### 3.459 $$\int \frac{a+c x^2}{(d+e x)^5} \, dx$$

Optimal. Leaf size=57 $-\frac{a e^2+c d^2}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2}+\frac{2 c d}{3 e^3 (d+e x)^3}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0302744, antiderivative size = 57, 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} $-\frac{a e^2+c d^2}{4 e^3 (d+e x)^4}-\frac{c}{2 e^3 (d+e x)^2}+\frac{2 c d}{3 e^3 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0144593, size = 40, normalized size = 0.7 $-\frac{3 a e^2+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)/(d + e*x)^5,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 52, normalized size = 0.9 \begin{align*} -{\frac{a{e}^{2}+c{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{2\,cd}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{c}{2\,{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)^5,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.25182, size = 101, normalized size = 1.77 \begin{align*} -\frac{6 \, c e^{2} x^{2} + 4 \, c d e x + c d^{2} + 3 \, a e^{2}}{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+a)/(e*x+d)^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.84587, size = 155, normalized size = 2.72 \begin{align*} -\frac{6 \, c e^{2} x^{2} + 4 \, c d e x + c d^{2} + 3 \, a e^{2}}{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+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.914188, size = 80, normalized size = 1.4 \begin{align*} - \frac{3 a e^{2} + c d^{2} + 4 c d e x + 6 c e^{2} x^{2}}{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+a)/(e*x+d)**5,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.34556, size = 80, normalized size = 1.4 \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{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+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 + 3*a/(x*e + d)^4)*e^(-1
)