3.2361 \(\int \frac{x (a+b x+c x^2)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=101 \[ -\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.0733396, antiderivative size = 101, 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} \[ -\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0308944, size = 77, normalized size = 0.76 \[ -\frac{e \left (a e (d+4 e x)+b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 c \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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

[Out]

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

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Maxima [A]  time = 1.06958, size = 157, normalized size = 1.55 \begin{align*} -\frac{12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \,{\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} 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)^5,x, algorithm="maxima")

[Out]

-1/12*(12*c*e^3*x^3 + 3*c*d^3 + b*d^2*e + a*d*e^2 + 6*(3*c*d*e^2 + b*e^3)*x^2 + 4*(3*c*d^2*e + b*d*e^2 + a*e^3
)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)

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Fricas [A]  time = 1.3139, size = 243, normalized size = 2.41 \begin{align*} -\frac{12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \,{\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

-1/12*(12*c*e^3*x^3 + 3*c*d^3 + b*d^2*e + a*d*e^2 + 6*(3*c*d*e^2 + b*e^3)*x^2 + 4*(3*c*d^2*e + b*d*e^2 + a*e^3
)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)

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Sympy [A]  time = 2.57528, size = 124, normalized size = 1.23 \begin{align*} - \frac{a d e^{2} + b d^{2} e + 3 c d^{3} + 12 c e^{3} x^{3} + x^{2} \left (6 b e^{3} + 18 c d e^{2}\right ) + x \left (4 a e^{3} + 4 b d e^{2} + 12 c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**2+b*x+a)/(e*x+d)**5,x)

[Out]

-(a*d*e**2 + b*d**2*e + 3*c*d**3 + 12*c*e**3*x**3 + x**2*(6*b*e**3 + 18*c*d*e**2) + x*(4*a*e**3 + 4*b*d*e**2 +
12*c*d**2*e))/(12*d**4*e**4 + 48*d**3*e**5*x + 72*d**2*e**6*x**2 + 48*d*e**7*x**3 + 12*e**8*x**4)

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Giac [A]  time = 1.10768, size = 173, normalized size = 1.71 \begin{align*} -\frac{1}{12} \,{\left (\frac{12 \, c e^{\left (-1\right )}}{x e + d} - \frac{18 \, c d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac{12 \, c d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{6 \, b}{{\left (x e + d\right )}^{2}} - \frac{8 \, b d}{{\left (x e + d\right )}^{3}} + \frac{3 \, b d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, a e}{{\left (x e + d\right )}^{3}} - \frac{3 \, a d e}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\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)^5,x, algorithm="giac")

[Out]

-1/12*(12*c*e^(-1)/(x*e + d) - 18*c*d*e^(-1)/(x*e + d)^2 + 12*c*d^2*e^(-1)/(x*e + d)^3 - 3*c*d^3*e^(-1)/(x*e +
d)^4 + 6*b/(x*e + d)^2 - 8*b*d/(x*e + d)^3 + 3*b*d^2/(x*e + d)^4 + 4*a*e/(x*e + d)^3 - 3*a*d*e/(x*e + d)^4)*e
^(-3)