3.1084 \(\int \frac{d+e x}{(c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0093726, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {629} \[ -\frac{1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

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

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=-\frac{1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0112833, size = 30, normalized size = 0.88 \[ -\frac{\sqrt{c (d+e x)^2}}{3 c^3 e (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[c*(d + e*x)^2]/(3*c^3*e*(d + e*x)^4)

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 35, normalized size = 1. \begin{align*} -{\frac{ \left ( ex+d \right ) ^{2}}{3\,e} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.57778, size = 41, normalized size = 1.21 \begin{align*} -\frac{1}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.18579, size = 167, normalized size = 4.91 \begin{align*} -\frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (c^{3} e^{5} x^{4} + 4 \, c^{3} d e^{4} x^{3} + 6 \, c^{3} d^{2} e^{3} x^{2} + 4 \, c^{3} d^{3} e^{2} x + c^{3} d^{4} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.64703, size = 124, normalized size = 3.65 \begin{align*} \begin{cases} - \frac{1}{3 c^{2} d^{2} e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}} + 6 c^{2} d e^{2} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}} + 3 c^{2} e^{3} x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{d x}{\left (c d^{2}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(3*c**2*d**2*e*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2) + 6*c**2*d*e**2*x*sqrt(c*d**2 + 2*c*d*e*x
+ c*e**2*x**2) + 3*c**2*e**3*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)), Ne(e, 0)), (d*x/(c*d**2)**(5/2), Tr
ue))

________________________________________________________________________________________

Giac [B]  time = 1.33218, size = 86, normalized size = 2.53 \begin{align*} \frac{6 \, C_{0} d^{3} e^{\left (-3\right )} + 6 \,{\left (3 \, C_{0} d^{2} e^{\left (-2\right )} +{\left (3 \, C_{0} d e^{\left (-1\right )} + C_{0} x\right )} x\right )} x - \frac{e^{\left (-1\right )}}{c}}{3 \,{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*C_0*d^3*e^(-3) + 6*(3*C_0*d^2*e^(-2) + (3*C_0*d*e^(-1) + C_0*x)*x)*x - e^(-1)/c)/(c*x^2*e^2 + 2*c*d*x*e
+ c*d^2)^(3/2)