### 3.1980 $$\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^{5/2}} \, dx$$

Optimal. Leaf size=39 $\frac{2 c d \sqrt{d+e x}}{e^2}-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{\sqrt{d+e x}}$

[Out]

(-2*(a - (c*d^2)/e^2))/Sqrt[d + e*x] + (2*c*d*Sqrt[d + e*x])/e^2

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Rubi [A]  time = 0.0231704, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {24, 43} $\frac{2 c d \sqrt{d+e x}}{e^2}-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{\sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(a - (c*d^2)/e^2))/Sqrt[d + e*x] + (2*c*d*Sqrt[d + e*x])/e^2

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0185505, size = 31, normalized size = 0.79 $\frac{2 c d (2 d+e x)-2 a e^2}{e^2 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*a*e^2 + 2*c*d*(2*d + e*x))/(e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.041, size = 31, normalized size = 0.8 \begin{align*} -2\,{\frac{-cdex+a{e}^{2}-2\,c{d}^{2}}{\sqrt{ex+d}{e}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(e*x+d)^(1/2)*(-c*d*e*x+a*e^2-2*c*d^2)/e^2

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Maxima [A]  time = 1.00864, size = 57, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{e x + d} c d}{e} + \frac{c d^{2} - a e^{2}}{\sqrt{e x + d} e}\right )}}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(e*x + d)*c*d/e + (c*d^2 - a*e^2)/(sqrt(e*x + d)*e))/e

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Fricas [A]  time = 2.0821, size = 82, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{e^{3} x + d e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.56779, size = 58, normalized size = 1.49 \begin{align*} \begin{cases} - \frac{2 a}{\sqrt{d + e x}} + \frac{4 c d^{2}}{e^{2} \sqrt{d + e x}} + \frac{2 c d x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 \sqrt{d}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a/sqrt(d + e*x) + 4*c*d**2/(e**2*sqrt(d + e*x)) + 2*c*d*x/(e*sqrt(d + e*x)), Ne(e, 0)), (c*x**2/
(2*sqrt(d)), True))

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Giac [A]  time = 1.16531, size = 68, normalized size = 1.74 \begin{align*} 2 \, \sqrt{x e + d} c d e^{\left (-2\right )} + \frac{2 \,{\left ({\left (x e + d\right )} c d^{2} -{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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