3.1063 \(\int \frac{(d+e x)^3}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0232877, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {643, 629} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c^2 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

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

Rule 643

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/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)^3}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac{\int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx}{c}\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c^2 e}\\ \end{align*}

Mathematica [A]  time = 0.0059435, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4}{3 e \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

(d + e*x)^4/(3*e*Sqrt[c*(d + e*x)^2])

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 49, normalized size = 1.4 \begin{align*}{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) \left ( ex+d \right ) }{3}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.21266, size = 197, normalized size = 5.79 \begin{align*} \frac{4 \, c^{2} d^{3} e^{4} \log \left (x + \frac{d}{e}\right )}{3 \, \left (c e^{2}\right )^{\frac{5}{2}}} - \frac{4 \, c d^{2} e^{3} x}{3 \, \left (c e^{2}\right )^{\frac{3}{2}}} + \frac{2 \, d e^{2} x^{2}}{3 \, \sqrt{c e^{2}}} - \frac{4}{3} \, d^{3} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) + \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e x^{2}}{3 \, c} + \frac{7 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{2}}{3 \, c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

4/3*c^2*d^3*e^4*log(x + d/e)/(c*e^2)^(5/2) - 4/3*c*d^2*e^3*x/(c*e^2)^(3/2) + 2/3*d*e^2*x^2/sqrt(c*e^2) - 4/3*d
^3*sqrt(1/(c*e^2))*log(x + d/e) + 1/3*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*e*x^2/c + 7/3*sqrt(c*e^2*x^2 + 2*c*d
*e*x + c*d^2)*d^2/(c*e)

________________________________________________________________________________________

Fricas [A]  time = 2.33509, size = 117, normalized size = 3.44 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \,{\left (c e x + c d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.0791, size = 114, normalized size = 3.35 \begin{align*} \begin{cases} \frac{d^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c e} + \frac{2 d x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c} + \frac{e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c} & \text{for}\: e \neq 0 \\\frac{d^{3} x}{\sqrt{c d^{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Piecewise((d**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*c*e) + 2*d*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/
(3*c) + e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*c), Ne(e, 0)), (d**3*x/sqrt(c*d**2), True))

________________________________________________________________________________________

Giac [A]  time = 1.3571, size = 68, normalized size = 2. \begin{align*} \frac{1}{3} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (x{\left (\frac{x e}{c} + \frac{2 \, d}{c}\right )} + \frac{d^{2} e^{\left (-1\right )}}{c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="giac")

[Out]

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