e^{3} - 10 \, B a^{3} b d e^{4} + 6 \, A a^{2} b^{2} d e^{4} + 3 \, B a^{4} e^{5} - 2 \, A a^{3} b e^{5}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{B a b^{5} d^{5} + A b^{6} d^{5} - 15 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 50 \, B a^{3} b^{3} d^{3} e^{2} - 30 \, A a^{2} b^{4} d^{3} e^{2} - 70 \, B a^{4} b^{2} d^{2} e^{3} + 50 \, A a^{3} b^{3} d^{2} e^{3} + 45 \, B a^{5} b d e^{4} - 35 \, A a^{4} b^{2} d e^{4} - 11 \, B a^{6} e^{5} + 9 \, A a^{5} b e^{5} + 2 \,{\left (B b^{6} d^{5} - 10 \, B a b^{5} d^{4} e + 5 \, A b^{6} d^{4} e + 30 \, B a^{2} b^{4} d^{3} e^{2} - 20 \, A a b^{5} d^{3} e^{2} - 40 \, B a^{3} b^{3} d^{2} e^{3} + 30 \, A a^{2} b^{4} d^{2} e^{3} + 25 \, B a^{4} b^{2} d e^{4} - 20 \, A a^{3} b^{3} d e^{4} - 6 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{7}} + \frac{3 \, B b^{9} x^{4} e^{5} + 20 \, B b^{9} d x^{3} e^{4} + 60 \, B b^{9} d^{2} x^{2} e^{3} + 120 \, B b^{9} d^{3} x e^{2} - 12 \, B a b^{8} x^{3} e^{5} + 4 \, A b^{9} x^{3} e^{5} - 90 \, B a b^{8} d x^{2} e^{4} + 30 \, A b^{9} d x^{2} e^{4} - 360 \, B a b^{8} d^{2} x e^{3} + 120 \, A b^{9} d^{2} x e^{3} + 36 \, B a^{2} b^{7} x^{2} e^{5} - 18 \, A a b^{8} x^{2} e^{5} + 360 \, B a^{2} b^{7} d x e^{4} - 180 \, A a b^{8} d x e^{4} - 120 \, B a^{3} b^{6} x e^{5} + 72 \, A a^{2} b^{7} x e^{5}}{12 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^5/(b*x + a)^3,x, algorithm="giac")
[Out]