3.1987 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11/2}} \, dx\)

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*S
qrt[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 = 0.167296, 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 \[ -\frac{6 c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}+\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]

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*S
qrt[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 in Sympy [A]  time = 45.9811, size = 107, normalized size = 0.93 \[ \frac{2 c^{3} d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{4}} + \frac{6 c^{2} d^{2} \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{e^{4}} - \frac{6 c d \left (a e^{2} - c d^{2}\right )^{2}}{e^{4} \sqrt{d + e x}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{3}}{3 e^{4} \left (d + e x\right )^{\frac{3}{2}}} \]

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)**(11/2),x)

[Out]

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

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Mathematica [A]  time = 0.237305, size = 93, normalized size = 0.81 \[ \frac{2 \sqrt{d+e x} \left (c^2 d^2 \left (9 a e^2-8 c d^2\right )-\frac{9 c d \left (c d^2-a e^2\right )^2}{d+e x}+\frac{\left (c d^2-a e^2\right )^3}{(d+e x)^2}+c^3 d^3 e x\right )}{3 e^4} \]

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)^(11/2),x]

[Out]

(2*Sqrt[d + e*x]*(c^2*d^2*(-8*c*d^2 + 9*a*e^2) + c^3*d^3*e*x + (c*d^2 - a*e^2)^3
/(d + e*x)^2 - (9*c*d*(c*d^2 - a*e^2)^2)/(d + e*x)))/(3*e^4)

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Maple [A]  time = 0.009, size = 130, normalized size = 1.1 \[ -{\frac{-2\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}-18\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}+12\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+18\,x{a}^{2}cd{e}^{5}-72\,xa{c}^{2}{d}^{3}{e}^{3}+48\,{c}^{3}{d}^{5}ex+2\,{a}^{3}{e}^{6}+12\,{a}^{2}c{d}^{2}{e}^{4}-48\,{c}^{2}{d}^{4}a{e}^{2}+32\,{c}^{3}{d}^{6}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(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 = 0.747761, size = 190, normalized size = 1.65 \[ \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c^{3} d^{3} - 9 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 9 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \]

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)^(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 = 0.220808, size = 189, normalized size = 1.64 \[ \frac{2 \,{\left (c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 24 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 3 \,{\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \,{\left (8 \, c^{3} d^{5} e - 12 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )}}{3 \,{\left (e^{5} x + d e^{4}\right )} \sqrt{e x + d}} \]

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)^(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)/((e^5*x + d*e^4)*sqrt(e*x + d))

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Sympy [A]  time = 69.2295, size = 450, normalized size = 3.91 \[ \begin{cases} - \frac{2 a^{3} e^{6}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 c^{3} d^{6}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 c^{3} d^{5} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} \sqrt{d} x^{4}}{4} & \text{otherwise} \end{cases} \]

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)) + 48*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*s
qrt(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/XCAS [A]  time = 0.223782, size = 261, normalized size = 2.27 \[ \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt{x e + d} c^{3} d^{4} e^{8} + 9 \, \sqrt{x e + d} a c^{2} d^{2} e^{10}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )}^{4} c^{3} d^{5} -{\left (x e + d\right )}^{3} c^{3} d^{6} - 18 \,{\left (x e + d\right )}^{4} a c^{2} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} + 9 \,{\left (x e + d\right )}^{4} a^{2} c d e^{4} - 3 \,{\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} +{\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{9}{2}}} \]

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)^(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)