### 3.2116 $$\int \frac{a+b x+c x^2}{(d+e x)^5} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0464884, 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} $-\frac{a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac{2 c d-b e}{3 e^3 (d+e x)^3}-\frac{c}{2 e^3 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0182494, size = 49, normalized size = 0.71 $-\frac{e (3 a e+b (d+4 e x))+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 63, normalized size = 0.9 \begin{align*} -{\frac{a{e}^{2}-bde+c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{c}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{be-2\,cd}{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)^5,x)

[Out]

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

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Maxima [A]  time = 1.00307, size = 116, normalized size = 1.68 \begin{align*} -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 3 \, a e^{2} + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} 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)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.899, size = 180, normalized size = 2.61 \begin{align*} -\frac{6 \, c e^{2} x^{2} + c d^{2} + b d e + 3 \, a e^{2} + 4 \,{\left (c d e + b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.49977, size = 92, normalized size = 1.33 \begin{align*} - \frac{3 a e^{2} + b d e + c d^{2} + 6 c e^{2} x^{2} + x \left (4 b e^{2} + 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07926, size = 116, normalized size = 1.68 \begin{align*} -\frac{1}{12} \,{\left (\frac{6 \, c e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{8 \, c d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac{3 \, c d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac{4 \, b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="giac")

[Out]

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