### 3.1053 $$\int (c d^2+2 c d e x+c e^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=36 $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e}$

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2))/(6*e)

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Rubi [A]  time = 0.0067982, 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} $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2))/(6*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 )^{5/2} \, dx &=\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e}\\ \end{align*}

Mathematica [A]  time = 0.0017505, size = 25, normalized size = 0.69 $\frac{(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^(5/2))/(6*e)

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Maple [B]  time = 0.041, size = 84, normalized size = 2.3 \begin{align*}{\frac{x \left ({e}^{5}{x}^{5}+6\,d{e}^{4}{x}^{4}+15\,{d}^{2}{e}^{3}{x}^{3}+20\,{d}^{3}{e}^{2}{x}^{2}+15\,{d}^{4}ex+6\,{d}^{5} \right ) }{6\, \left ( ex+d \right ) ^{5}} \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((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

1/6*x*(e^5*x^5+6*d*e^4*x^4+15*d^2*e^3*x^3+20*d^3*e^2*x^2+15*d^4*e*x+6*d^5)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(
e*x+d)^5

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.35627, size = 213, normalized size = 5.92 \begin{align*} \frac{{\left (c^{2} e^{5} x^{6} + 6 \, c^{2} d e^{4} x^{5} + 15 \, c^{2} d^{2} e^{3} x^{4} + 20 \, c^{2} d^{3} e^{2} x^{3} + 15 \, c^{2} d^{4} e x^{2} + 6 \, c^{2} d^{5} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \,{\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)^(5/2),x, algorithm="fricas")

[Out]

1/6*(c^2*e^5*x^6 + 6*c^2*d*e^4*x^5 + 15*c^2*d^2*e^3*x^4 + 20*c^2*d^3*e^2*x^3 + 15*c^2*d^4*e*x^2 + 6*c^2*d^5*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{5}{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)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.31023, size = 120, normalized size = 3.33 \begin{align*} \frac{1}{6} \,{\left (c^{2} d^{5} e^{\left (-1\right )} +{\left (5 \, c^{2} d^{4} +{\left (10 \, c^{2} d^{3} e +{\left (10 \, c^{2} d^{2} e^{2} +{\left (c^{2} x e^{4} + 5 \, c^{2} d e^{3}\right )} x\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((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="giac")

[Out]

1/6*(c^2*d^5*e^(-1) + (5*c^2*d^4 + (10*c^2*d^3*e + (10*c^2*d^2*e^2 + (c^2*x*e^4 + 5*c^2*d*e^3)*x)*x)*x)*x)*sqr
t(c*x^2*e^2 + 2*c*d*x*e + c*d^2)