### 3.2117 $$\int \frac{a+b x+c x^2}{(d+e x)^6} \, dx$$

Optimal. Leaf size=69 $-\frac{a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac{2 c d-b e}{4 e^3 (d+e x)^4}-\frac{c}{3 e^3 (d+e x)^3}$

[Out]

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

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Rubi [A]  time = 0.0442678, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {698} $-\frac{a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac{2 c d-b e}{4 e^3 (d+e x)^4}-\frac{c}{3 e^3 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)/(d + e*x)^6,x]

[Out]

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

Rule 698

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

Rubi steps

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

Mathematica [A]  time = 0.0204608, size = 51, normalized size = 0.74 $-\frac{3 e (4 a e+b (d+5 e x))+2 c \left (d^2+5 d e x+10 e^2 x^2\right )}{60 e^3 (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 63, normalized size = 0.9 \begin{align*} -{\frac{be-2\,cd}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{a{e}^{2}-bde+c{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{c}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00236, size = 136, normalized size = 1.97 \begin{align*} -\frac{20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \,{\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*c*e^2*x^2 + 2*c*d^2 + 3*b*d*e + 12*a*e^2 + 5*(2*c*d*e + 3*b*e^2)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*
e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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Fricas [A]  time = 2.01564, size = 217, normalized size = 3.14 \begin{align*} -\frac{20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \,{\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*c*e^2*x^2 + 2*c*d^2 + 3*b*d*e + 12*a*e^2 + 5*(2*c*d*e + 3*b*e^2)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*
e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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Sympy [A]  time = 2.10594, size = 107, normalized size = 1.55 \begin{align*} - \frac{12 a e^{2} + 3 b d e + 2 c d^{2} + 20 c e^{2} x^{2} + x \left (15 b e^{2} + 10 c d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12*a*e**2 + 3*b*d*e + 2*c*d**2 + 20*c*e**2*x**2 + x*(15*b*e**2 + 10*c*d*e))/(60*d**5*e**3 + 300*d**4*e**4*x
+ 600*d**3*e**5*x**2 + 600*d**2*e**6*x**3 + 300*d*e**7*x**4 + 60*e**8*x**5)

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Giac [A]  time = 1.11487, size = 69, normalized size = 1. \begin{align*} -\frac{{\left (20 \, c x^{2} e^{2} + 10 \, c d x e + 2 \, c d^{2} + 15 \, b x e^{2} + 3 \, b d e + 12 \, a e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*c*x^2*e^2 + 10*c*d*x*e + 2*c*d^2 + 15*b*x*e^2 + 3*b*d*e + 12*a*e^2)*e^(-3)/(x*e + d)^5