### 3.1981 $$\int \frac{a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=41 $-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}$

[Out]

(-2*(a - (c*d^2)/e^2))/(3*(d + e*x)^(3/2)) - (2*c*d)/(e^2*Sqrt[d + e*x])

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Rubi [A]  time = 0.0221359, antiderivative size = 41, 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} $-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(a - (c*d^2)/e^2))/(3*(d + e*x)^(3/2)) - (2*c*d)/(e^2*Sqrt[d + e*x])

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)^{7/2}} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^{5/2}} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^{5/2}}+\frac{c d e}{(d+e x)^{3/2}}\right ) \, dx}{e^2}\\ &=-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0207019, size = 33, normalized size = 0.8 $-\frac{2 \left (a e^2+c d (2 d+3 e x)\right )}{3 e^2 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.041, size = 31, normalized size = 0.8 \begin{align*} -{\frac{6\,cdex+2\,a{e}^{2}+4\,c{d}^{2}}{3\,{e}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{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)^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.03301, size = 45, normalized size = 1.1 \begin{align*} -\frac{2 \,{\left (3 \,{\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}}{3 \,{\left (e x + d\right )}^{\frac{3}{2}} 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)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.1431, size = 111, normalized size = 2.71 \begin{align*} -\frac{2 \,{\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \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)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.62371, size = 126, normalized size = 3.07 \begin{align*} \begin{cases} - \frac{2 a e^{2}}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} - \frac{4 c d^{2}}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} - \frac{6 c d e x}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 d^{\frac{3}{2}}} & \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)**(7/2),x)

[Out]

Piecewise((-2*a*e**2/(3*d*e**2*sqrt(d + e*x) + 3*e**3*x*sqrt(d + e*x)) - 4*c*d**2/(3*d*e**2*sqrt(d + e*x) + 3*
e**3*x*sqrt(d + e*x)) - 6*c*d*e*x/(3*d*e**2*sqrt(d + e*x) + 3*e**3*x*sqrt(d + e*x)), Ne(e, 0)), (c*x**2/(2*d**
(3/2)), True))

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Giac [A]  time = 1.15023, size = 63, normalized size = 1.54 \begin{align*} -\frac{2 \,{\left (3 \,{\left (x e + d\right )}^{2} c d -{\left (x e + d\right )} c d^{2} +{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{5}{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)^(7/2),x, algorithm="giac")

[Out]

-2/3*(3*(x*e + d)^2*c*d - (x*e + d)*c*d^2 + (x*e + d)*a*e^2)*e^(-2)/(x*e + d)^(5/2)