3.2362 \(\int \frac{x (a+b x+c x^2)}{(d+e x)^6} \, dx\)

Optimal. Leaf size=103 \[ -\frac{3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac{d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac{3 c d-b e}{3 e^4 (d+e x)^3}-\frac{c}{2 e^4 (d+e x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0749924, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ -\frac{3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac{d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac{3 c d-b e}{3 e^4 (d+e x)^3}-\frac{c}{2 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0327718, size = 79, normalized size = 0.77 \[ -\frac{e \left (3 a e (d+5 e x)+2 b \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 c \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )}{60 e^4 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 93, normalized size = 0.9 \begin{align*}{\frac{d \left ( a{e}^{2}-bde+c{d}^{2} \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{be-3\,cd}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{a{e}^{2}-2\,bde+3\,c{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.2618, size = 178, normalized size = 1.73 \begin{align*} -\frac{30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \,{\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \,{\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/60*(30*c*e^3*x^3 + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2 + 10*(3*c*d*e^2 + 2*b*e^3)*x^2 + 5*(3*c*d^2*e + 2*b*d*e^
2 + 3*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)

________________________________________________________________________________________

Fricas [A]  time = 1.35966, size = 282, normalized size = 2.74 \begin{align*} -\frac{30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \,{\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \,{\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/60*(30*c*e^3*x^3 + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2 + 10*(3*c*d*e^2 + 2*b*e^3)*x^2 + 5*(3*c*d^2*e + 2*b*d*e^
2 + 3*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)

________________________________________________________________________________________

Sympy [A]  time = 4.0645, size = 139, normalized size = 1.35 \begin{align*} - \frac{3 a d e^{2} + 2 b d^{2} e + 3 c d^{3} + 30 c e^{3} x^{3} + x^{2} \left (20 b e^{3} + 30 c d e^{2}\right ) + x \left (15 a e^{3} + 10 b d e^{2} + 15 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(x*(c*x**2+b*x+a)/(e*x+d)**6,x)

[Out]

-(3*a*d*e**2 + 2*b*d**2*e + 3*c*d**3 + 30*c*e**3*x**3 + x**2*(20*b*e**3 + 30*c*d*e**2) + x*(15*a*e**3 + 10*b*d
*e**2 + 15*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)

________________________________________________________________________________________

Giac [A]  time = 1.10845, size = 108, normalized size = 1.05 \begin{align*} -\frac{{\left (30 \, c x^{3} e^{3} + 30 \, c d x^{2} e^{2} + 15 \, c d^{2} x e + 3 \, c d^{3} + 20 \, b x^{2} e^{3} + 10 \, b d x e^{2} + 2 \, b d^{2} e + 15 \, a x e^{3} + 3 \, a d e^{2}\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(x*(c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="giac")

[Out]

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