### 3.458 $$\int \frac{a+c x^2}{(d+e x)^4} \, dx$$

Optimal. Leaf size=52 $-\frac{a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2}$

[Out]

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

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Rubi [A]  time = 0.0299414, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {697} $-\frac{a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac{c}{e^3 (d+e x)}+\frac{c d}{e^3 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)/(d + e*x)^4,x]

[Out]

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

Rule 697

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

Rubi steps

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

Mathematica [A]  time = 0.0134262, size = 39, normalized size = 0.75 $-\frac{a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)/(d + e*x)^4,x]

[Out]

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

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Maple [A]  time = 0.048, size = 51, normalized size = 1. \begin{align*} -{\frac{a{e}^{2}+c{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{{e}^{3} \left ( ex+d \right ) }}+{\frac{cd}{{e}^{3} \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 1.18936, size = 85, normalized size = 1.63 \begin{align*} -\frac{3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.91301, size = 130, normalized size = 2.5 \begin{align*} -\frac{3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.77414, size = 66, normalized size = 1.27 \begin{align*} - \frac{a e^{2} + c d^{2} + 3 c d e x + 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(e*x+d)**4,x)

[Out]

-(a*e**2 + c*d**2 + 3*c*d*e*x + 3*c*e**2*x**2)/(3*d**3*e**3 + 9*d**2*e**4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

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Giac [A]  time = 1.28112, size = 50, normalized size = 0.96 \begin{align*} -\frac{{\left (3 \, c x^{2} e^{2} + 3 \, c d x e + c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^4,x, algorithm="giac")

[Out]

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