### 3.1998 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{11/2}} \, dx$$

Optimal. Leaf size=115 $-\frac{6 c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}+\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}$

[Out]

(2*(c*d^2 - a*e^2)^3)/(3*e^4*(d + e*x)^(3/2)) - (6*c*d*(c*d^2 - a*e^2)^2)/(e^4*Sqrt[d + e*x]) - (6*c^2*d^2*(c*
d^2 - a*e^2)*Sqrt[d + e*x])/e^4 + (2*c^3*d^3*(d + e*x)^(3/2))/(3*e^4)

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Rubi [A]  time = 0.0520726, antiderivative size = 115, 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} $-\frac{6 c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}+\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(11/2),x]

[Out]

(2*(c*d^2 - a*e^2)^3)/(3*e^4*(d + e*x)^(3/2)) - (6*c*d*(c*d^2 - a*e^2)^2)/(e^4*Sqrt[d + e*x]) - (6*c^2*d^2*(c*
d^2 - a*e^2)*Sqrt[d + e*x])/e^4 + (2*c^3*d^3*(d + e*x)^(3/2))/(3*e^4)

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 )^3}{(d+e x)^{11/2}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{5/2}}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{3/2}}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 \sqrt{d+e x}}+\frac{c^3 d^3 \sqrt{d+e x}}{e^3}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{e^4}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0598344, size = 110, normalized size = 0.96 $-\frac{2 \left (3 a^2 c d e^4 (2 d+3 e x)+a^3 e^6-3 a c^2 d^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^3 d^3 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(11/2),x]

[Out]

(-2*(a^3*e^6 + 3*a^2*c*d*e^4*(2*d + 3*e*x) - 3*a*c^2*d^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^3*d^3*(16*d^3
+ 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)))/(3*e^4*(d + e*x)^(3/2))

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Maple [A]  time = 0.044, size = 130, normalized size = 1.1 \begin{align*} -{\frac{-2\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}-18\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+12\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+18\,{a}^{2}cd{e}^{5}x-72\,a{c}^{2}{d}^{3}{e}^{3}x+48\,{c}^{3}{d}^{5}ex+2\,{a}^{3}{e}^{6}+12\,{a}^{2}c{d}^{2}{e}^{4}-48\,a{c}^{2}{d}^{4}{e}^{2}+32\,{c}^{3}{d}^{6}}{3\,{e}^{4}} \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)^3/(e*x+d)^(11/2),x)

[Out]

-2/3/(e*x+d)^(3/2)*(-c^3*d^3*e^3*x^3-9*a*c^2*d^2*e^4*x^2+6*c^3*d^4*e^2*x^2+9*a^2*c*d*e^5*x-36*a*c^2*d^3*e^3*x+
24*c^3*d^5*e*x+a^3*e^6+6*a^2*c*d^2*e^4-24*a*c^2*d^4*e^2+16*c^3*d^6)/e^4

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Maxima [A]  time = 1.01749, size = 190, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c^{3} d^{3} - 9 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 9 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, 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)^3/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

2/3*(((e*x + d)^(3/2)*c^3*d^3 - 9*(c^3*d^4 - a*c^2*d^2*e^2)*sqrt(e*x + d))/e^3 + (c^3*d^6 - 3*a*c^2*d^4*e^2 +
3*a^2*c*d^2*e^4 - a^3*e^6 - 9*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e

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Fricas [A]  time = 1.87473, size = 305, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 24 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 3 \,{\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \,{\left (8 \, c^{3} d^{5} e - 12 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\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)^3/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

2/3*(c^3*d^3*e^3*x^3 - 16*c^3*d^6 + 24*a*c^2*d^4*e^2 - 6*a^2*c*d^2*e^4 - a^3*e^6 - 3*(2*c^3*d^4*e^2 - 3*a*c^2*
d^2*e^4)*x^2 - 3*(8*c^3*d^5*e - 12*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*
e^4)

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Sympy [A]  time = 31.4873, size = 450, normalized size = 3.91 \begin{align*} \begin{cases} - \frac{2 a^{3} e^{6}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 c^{3} d^{6}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 c^{3} d^{5} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} \sqrt{d} x^{4}}{4} & \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)**3/(e*x+d)**(11/2),x)

[Out]

Piecewise((-2*a**3*e**6/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*a**2*c*d**2*e**4/(3*d*e**4*sqrt
(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 18*a**2*c*d*e**5*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 4
8*a*c**2*d**4*e**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 72*a*c**2*d**3*e**3*x/(3*d*e**4*sqrt(d
+ e*x) + 3*e**5*x*sqrt(d + e*x)) + 18*a*c**2*d**2*e**4*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x))
- 32*c**3*d**6/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*c**3*d**5*e*x/(3*d*e**4*sqrt(d + e*x) +
3*e**5*x*sqrt(d + e*x)) - 12*c**3*d**4*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*c**3*d*
*3*e**3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), (c**3*sqrt(d)*x**4/4, True))

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Giac [A]  time = 1.19244, size = 261, normalized size = 2.27 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt{x e + d} c^{3} d^{4} e^{8} + 9 \, \sqrt{x e + d} a c^{2} d^{2} e^{10}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )}^{4} c^{3} d^{5} -{\left (x e + d\right )}^{3} c^{3} d^{6} - 18 \,{\left (x e + d\right )}^{4} a c^{2} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} + 9 \,{\left (x e + d\right )}^{4} a^{2} c d e^{4} - 3 \,{\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} +{\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{3 \,{\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)^3/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*c^3*d^3*e^8 - 9*sqrt(x*e + d)*c^3*d^4*e^8 + 9*sqrt(x*e + d)*a*c^2*d^2*e^10)*e^(-12) - 2/3
*(9*(x*e + d)^4*c^3*d^5 - (x*e + d)^3*c^3*d^6 - 18*(x*e + d)^4*a*c^2*d^3*e^2 + 3*(x*e + d)^3*a*c^2*d^4*e^2 + 9
*(x*e + d)^4*a^2*c*d*e^4 - 3*(x*e + d)^3*a^2*c*d^2*e^4 + (x*e + d)^3*a^3*e^6)*e^(-4)/(x*e + d)^(9/2)