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

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

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Rubi [A]  time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 77, normalized size = 0.76 \begin {gather*} -\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 (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(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)))/
(e^4*(d + e*x)^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 116, normalized size = 1.15 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.16, size = 128, normalized size = 1.27 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.05, size = 93, normalized size = 0.92 \begin {gather*} -\frac {c}{\left (e x +d \right ) e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {b e -3 c d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{3 \left (e x +d \right )^{3} e^{4}} \end {gather*}

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]

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

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maxima [A]  time = 0.47, size = 116, normalized size = 1.15 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.06, size = 111, normalized size = 1.10 \begin {gather*} -\frac {\frac {c\,x^3}{e}+\frac {d\,\left (3\,c\,d^2+b\,d\,e+a\,e^2\right )}{12\,e^4}+\frac {x\,\left (3\,c\,d^2+b\,d\,e+a\,e^2\right )}{3\,e^3}+\frac {x^2\,\left (b\,e+3\,c\,d\right )}{2\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.21, size = 126, normalized size = 1.25 \begin {gather*} \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 {gather*}

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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