### 3.1996 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=115 $-\frac{6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac{2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4}$

[Out]

(-2*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/e^4 + (2*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2))/e^4 - (6*c^2*d^2*(c*d^2 -
a*e^2)*(d + e*x)^(5/2))/(5*e^4) + (2*c^3*d^3*(d + e*x)^(7/2))/(7*e^4)

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Rubi [A]  time = 0.0545124, 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} $-\frac{6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac{2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4}$

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)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0695736, size = 110, normalized size = 0.96 $\frac{2 \sqrt{d+e x} \left (35 a^2 c d e^4 (e x-2 d)+35 a^3 e^6+7 a c^2 d^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^3 d^3 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )\right )}{35 e^4}$

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)^(7/2),x]

[Out]

(2*Sqrt[d + e*x]*(35*a^3*e^6 + 35*a^2*c*d*e^4*(-2*d + e*x) + 7*a*c^2*d^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c
^3*d^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)))/(35*e^4)

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Maple [A]  time = 0.043, size = 131, normalized size = 1.1 \begin{align*}{\frac{10\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+42\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-12\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+70\,{a}^{2}cd{e}^{5}x-56\,a{c}^{2}{d}^{3}{e}^{3}x+16\,{c}^{3}{d}^{5}ex+70\,{a}^{3}{e}^{6}-140\,{a}^{2}c{d}^{2}{e}^{4}+112\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{35\,{e}^{4}}\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)^(7/2),x)

[Out]

2/35*(e*x+d)^(1/2)*(5*c^3*d^3*e^3*x^3+21*a*c^2*d^2*e^4*x^2-6*c^3*d^4*e^2*x^2+35*a^2*c*d*e^5*x-28*a*c^2*d^3*e^3
*x+8*c^3*d^5*e*x+35*a^3*e^6-70*a^2*c*d^2*e^4+56*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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Maxima [A]  time = 1.01014, size = 185, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} d^{3} - 21 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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 )}^{\frac{3}{2}} - 35 \,{\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}\right )}}{35 \, e^{4}} \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)^(7/2),x, algorithm="maxima")

[Out]

2/35*(5*(e*x + d)^(7/2)*c^3*d^3 - 21*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(5/2) + 35*(c^3*d^5 - 2*a*c^2*d^3*e^2
+ a^2*c*d*e^4)*(e*x + d)^(3/2) - 35*(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^
4

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Fricas [A]  time = 1.94957, size = 275, normalized size = 2.39 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 56 \, a c^{2} d^{4} e^{2} - 70 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} - 3 \,{\left (2 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (8 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{4}} \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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*d^3*e^3*x^3 - 16*c^3*d^6 + 56*a*c^2*d^4*e^2 - 70*a^2*c*d^2*e^4 + 35*a^3*e^6 - 3*(2*c^3*d^4*e^2 - 7
*a*c^2*d^2*e^4)*x^2 + (8*c^3*d^5*e - 28*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 138.656, size = 376, normalized size = 3.27 \begin{align*} \begin{cases} - \frac{\frac{2 a^{3} d e^{3}}{\sqrt{d + e x}} + 2 a^{3} e^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} c d^{2} e \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} c d e \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + \frac{6 a c^{2} d^{3} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{6 a c^{2} d^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 c^{3} d^{4} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 c^{3} d^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{3} d^{\frac{5}{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)**(7/2),x)

[Out]

Piecewise((-(2*a**3*d*e**3/sqrt(d + e*x) + 2*a**3*e**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 6*a**2*c*d**2*e*(-
d/sqrt(d + e*x) - sqrt(d + e*x)) + 6*a**2*c*d*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)
+ 6*a*c**2*d**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 6*a*c**2*d**2*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e + 2*c**3*d**4*(-d**3/sqrt(d + e*x)
- 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 2*c**3*d**3*(d**4/sqrt(d + e*x) + 4*d
**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3)/e, Ne(e, 0)),
(c**3*d**(5/2)*x**4/4, True))

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Giac [A]  time = 1.2541, size = 250, normalized size = 2.17 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{3} e^{24} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{4} e^{24} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{5} e^{24} - 35 \, \sqrt{x e + d} c^{3} d^{6} e^{24} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} d^{2} e^{26} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d^{3} e^{26} + 105 \, \sqrt{x e + d} a c^{2} d^{4} e^{26} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} c d e^{28} - 105 \, \sqrt{x e + d} a^{2} c d^{2} e^{28} + 35 \, \sqrt{x e + d} a^{3} e^{30}\right )} e^{\left (-28\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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*d^3*e^24 - 21*(x*e + d)^(5/2)*c^3*d^4*e^24 + 35*(x*e + d)^(3/2)*c^3*d^5*e^24 - 35*
sqrt(x*e + d)*c^3*d^6*e^24 + 21*(x*e + d)^(5/2)*a*c^2*d^2*e^26 - 70*(x*e + d)^(3/2)*a*c^2*d^3*e^26 + 105*sqrt(
x*e + d)*a*c^2*d^4*e^26 + 35*(x*e + d)^(3/2)*a^2*c*d*e^28 - 105*sqrt(x*e + d)*a^2*c*d^2*e^28 + 35*sqrt(x*e + d
)*a^3*e^30)*e^(-28)