### 3.1029 $$\int (d+e x)^3 \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx$$

Optimal. Leaf size=34 $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e}$

[Out]

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

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Rubi [A]  time = 0.0235951, 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} $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e}$

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

Mathematica [A]  time = 0.0111601, size = 27, normalized size = 0.79 $\frac{(d+e x)^4 \sqrt{c (d+e x)^2}}{5 e}$

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*Sqrt[c*(d + e*x)^2])/(5*e)

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Maple [B]  time = 0.042, size = 73, normalized size = 2.2 \begin{align*}{\frac{x \left ({e}^{4}{x}^{4}+5\,d{e}^{3}{x}^{3}+10\,{d}^{2}{e}^{2}{x}^{2}+10\,{d}^{3}ex+5\,{d}^{4} \right ) }{5\,ex+5\,d}\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]

1/5*x*(e^4*x^4+5*d*e^3*x^3+10*d^2*e^2*x^2+10*d^3*e*x+5*d^4)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/(e*x+d)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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]

Exception raised: ValueError

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Fricas [B]  time = 2.15252, size = 158, normalized size = 4.65 \begin{align*} \frac{{\left (e^{4} x^{5} + 5 \, d e^{3} x^{4} + 10 \, d^{2} e^{2} x^{3} + 10 \, d^{3} e x^{2} + 5 \, d^{4} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \,{\left (e x + 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]

1/5*(e^4*x^5 + 5*d*e^3*x^4 + 10*d^2*e^2*x^3 + 10*d^3*e*x^2 + 5*d^4*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x
+ d)

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Sympy [A]  time = 0.823304, size = 187, normalized size = 5.5 \begin{align*} \begin{cases} \frac{d^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac{4 d^{3} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{6 d^{2} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{4 d e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\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**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(5*e) + 4*d**3*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)
/5 + 6*d**2*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5 + 4*d*e**2*x**3*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x
**2)/5 + e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5, Ne(e, 0)), (d**3*x*sqrt(c*d**2), True))

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Giac [A]  time = 1.20402, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{5} \,{\left (d^{4} e^{\left (-1\right )} +{\left (4 \, d^{3} +{\left (6 \, d^{2} e +{\left (x e^{3} + 4 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \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]

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