### 3.1050 $$\int (d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=34 $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e}$

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(9/2)/(9*c^2*e)

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Rubi [A]  time = 0.0238558, 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} $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(9/2)/(9*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 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac{\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e}\\ \end{align*}

Mathematica [A]  time = 0.0264728, size = 27, normalized size = 0.79 $\frac{(d+e x)^4 \left (c (d+e x)^2\right )^{5/2}}{9 e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

((d + e*x)^4*(c*(d + e*x)^2)^(5/2))/(9*e)

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Maple [B]  time = 0.04, size = 117, normalized size = 3.4 \begin{align*}{\frac{x \left ({e}^{8}{x}^{8}+9\,d{e}^{7}{x}^{7}+36\,{d}^{2}{e}^{6}{x}^{6}+84\,{d}^{3}{e}^{5}{x}^{5}+126\,{d}^{4}{e}^{4}{x}^{4}+126\,{d}^{5}{e}^{3}{x}^{3}+84\,{d}^{6}{e}^{2}{x}^{2}+36\,{d}^{7}ex+9\,{d}^{8} \right ) }{9\, \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((e*x+d)^3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

1/9*x*(e^8*x^8+9*d*e^7*x^7+36*d^2*e^6*x^6+84*d^3*e^5*x^5+126*d^4*e^4*x^4+126*d^5*e^3*x^3+84*d^6*e^2*x^2+36*d^7
*e*x+9*d^8)*(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((e*x+d)^3*(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 = 2.37562, size = 301, normalized size = 8.85 \begin{align*} \frac{{\left (c^{2} e^{8} x^{9} + 9 \, c^{2} d e^{7} x^{8} + 36 \, c^{2} d^{2} e^{6} x^{7} + 84 \, c^{2} d^{3} e^{5} x^{6} + 126 \, c^{2} d^{4} e^{4} x^{5} + 126 \, c^{2} d^{5} e^{3} x^{4} + 84 \, c^{2} d^{6} e^{2} x^{3} + 36 \, c^{2} d^{7} e x^{2} + 9 \, c^{2} d^{8} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{9 \,{\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)^(5/2),x, algorithm="fricas")

[Out]

1/9*(c^2*e^8*x^9 + 9*c^2*d*e^7*x^8 + 36*c^2*d^2*e^6*x^7 + 84*c^2*d^3*e^5*x^6 + 126*c^2*d^4*e^4*x^5 + 126*c^2*d
^5*e^3*x^4 + 84*c^2*d^6*e^2*x^3 + 36*c^2*d^7*e*x^2 + 9*c^2*d^8*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 = 19.2363, size = 374, normalized size = 11. \begin{align*} \begin{cases} \frac{c^{2} d^{8} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9 e} + \frac{8 c^{2} d^{7} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{28 c^{2} d^{6} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{56 c^{2} d^{5} e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{70 c^{2} d^{4} e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{56 c^{2} d^{3} e^{4} x^{5} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{28 c^{2} d^{2} e^{5} x^{6} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{8 c^{2} d e^{6} x^{7} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac{c^{2} e^{7} x^{8} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} & \text{for}\: e \neq 0 \\d^{3} 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)**3*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Piecewise((c**2*d**8*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(9*e) + 8*c**2*d**7*x*sqrt(c*d**2 + 2*c*d*e*x + c*
e**2*x**2)/9 + 28*c**2*d**6*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d**5*e**2*x**3*sqrt(c*d*
*2 + 2*c*d*e*x + c*e**2*x**2)/9 + 70*c**2*d**4*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d*
*3*e**4*x**5*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 28*c**2*d**2*e**5*x**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**
2*x**2)/9 + 8*c**2*d*e**6*x**7*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + c**2*e**7*x**8*sqrt(c*d**2 + 2*c*d*e
*x + c*e**2*x**2)/9, Ne(e, 0)), (d**3*x*(c*d**2)**(5/2), True))

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Giac [B]  time = 1.20719, size = 173, normalized size = 5.09 \begin{align*} \frac{1}{9} \,{\left (c^{2} d^{8} e^{\left (-1\right )} +{\left (8 \, c^{2} d^{7} +{\left (28 \, c^{2} d^{6} e +{\left (56 \, c^{2} d^{5} e^{2} +{\left (70 \, c^{2} d^{4} e^{3} +{\left (56 \, c^{2} d^{3} e^{4} +{\left (28 \, c^{2} d^{2} e^{5} +{\left (c^{2} x e^{7} + 8 \, c^{2} d e^{6}\right )} x\right )} x\right )} x\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((e*x+d)^3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="giac")

[Out]

1/9*(c^2*d^8*e^(-1) + (8*c^2*d^7 + (28*c^2*d^6*e + (56*c^2*d^5*e^2 + (70*c^2*d^4*e^3 + (56*c^2*d^3*e^4 + (28*c
^2*d^2*e^5 + (c^2*x*e^7 + 8*c^2*d*e^6)*x)*x)*x)*x)*x)*x)*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)