3.2316 \(\int \frac{(A+B x) (a+b x+c x^2)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=134 \[ \frac{A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{4 e^4 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac{-A c e-b B e+3 B c d}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.107743, antiderivative size = 133, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ -\frac{-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{4 e^4 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac{-A c e-b B e+3 B c d}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^6}+\frac{3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{e^3 (d+e x)^5}+\frac{-3 B c d+b B e+A c e}{e^3 (d+e x)^4}+\frac{B c}{e^3 (d+e x)^3}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )}{5 e^4 (d+e x)^5}-\frac{3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{4 e^4 (d+e x)^4}+\frac{3 B c d-b B e-A c e}{3 e^4 (d+e x)^3}-\frac{B c}{2 e^4 (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0702298, size = 122, normalized size = 0.91 \[ -\frac{A e \left (3 e (4 a e+b d+5 b e x)+2 c \left (d^2+5 d e x+10 e^2 x^2\right )\right )+B \left (e \left (3 a e (d+5 e x)+2 b \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 c \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 142, normalized size = 1.1 \begin{align*} -{\frac{aA{e}^{3}-Abd{e}^{2}+Ac{d}^{2}e-aBd{e}^{2}+B{d}^{2}be-Bc{d}^{3}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{Ace+bBe-3\,Bcd}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Bc}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{Ab{e}^{2}-2\,Acde+aB{e}^{2}-2\,Bbde+3\,Bc{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03208, size = 234, normalized size = 1.75 \begin{align*} -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 12 \, A a e^{3} + 2 \,{\left (B b + A c\right )} d^{2} e + 3 \,{\left (B a + A b\right )} d e^{2} + 10 \,{\left (3 \, B c d e^{2} + 2 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 2 \,{\left (B b + A c\right )} d e^{2} + 3 \,{\left (B a + A b\right )} 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 + 12*A*a*e^3 + 2*(B*b + A*c)*d^2*e + 3*(B*a + A*b)*d*e^2 + 10*(3*B*c*d*e^2 +
2*(B*b + A*c)*e^3)*x^2 + 5*(3*B*c*d^2*e + 2*(B*b + A*c)*d*e^2 + 3*(B*a + A*b)*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 = 1.32, size = 378, normalized size = 2.82 \begin{align*} -\frac{30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 12 \, A a e^{3} + 2 \,{\left (B b + A c\right )} d^{2} e + 3 \,{\left (B a + A b\right )} d e^{2} + 10 \,{\left (3 \, B c d e^{2} + 2 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B c d^{2} e + 2 \,{\left (B b + A c\right )} d e^{2} + 3 \,{\left (B a + A b\right )} 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+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 + 12*A*a*e^3 + 2*(B*b + A*c)*d^2*e + 3*(B*a + A*b)*d*e^2 + 10*(3*B*c*d*e^2 +
2*(B*b + A*c)*e^3)*x^2 + 5*(3*B*c*d^2*e + 2*(B*b + A*c)*d*e^2 + 3*(B*a + A*b)*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 = 147.672, size = 211, normalized size = 1.57 \begin{align*} - \frac{12 A a e^{3} + 3 A b d e^{2} + 2 A c d^{2} e + 3 B a d e^{2} + 2 B b d^{2} e + 3 B c d^{3} + 30 B c e^{3} x^{3} + x^{2} \left (20 A c e^{3} + 20 B b e^{3} + 30 B c d e^{2}\right ) + x \left (15 A b e^{3} + 10 A c d e^{2} + 15 B a e^{3} + 10 B b d e^{2} + 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12*A*a*e**3 + 3*A*b*d*e**2 + 2*A*c*d**2*e + 3*B*a*d*e**2 + 2*B*b*d**2*e + 3*B*c*d**3 + 30*B*c*e**3*x**3 + x*
*2*(20*A*c*e**3 + 20*B*b*e**3 + 30*B*c*d*e**2) + x*(15*A*b*e**3 + 10*A*c*d*e**2 + 15*B*a*e**3 + 10*B*b*d*e**2
+ 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 [A]  time = 1.11754, size = 182, normalized size = 1.36 \begin{align*} -\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 \, B b x^{2} e^{3} + 20 \, A c x^{2} e^{3} + 10 \, B b d x e^{2} + 10 \, A c d x e^{2} + 2 \, B b d^{2} e + 2 \, A c d^{2} e + 15 \, B a x e^{3} + 15 \, A b x e^{3} + 3 \, B a d e^{2} + 3 \, A b d e^{2} + 12 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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