3.2363 \(\int \frac{x (a+b x+c x^2)}{(d+e x)^7} \, dx\)

Optimal. Leaf size=103 \[ -\frac{3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac{d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{3 c d-b e}{4 e^4 (d+e x)^4}-\frac{c}{3 e^4 (d+e x)^3} \]

[Out]

(d*(c*d^2 - b*d*e + a*e^2))/(6*e^4*(d + e*x)^6) - (3*c*d^2 - e*(2*b*d - a*e))/(5*e^4*(d + e*x)^5) + (3*c*d - b
*e)/(4*e^4*(d + e*x)^4) - c/(3*e^4*(d + e*x)^3)

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Rubi [A]  time = 0.0731911, 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} \[ -\frac{3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac{d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{3 c d-b e}{4 e^4 (d+e x)^4}-\frac{c}{3 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x + c*x^2))/(d + e*x)^7,x]

[Out]

(d*(c*d^2 - b*d*e + a*e^2))/(6*e^4*(d + e*x)^6) - (3*c*d^2 - e*(2*b*d - a*e))/(5*e^4*(d + e*x)^5) + (3*c*d - b
*e)/(4*e^4*(d + e*x)^4) - c/(3*e^4*(d + e*x)^3)

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)^7} \, dx &=\int \left (-\frac{d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^7}+\frac{3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^6}+\frac{-3 c d+b e}{e^3 (d+e x)^5}+\frac{c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac{d \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}-\frac{3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac{3 c d-b e}{4 e^4 (d+e x)^4}-\frac{c}{3 e^4 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0312265, size = 77, normalized size = 0.75 \[ -\frac{e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*x + c*x^2))/(d + e*x)^7,x]

[Out]

-(c*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + e*(2*a*e*(d + 6*e*x) + b*(d^2 + 6*d*e*x + 15*e^2*x^2)))/(6
0*e^4*(d + e*x)^6)

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Maple [A]  time = 0.006, size = 93, normalized size = 0.9 \begin{align*}{\frac{d \left ( a{e}^{2}-bde+c{d}^{2} \right ) }{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{a{e}^{2}-2\,bde+3\,c{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{c}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{be-3\,cd}{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)^7,x)

[Out]

1/6*d*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^6-1/5*(a*e^2-2*b*d*e+3*c*d^2)/e^4/(e*x+d)^5-1/3*c/e^4/(e*x+d)^3-1/4*(b*e
-3*c*d)/e^4/(e*x+d)^4

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Maxima [A]  time = 1.18404, size = 185, normalized size = 1.8 \begin{align*} -\frac{20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \,{\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \,{\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/60*(20*c*e^3*x^3 + c*d^3 + b*d^2*e + 2*a*d*e^2 + 15*(c*d*e^2 + b*e^3)*x^2 + 6*(c*d^2*e + b*d*e^2 + 2*a*e^3)
*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

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Fricas [A]  time = 1.23896, size = 290, normalized size = 2.82 \begin{align*} -\frac{20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \,{\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \,{\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/60*(20*c*e^3*x^3 + c*d^3 + b*d^2*e + 2*a*d*e^2 + 15*(c*d*e^2 + b*e^3)*x^2 + 6*(c*d^2*e + b*d*e^2 + 2*a*e^3)
*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

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Sympy [A]  time = 6.58419, size = 148, normalized size = 1.44 \begin{align*} - \frac{2 a d e^{2} + b d^{2} e + c d^{3} + 20 c e^{3} x^{3} + x^{2} \left (15 b e^{3} + 15 c d e^{2}\right ) + x \left (12 a e^{3} + 6 b d e^{2} + 6 c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**2+b*x+a)/(e*x+d)**7,x)

[Out]

-(2*a*d*e**2 + b*d**2*e + c*d**3 + 20*c*e**3*x**3 + x**2*(15*b*e**3 + 15*c*d*e**2) + x*(12*a*e**3 + 6*b*d*e**2
+ 6*c*d**2*e))/(60*d**6*e**4 + 360*d**5*e**5*x + 900*d**4*e**6*x**2 + 1200*d**3*e**7*x**3 + 900*d**2*e**8*x**
4 + 360*d*e**9*x**5 + 60*e**10*x**6)

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Giac [A]  time = 1.09279, size = 105, normalized size = 1.02 \begin{align*} -\frac{{\left (20 \, c x^{3} e^{3} + 15 \, c d x^{2} e^{2} + 6 \, c d^{2} x e + c d^{3} + 15 \, b x^{2} e^{3} + 6 \, b d x e^{2} + b d^{2} e + 12 \, a x e^{3} + 2 \, a d e^{2}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(20*c*x^3*e^3 + 15*c*d*x^2*e^2 + 6*c*d^2*x*e + c*d^3 + 15*b*x^2*e^3 + 6*b*d*x*e^2 + b*d^2*e + 12*a*x*e^3
+ 2*a*d*e^2)*e^(-4)/(x*e + d)^6