3.1988 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx\)

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*S
qrt[d + e*x])/e^4

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Rubi [A]  time = 0.1665, 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 \[ \frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4} \]

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*S
qrt[d + e*x])/e^4

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Rubi in Sympy [A]  time = 46.2306, size = 105, normalized size = 0.93 \[ \frac{2 c^{3} d^{3} \sqrt{d + e x}}{e^{4}} - \frac{6 c^{2} d^{2} \left (a e^{2} - c d^{2}\right )}{e^{4} \sqrt{d + e x}} - \frac{2 c d \left (a e^{2} - c d^{2}\right )^{2}}{e^{4} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{3}}{5 e^{4} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116015, size = 108, normalized size = 0.96 \[ \sqrt{d+e x} \left (-\frac{6 c^2 d^2 \left (a e^2-c d^2\right )}{e^4 (d+e x)}-\frac{2 c d \left (a e^2-c d^2\right )^2}{e^4 (d+e x)^2}-\frac{2 \left (a e^2-c d^2\right )^3}{5 e^4 (d+e x)^3}+\frac{2 c^3 d^3}{e^4}\right ) \]

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]

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

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Maple [A]  time = 0.009, size = 130, normalized size = 1.2 \[ -{\frac{-10\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+30\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-60\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+10\,x{a}^{2}cd{e}^{5}+40\,xa{c}^{2}{d}^{3}{e}^{3}-80\,{c}^{3}{d}^{5}ex+2\,{a}^{3}{e}^{6}+4\,{a}^{2}c{d}^{2}{e}^{4}+16\,{c}^{2}{d}^{4}a{e}^{2}-32\,{c}^{3}{d}^{6}}{5\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(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 = 0.727918, size = 189, normalized size = 1.67 \[ \frac{2 \,{\left (\frac{5 \, \sqrt{e x + d} c^{3} d^{3}}{e^{3}} + \frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{3}}\right )}}{5 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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 = 0.224111, size = 205, normalized size = 1.81 \[ \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} + 16 \, c^{3} d^{6} - 8 \, a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \,{\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (8 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )}}{5 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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)/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(e*x + d))

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Sympy [A]  time = 123.896, size = 654, normalized size = 5.79 \[ \begin{cases} - \frac{2 a^{3} e^{6}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{4 a^{2} c d^{2} e^{4}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{10 a^{2} c d e^{5} x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{16 a c^{2} d^{4} e^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{40 a c^{2} d^{3} e^{3} x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{30 a c^{2} d^{2} e^{4} x^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{32 c^{3} d^{6}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{80 c^{3} d^{5} e x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{60 c^{3} d^{4} e^{2} x^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{10 c^{3} d^{3} e^{3} x^{3}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} x^{4}}{4 \sqrt{d}} & \text{otherwise} \end{cases} \]

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)), True))

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GIAC/XCAS [A]  time = 0.222449, size = 252, normalized size = 2.23 \[ 2 \, \sqrt{x e + d} c^{3} d^{3} e^{\left (-4\right )} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{5} c^{3} d^{4} - 5 \,{\left (x e + d\right )}^{4} c^{3} d^{5} +{\left (x e + d\right )}^{3} c^{3} d^{6} - 15 \,{\left (x e + d\right )}^{5} a c^{2} d^{2} e^{2} + 10 \,{\left (x e + d\right )}^{4} a c^{2} d^{3} e^{2} - 3 \,{\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} - 5 \,{\left (x e + d\right )}^{4} a^{2} c d e^{4} + 3 \,{\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} -{\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{5 \,{\left (x e + d\right )}^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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 - 15*(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)