### 3.1042 $$\int (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx$$

Optimal. Leaf size=36 $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e}$

[Out]

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

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Rubi [A]  time = 0.0060748, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {609} $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

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

Mathematica [A]  time = 0.0018106, size = 25, normalized size = 0.69 $\frac{(d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 62, normalized size = 1.7 \begin{align*}{\frac{x \left ({e}^{3}{x}^{3}+4\,d{e}^{2}{x}^{2}+6\,{d}^{2}ex+4\,{d}^{3} \right ) }{4\, \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.18211, size = 144, normalized size = 4. \begin{align*} \frac{{\left (c e^{3} x^{4} + 4 \, c d e^{2} x^{3} + 6 \, c d^{2} e x^{2} + 4 \, c d^{3} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (e x + d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29678, size = 74, normalized size = 2.06 \begin{align*} \frac{1}{4} \,{\left (c d^{3} e^{\left (-1\right )} +{\left (3 \, c d^{2} +{\left (c x e^{2} + 3 \, c d e\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((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

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