3.1990 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac{2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 c^2 d^2}{e^3 \sqrt{d+e x}} \]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*c*d*(c*d^2 - a*e^2))/(3*e^3*(d + e*x)^(3/2)) - (2*c^2*d^2)
/(e^3*Sqrt[d + e*x])

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Rubi [A]  time = 0.0395864, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {626, 43} \[ \frac{4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac{2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 c^2 d^2}{e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(11/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*c*d*(c*d^2 - a*e^2))/(3*e^3*(d + e*x)^(3/2)) - (2*c^2*d^2)
/(e^3*Sqrt[d + e*x])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{11/2}} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^{7/2}}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^{5/2}}+\frac{c^2 d^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^{5/2}}+\frac{4 c d \left (c d^2-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac{2 c^2 d^2}{e^3 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0370368, size = 67, normalized size = 0.83 \[ -\frac{2 \left (3 a^2 e^4+2 a c d e^2 (2 d+5 e x)+c^2 d^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(11/2),x]

[Out]

(-2*(3*a^2*e^4 + 2*a*c*d*e^2*(2*d + 5*e*x) + c^2*d^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2)
)

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Maple [A]  time = 0.044, size = 73, normalized size = 0.9 \begin{align*} -{\frac{30\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+20\,acd{e}^{3}x+40\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}+8\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{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)^2/(e*x+d)^(11/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(15*c^2*d^2*e^2*x^2+10*a*c*d*e^3*x+20*c^2*d^3*e*x+3*a^2*e^4+4*a*c*d^2*e^2+8*c^2*d^4)/e^3

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Maxima [A]  time = 1.02547, size = 104, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} c^{2} d^{2} + 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} - 10 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \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)^2/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

-2/15*(15*(e*x + d)^2*c^2*d^2 + 3*c^2*d^4 - 6*a*c*d^2*e^2 + 3*a^2*e^4 - 10*(c^2*d^3 - a*c*d*e^2)*(e*x + d))/((
e*x + d)^(5/2)*e^3)

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Fricas [A]  time = 1.72431, size = 220, normalized size = 2.72 \begin{align*} -\frac{2 \,{\left (15 \, c^{2} d^{2} e^{2} x^{2} + 8 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\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)^2/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

-2/15*(15*c^2*d^2*e^2*x^2 + 8*c^2*d^4 + 4*a*c*d^2*e^2 + 3*a^2*e^4 + 10*(2*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(e*x +
d)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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Sympy [A]  time = 27.4767, size = 388, normalized size = 4.79 \begin{align*} \begin{cases} - \frac{6 a^{2} e^{4}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{8 a c d^{2} e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{20 a c d e^{3} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 c^{2} d^{4}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 c^{2} d^{3} e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 c^{2} d^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{2} x^{3}}{3 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)**2/(e*x+d)**(11/2),x)

[Out]

Piecewise((-6*a**2*e**4/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x))
- 8*a*c*d**2*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 20*a
*c*d*e**3*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*c**2*d*
*4/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*c**2*d**3*e*x/(1
5*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*c**2*d**2*e**2*x**2/(
15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), (c**2*x**3/(3
*d**(3/2)), True))

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Giac [A]  time = 1.16608, size = 146, normalized size = 1.8 \begin{align*} -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{4} c^{2} d^{2} - 10 \,{\left (x e + d\right )}^{3} c^{2} d^{3} + 3 \,{\left (x e + d\right )}^{2} c^{2} d^{4} + 10 \,{\left (x e + d\right )}^{3} a c d e^{2} - 6 \,{\left (x e + d\right )}^{2} a c d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{2} a^{2} e^{4}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{9}{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)^2/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^4*c^2*d^2 - 10*(x*e + d)^3*c^2*d^3 + 3*(x*e + d)^2*c^2*d^4 + 10*(x*e + d)^3*a*c*d*e^2 - 6*
(x*e + d)^2*a*c*d^2*e^2 + 3*(x*e + d)^2*a^2*e^4)*e^(-3)/(x*e + d)^(9/2)