3.1295 \(\int \frac{(A+B x) \left (a+c x^2\right )}{(d+e x)^6} \, dx\)

Optimal. Leaf size=108 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{4 e^4 (d+e x)^4}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^5}+\frac{c (3 B d-A e)}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2} \]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2))/(5*e^4*(d + e*x)^5) - (3*B*c*d^2 - 2*A*c*d*e + a*B
*e^2)/(4*e^4*(d + e*x)^4) + (c*(3*B*d - A*e))/(3*e^4*(d + e*x)^3) - (B*c)/(2*e^4
*(d + e*x)^2)

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Rubi [A]  time = 0.177615, antiderivative size = 108, 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 \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{4 e^4 (d+e x)^4}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^5}+\frac{c (3 B d-A e)}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2))/(d + e*x)^6,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2))/(5*e^4*(d + e*x)^5) - (3*B*c*d^2 - 2*A*c*d*e + a*B
*e^2)/(4*e^4*(d + e*x)^4) + (c*(3*B*d - A*e))/(3*e^4*(d + e*x)^3) - (B*c)/(2*e^4
*(d + e*x)^2)

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Rubi in Sympy [A]  time = 28.7684, size = 102, normalized size = 0.94 \[ - \frac{B c}{2 e^{4} \left (d + e x\right )^{2}} - \frac{c \left (A e - 3 B d\right )}{3 e^{4} \left (d + e x\right )^{3}} - \frac{- 2 A c d e + B a e^{2} + 3 B c d^{2}}{4 e^{4} \left (d + e x\right )^{4}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{5 e^{4} \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)/(e*x+d)**6,x)

[Out]

-B*c/(2*e**4*(d + e*x)**2) - c*(A*e - 3*B*d)/(3*e**4*(d + e*x)**3) - (-2*A*c*d*e
+ B*a*e**2 + 3*B*c*d**2)/(4*e**4*(d + e*x)**4) - (A*e - B*d)*(a*e**2 + c*d**2)/
(5*e**4*(d + e*x)**5)

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Mathematica [A]  time = 0.0834445, size = 90, normalized size = 0.83 \[ -\frac{2 A e \left (6 a e^2+c \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 B \left (a e^2 (d+5 e x)+c \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^6,x]

[Out]

-(2*A*e*(6*a*e^2 + c*(d^2 + 5*d*e*x + 10*e^2*x^2)) + 3*B*(a*e^2*(d + 5*e*x) + c*
(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)))/(60*e^4*(d + e*x)^5)

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Maple [A]  time = 0.009, size = 110, normalized size = 1. \[ -{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{c \left ( Ae-3\,Bd \right ) }{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Bc}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-Bc{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)/(e*x+d)^6,x)

[Out]

-1/4*(-2*A*c*d*e+B*a*e^2+3*B*c*d^2)/e^4/(e*x+d)^4-1/3*c*(A*e-3*B*d)/e^4/(e*x+d)^
3-1/2*B*c/e^4/(e*x+d)^2-1/5*(A*a*e^3+A*c*d^2*e-B*a*d*e^2-B*c*d^3)/e^4/(e*x+d)^5

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Maxima [A]  time = 0.706659, size = 200, normalized size = 1.85 \[ -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 2 \, A c d^{2} e + 3 \, B a d e^{2} + 12 \, A a e^{3} + 10 \,{\left (3 \, B c d e^{2} + 2 \, A c e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 2 \, A c d e^{2} + 3 \, B a e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^6,x, algorithm="maxima")

[Out]

-1/60*(30*B*c*e^3*x^3 + 3*B*c*d^3 + 2*A*c*d^2*e + 3*B*a*d*e^2 + 12*A*a*e^3 + 10*
(3*B*c*d*e^2 + 2*A*c*e^3)*x^2 + 5*(3*B*c*d^2*e + 2*A*c*d*e^2 + 3*B*a*e^3)*x)/(e^
9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Fricas [A]  time = 0.266864, size = 200, normalized size = 1.85 \[ -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 2 \, A c d^{2} e + 3 \, B a d e^{2} + 12 \, A a e^{3} + 10 \,{\left (3 \, B c d e^{2} + 2 \, A c e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 2 \, A c d e^{2} + 3 \, B a e^{3}\right )} x}{60 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^6,x, algorithm="fricas")

[Out]

-1/60*(30*B*c*e^3*x^3 + 3*B*c*d^3 + 2*A*c*d^2*e + 3*B*a*d*e^2 + 12*A*a*e^3 + 10*
(3*B*c*d*e^2 + 2*A*c*e^3)*x^2 + 5*(3*B*c*d^2*e + 2*A*c*d*e^2 + 3*B*a*e^3)*x)/(e^
9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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Sympy [A]  time = 36.7262, size = 163, normalized size = 1.51 \[ - \frac{12 A a e^{3} + 2 A c d^{2} e + 3 B a d e^{2} + 3 B c d^{3} + 30 B c e^{3} x^{3} + x^{2} \left (20 A c e^{3} + 30 B c d e^{2}\right ) + x \left (10 A c d e^{2} + 15 B a e^{3} + 15 B c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)/(e*x+d)**6,x)

[Out]

-(12*A*a*e**3 + 2*A*c*d**2*e + 3*B*a*d*e**2 + 3*B*c*d**3 + 30*B*c*e**3*x**3 + x*
*2*(20*A*c*e**3 + 30*B*c*d*e**2) + x*(10*A*c*d*e**2 + 15*B*a*e**3 + 15*B*c*d**2*
e))/(60*d**5*e**4 + 300*d**4*e**5*x + 600*d**3*e**6*x**2 + 600*d**2*e**7*x**3 +
300*d*e**8*x**4 + 60*e**9*x**5)

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GIAC/XCAS [A]  time = 0.280367, size = 128, normalized size = 1.19 \[ -\frac{{\left (30 \, B c x^{3} e^{3} + 30 \, B c d x^{2} e^{2} + 15 \, B c d^{2} x e + 3 \, B c d^{3} + 20 \, A c x^{2} e^{3} + 10 \, A c d x e^{2} + 2 \, A c d^{2} e + 15 \, B a x e^{3} + 3 \, B a d e^{2} + 12 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^6,x, algorithm="giac")

[Out]

-1/60*(30*B*c*x^3*e^3 + 30*B*c*d*x^2*e^2 + 15*B*c*d^2*x*e + 3*B*c*d^3 + 20*A*c*x
^2*e^3 + 10*A*c*d*x*e^2 + 2*A*c*d^2*e + 15*B*a*x*e^3 + 3*B*a*d*e^2 + 12*A*a*e^3)
*e^(-4)/(x*e + d)^5