### 3.1999 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{13/2}} \, dx$$

Optimal. Leaf size=113 $\frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4}$

[Out]

(2*(c*d^2 - a*e^2)^3)/(5*e^4*(d + e*x)^(5/2)) - (2*c*d*(c*d^2 - a*e^2)^2)/(e^4*(d + e*x)^(3/2)) + (6*c^2*d^2*(
c*d^2 - a*e^2))/(e^4*Sqrt[d + e*x]) + (2*c^3*d^3*Sqrt[d + e*x])/e^4

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Rubi [A]  time = 0.0530466, antiderivative size = 113, 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} $\frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4}$

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)^(13/2),x]

[Out]

(2*(c*d^2 - a*e^2)^3)/(5*e^4*(d + e*x)^(5/2)) - (2*c*d*(c*d^2 - a*e^2)^2)/(e^4*(d + e*x)^(3/2)) + (6*c^2*d^2*(
c*d^2 - a*e^2))/(e^4*Sqrt[d + e*x]) + (2*c^3*d^3*Sqrt[d + e*x])/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)^{13/2}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{7/2}}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{5/2}}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^{3/2}}+\frac{c^3 d^3}{e^3 \sqrt{d+e x}}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.0561081, size = 109, normalized size = 0.96 $-\frac{2 \left (a^2 c d e^4 (2 d+5 e x)+a^3 e^6+a c^2 d^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-c^3 d^3 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )}{5 e^4 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(13/2),x]

[Out]

(-2*(a^3*e^6 + a^2*c*d*e^4*(2*d + 5*e*x) + a*c^2*d^2*e^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - c^3*d^3*(16*d^3 + 4
0*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)))/(5*e^4*(d + e*x)^(5/2))

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Maple [A]  time = 0.044, size = 130, normalized size = 1.2 \begin{align*} -{\frac{-10\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+30\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-60\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+10\,{a}^{2}cd{e}^{5}x+40\,a{c}^{2}{d}^{3}{e}^{3}x-80\,{c}^{3}{d}^{5}ex+2\,{a}^{3}{e}^{6}+4\,{a}^{2}c{d}^{2}{e}^{4}+16\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{5\,{e}^{4}} \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)^3/(e*x+d)^(13/2),x)

[Out]

-2/5/(e*x+d)^(5/2)*(-5*c^3*d^3*e^3*x^3+15*a*c^2*d^2*e^4*x^2-30*c^3*d^4*e^2*x^2+5*a^2*c*d*e^5*x+20*a*c^2*d^3*e^
3*x-40*c^3*d^5*e*x+a^3*e^6+2*a^2*c*d^2*e^4+8*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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Maxima [A]  time = 1.04851, size = 189, normalized size = 1.67 \begin{align*} \frac{2 \,{\left (\frac{5 \, \sqrt{e x + d} c^{3} d^{3}}{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} + 15 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 5 \,{\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{5}{2}} e^{3}}\right )}}{5 \, 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)^(13/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.93331, size = 323, normalized size = 2.86 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} + 16 \, c^{3} d^{6} - 8 \, a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \,{\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (8 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{5 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} 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)^(13/2),x, algorithm="fricas")

[Out]

2/5*(5*c^3*d^3*e^3*x^3 + 16*c^3*d^6 - 8*a*c^2*d^4*e^2 - 2*a^2*c*d^2*e^4 - a^3*e^6 + 15*(2*c^3*d^4*e^2 - a*c^2*
d^2*e^4)*x^2 + 5*(8*c^3*d^5*e - 4*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)*sqrt(e*x + d)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2
*e^5*x + d^3*e^4)

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Sympy [A]  time = 47.2321, size = 654, normalized size = 5.79 \begin{align*} \begin{cases} - \frac{2 a^{3} e^{6}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{4 a^{2} c d^{2} e^{4}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{10 a^{2} c d e^{5} x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{16 a c^{2} d^{4} e^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{40 a c^{2} d^{3} e^{3} x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{30 a c^{2} d^{2} e^{4} x^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{32 c^{3} d^{6}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{80 c^{3} d^{5} e x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{60 c^{3} d^{4} e^{2} x^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{10 c^{3} d^{3} e^{3} x^{3}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} x^{4}}{4 \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)**3/(e*x+d)**(13/2),x)

[Out]

Piecewise((-2*a**3*e**6/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) -
4*a**2*c*d**2*e**4/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 10*a*
*2*c*d*e**5*x/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 16*a*c**2*
d**4*e**2/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 40*a*c**2*d**3
*e**3*x/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) - 30*a*c**2*d**2*e
**4*x**2/(5*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 32*c**3*d**6/(5
*d**2*e**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 80*c**3*d**5*e*x/(5*d**2*e
**4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 60*c**3*d**4*e**2*x**2/(5*d**2*e*
*4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)) + 10*c**3*d**3*e**3*x**3/(5*d**2*e**
4*sqrt(d + e*x) + 10*d*e**5*x*sqrt(d + e*x) + 5*e**6*x**2*sqrt(d + e*x)), Ne(e, 0)), (c**3*x**4/(4*sqrt(d)), T
rue))

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Giac [A]  time = 1.23598, size = 252, normalized size = 2.23 \begin{align*} 2 \, \sqrt{x e + d} c^{3} d^{3} e^{\left (-4\right )} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{5} c^{3} d^{4} - 5 \,{\left (x e + d\right )}^{4} c^{3} d^{5} +{\left (x e + d\right )}^{3} c^{3} d^{6} - 15 \,{\left (x e + d\right )}^{5} a c^{2} d^{2} e^{2} + 10 \,{\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} - 5 \,{\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 )}}{5 \,{\left (x e + d\right )}^{\frac{11}{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)^(13/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*c^3*d^3*e^(-4) + 2/5*(15*(x*e + d)^5*c^3*d^4 - 5*(x*e + d)^4*c^3*d^5 + (x*e + d)^3*c^3*d^6 - 1
5*(x*e + d)^5*a*c^2*d^2*e^2 + 10*(x*e + d)^4*a*c^2*d^3*e^2 - 3*(x*e + d)^3*a*c^2*d^4*e^2 - 5*(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)^(11/2)