3.1997 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{9/2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0515285, 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} \[ -\frac{2 c^2 d^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{e^4}+\frac{6 c d \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{e^4}+\frac{2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt{d+e x}}+\frac{2 c^3 d^3 (d+e x)^{5/2}}{5 e^4} \]

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)^(9/2),x]

[Out]

(2*(c*d^2 - a*e^2)^3)/(e^4*Sqrt[d + e*x]) + (6*c*d*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/e^4 - (2*c^2*d^2*(c*d^2 -
a*e^2)*(d + e*x)^(3/2))/e^4 + (2*c^3*d^3*(d + e*x)^(5/2))/(5*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)^{9/2}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{3/2}}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 \sqrt{d+e x}}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{e^3}+\frac{c^3 d^3 (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt{d+e x}}+\frac{6 c d \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}{e^4}-\frac{2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^4}+\frac{2 c^3 d^3 (d+e x)^{5/2}}{5 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0568238, size = 109, normalized size = 0.96 \[ \frac{2 \left (15 a^2 c d e^4 (2 d+e x)-5 a^3 e^6-5 a c^2 d^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+c^3 d^3 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )}{5 e^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(9/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 131, normalized size = 1.2 \begin{align*} -{\frac{-2\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}-10\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+4\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-30\,{a}^{2}cd{e}^{5}x+40\,a{c}^{2}{d}^{3}{e}^{3}x-16\,{c}^{3}{d}^{5}ex+10\,{a}^{3}{e}^{6}-60\,{a}^{2}c{d}^{2}{e}^{4}+80\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{5\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \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)^(9/2),x)

[Out]

-2/5/(e*x+d)^(1/2)*(-c^3*d^3*e^3*x^3-5*a*c^2*d^2*e^4*x^2+2*c^3*d^4*e^2*x^2-15*a^2*c*d*e^5*x+20*a*c^2*d^3*e^3*x
-8*c^3*d^5*e*x+5*a^3*e^6-30*a^2*c*d^2*e^4+40*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

________________________________________________________________________________________

Maxima [A]  time = 1.05078, size = 194, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{5}{2}} c^{3} d^{3} - 5 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \sqrt{e x + d}}{e^{3}} + \frac{5 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}}{\sqrt{e x + d} 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)^(9/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.86707, size = 284, normalized size = 2.51 \begin{align*} \frac{2 \,{\left (c^{3} d^{3} e^{3} x^{3} + 16 \, c^{3} d^{6} - 40 \, a c^{2} d^{4} e^{2} + 30 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} -{\left (2 \, c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (8 \, c^{3} d^{5} e - 20 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{5 \,{\left (e^{5} x + d 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)^(9/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 16.4614, size = 230, normalized size = 2.04 \begin{align*} \begin{cases} - \frac{2 a^{3} e^{2}}{\sqrt{d + e x}} + \frac{12 a^{2} c d^{2}}{\sqrt{d + e x}} + \frac{6 a^{2} c d e x}{\sqrt{d + e x}} - \frac{16 a c^{2} d^{4}}{e^{2} \sqrt{d + e x}} - \frac{8 a c^{2} d^{3} x}{e \sqrt{d + e x}} + \frac{2 a c^{2} d^{2} x^{2}}{\sqrt{d + e x}} + \frac{32 c^{3} d^{6}}{5 e^{4} \sqrt{d + e x}} + \frac{16 c^{3} d^{5} x}{5 e^{3} \sqrt{d + e x}} - \frac{4 c^{3} d^{4} x^{2}}{5 e^{2} \sqrt{d + e x}} + \frac{2 c^{3} d^{3} x^{3}}{5 e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} d^{\frac{3}{2}} 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)**(9/2),x)

[Out]

Piecewise((-2*a**3*e**2/sqrt(d + e*x) + 12*a**2*c*d**2/sqrt(d + e*x) + 6*a**2*c*d*e*x/sqrt(d + e*x) - 16*a*c**
2*d**4/(e**2*sqrt(d + e*x)) - 8*a*c**2*d**3*x/(e*sqrt(d + e*x)) + 2*a*c**2*d**2*x**2/sqrt(d + e*x) + 32*c**3*d
**6/(5*e**4*sqrt(d + e*x)) + 16*c**3*d**5*x/(5*e**3*sqrt(d + e*x)) - 4*c**3*d**4*x**2/(5*e**2*sqrt(d + e*x)) +
2*c**3*d**3*x**3/(5*e*sqrt(d + e*x)), Ne(e, 0)), (c**3*d**(3/2)*x**4/4, True))

________________________________________________________________________________________

Giac [A]  time = 1.21464, size = 263, normalized size = 2.33 \begin{align*} \frac{2}{5} \,{\left ({\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{3} e^{16} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{4} e^{16} + 15 \, \sqrt{x e + d} c^{3} d^{5} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d^{2} e^{18} - 30 \, \sqrt{x e + d} a c^{2} d^{3} e^{18} + 15 \, \sqrt{x e + d} a^{2} c d e^{20}\right )} e^{\left (-20\right )} + \frac{2 \,{\left ({\left (x e + d\right )}^{3} c^{3} d^{6} - 3 \,{\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} + 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 )}}{{\left (x e + d\right )}^{\frac{7}{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)^(9/2),x, algorithm="giac")

[Out]

2/5*((x*e + d)^(5/2)*c^3*d^3*e^16 - 5*(x*e + d)^(3/2)*c^3*d^4*e^16 + 15*sqrt(x*e + d)*c^3*d^5*e^16 + 5*(x*e +
d)^(3/2)*a*c^2*d^2*e^18 - 30*sqrt(x*e + d)*a*c^2*d^3*e^18 + 15*sqrt(x*e + d)*a^2*c*d*e^20)*e^(-20) + 2*((x*e +
d)^3*c^3*d^6 - 3*(x*e + d)^3*a*c^2*d^4*e^2 + 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)^(7/2)