### 3.1051 $$\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^{5/2} \, dx$$

Optimal. Leaf size=39 $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e}$

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))/(8*c*e)

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Rubi [A]  time = 0.0207566, antiderivative size = 39, 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 = {642, 609} $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(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)^(7/2))/(8*c*e)

Rule 642

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/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/2]

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 (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac{\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e}\\ \end{align*}

Mathematica [A]  time = 0.0200629, size = 28, normalized size = 0.72 $\frac{(d+e x) \left (c (d+e x)^2\right )^{7/2}}{8 c e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*(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)^(7/2))/(8*c*e)

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Maple [B]  time = 0.04, size = 106, normalized size = 2.7 \begin{align*}{\frac{x \left ({e}^{7}{x}^{7}+8\,d{e}^{6}{x}^{6}+28\,{d}^{2}{e}^{5}{x}^{5}+56\,{d}^{3}{e}^{4}{x}^{4}+70\,{d}^{4}{e}^{3}{x}^{3}+56\,{d}^{5}{e}^{2}{x}^{2}+28\,{d}^{6}ex+8\,{d}^{7} \right ) }{8\, \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)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

1/8*x*(e^7*x^7+8*d*e^6*x^6+28*d^2*e^5*x^5+56*d^3*e^4*x^4+70*d^4*e^3*x^3+56*d^5*e^2*x^2+28*d^6*e*x+8*d^7)*(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)^2*(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 = 2.50481, size = 270, normalized size = 6.92 \begin{align*} \frac{{\left (c^{2} e^{7} x^{8} + 8 \, c^{2} d e^{6} x^{7} + 28 \, c^{2} d^{2} e^{5} x^{6} + 56 \, c^{2} d^{3} e^{4} x^{5} + 70 \, c^{2} d^{4} e^{3} x^{4} + 56 \, c^{2} d^{5} e^{2} x^{3} + 28 \, c^{2} d^{6} e x^{2} + 8 \, c^{2} d^{7} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{8 \,{\left (e x + d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

1/8*(c^2*e^7*x^8 + 8*c^2*d*e^6*x^7 + 28*c^2*d^2*e^5*x^6 + 56*c^2*d^3*e^4*x^5 + 70*c^2*d^4*e^3*x^4 + 56*c^2*d^5
*e^2*x^3 + 28*c^2*d^6*e*x^2 + 8*c^2*d^7*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 \left (d + e x\right )^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20787, size = 155, normalized size = 3.97 \begin{align*} \frac{1}{8} \,{\left (c^{2} d^{7} e^{\left (-1\right )} +{\left (7 \, c^{2} d^{6} +{\left (21 \, c^{2} d^{5} e +{\left (35 \, c^{2} d^{4} e^{2} +{\left (35 \, c^{2} d^{3} e^{3} +{\left (21 \, c^{2} d^{2} e^{4} +{\left (c^{2} x e^{6} + 7 \, c^{2} d e^{5}\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)^2*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="giac")

[Out]

1/8*(c^2*d^7*e^(-1) + (7*c^2*d^6 + (21*c^2*d^5*e + (35*c^2*d^4*e^2 + (35*c^2*d^3*e^3 + (21*c^2*d^2*e^4 + (c^2*
x*e^6 + 7*c^2*d*e^5)*x)*x)*x)*x)*x)*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)