3.1983 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 c^2 d^2 (d+e x)^{9/2} \left (c d^2-a e^2\right )}{3 e^4}+\frac{6 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \]

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(5/2))/(5*e^4) + (6*c*d*(c*d^2 - a*e^2)^2*(d + e
*x)^(7/2))/(7*e^4) - (2*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(9/2))/(3*e^4) + (2*c^
3*d^3*(d + e*x)^(11/2))/(11*e^4)

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Rubi [A]  time = 0.165187, antiderivative size = 119, 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 \[ -\frac{2 c^2 d^2 (d+e x)^{9/2} \left (c d^2-a e^2\right )}{3 e^4}+\frac{6 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{2 c^3 d^3 (d+e x)^{11/2}}{11 e^4} \]

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)^(3/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(5/2))/(5*e^4) + (6*c*d*(c*d^2 - a*e^2)^2*(d + e
*x)^(7/2))/(7*e^4) - (2*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(9/2))/(3*e^4) + (2*c^
3*d^3*(d + e*x)^(11/2))/(11*e^4)

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Rubi in Sympy [A]  time = 46.6786, size = 110, normalized size = 0.92 \[ \frac{2 c^{3} d^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{2 c^{2} d^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} - c d^{2}\right )}{3 e^{4}} + \frac{6 c d \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{3}}{5 e^{4}} \]

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)**(3/2),x)

[Out]

2*c**3*d**3*(d + e*x)**(11/2)/(11*e**4) + 2*c**2*d**2*(d + e*x)**(9/2)*(a*e**2 -
c*d**2)/(3*e**4) + 6*c*d*(d + e*x)**(7/2)*(a*e**2 - c*d**2)**2/(7*e**4) + 2*(d
+ e*x)**(5/2)*(a*e**2 - c*d**2)**3/(5*e**4)

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Mathematica [A]  time = 0.166367, size = 111, normalized size = 0.93 \[ \frac{2 (d+e x)^{5/2} \left (231 a^3 e^6-99 a^2 c d e^4 (2 d-5 e x)+11 a c^2 d^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4} \]

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)^(3/2),x]

[Out]

(2*(d + e*x)^(5/2)*(231*a^3*e^6 - 99*a^2*c*d*e^4*(2*d - 5*e*x) + 11*a*c^2*d^2*e^
2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + c^3*d^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2
+ 105*e^3*x^3)))/(1155*e^4)

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Maple [A]  time = 0.009, size = 131, normalized size = 1.1 \[{\frac{210\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+770\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-140\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+990\,x{a}^{2}cd{e}^{5}-440\,xa{c}^{2}{d}^{3}{e}^{3}+80\,{c}^{3}{d}^{5}ex+462\,{a}^{3}{e}^{6}-396\,{a}^{2}c{d}^{2}{e}^{4}+176\,{c}^{2}{d}^{4}a{e}^{2}-32\,{c}^{3}{d}^{6}}{1155\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

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)^(3/2),x)

[Out]

2/1155*(e*x+d)^(5/2)*(105*c^3*d^3*e^3*x^3+385*a*c^2*d^2*e^4*x^2-70*c^3*d^4*e^2*x
^2+495*a^2*c*d*e^5*x-220*a*c^2*d^3*e^3*x+40*c^3*d^5*e*x+231*a^3*e^6-198*a^2*c*d^
2*e^4+88*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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Maxima [A]  time = 0.735028, size = 185, normalized size = 1.55 \[ \frac{2 \,{\left (105 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{3} d^{3} - 385 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 231 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{1155 \, e^{4}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/1155*(105*(e*x + d)^(11/2)*c^3*d^3 - 385*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(
9/2) + 495*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(7/2) - 231*(c^3*
d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(e*x + d)^(5/2))/e^4

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Fricas [A]  time = 0.217074, size = 312, normalized size = 2.62 \[ \frac{2 \,{\left (105 \, c^{3} d^{3} e^{5} x^{5} - 16 \, c^{3} d^{8} + 88 \, a c^{2} d^{6} e^{2} - 198 \, a^{2} c d^{4} e^{4} + 231 \, a^{3} d^{2} e^{6} + 35 \,{\left (4 \, c^{3} d^{4} e^{4} + 11 \, a c^{2} d^{2} e^{6}\right )} x^{4} + 5 \,{\left (c^{3} d^{5} e^{3} + 110 \, a c^{2} d^{3} e^{5} + 99 \, a^{2} c d e^{7}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{6} e^{2} - 11 \, a c^{2} d^{4} e^{4} - 264 \, a^{2} c d^{2} e^{6} - 77 \, a^{3} e^{8}\right )} x^{2} +{\left (8 \, c^{3} d^{7} e - 44 \, a c^{2} d^{5} e^{3} + 99 \, a^{2} c d^{3} e^{5} + 462 \, a^{3} d e^{7}\right )} x\right )} \sqrt{e x + d}}{1155 \, e^{4}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/1155*(105*c^3*d^3*e^5*x^5 - 16*c^3*d^8 + 88*a*c^2*d^6*e^2 - 198*a^2*c*d^4*e^4
+ 231*a^3*d^2*e^6 + 35*(4*c^3*d^4*e^4 + 11*a*c^2*d^2*e^6)*x^4 + 5*(c^3*d^5*e^3 +
110*a*c^2*d^3*e^5 + 99*a^2*c*d*e^7)*x^3 - 3*(2*c^3*d^6*e^2 - 11*a*c^2*d^4*e^4 -
264*a^2*c*d^2*e^6 - 77*a^3*e^8)*x^2 + (8*c^3*d^7*e - 44*a*c^2*d^5*e^3 + 99*a^2*
c*d^3*e^5 + 462*a^3*d*e^7)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 63.7834, size = 971, normalized size = 8.16 \[ \text{result too large to display} \]

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)**(3/2),x)

[Out]

Piecewise((-(2*a**3*d**3*e**3/sqrt(d + e*x) + 6*a**3*d**2*e**3*(-d/sqrt(d + e*x)
- sqrt(d + e*x)) + 6*a**3*d*e**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d +
e*x)**(3/2)/3) + 2*a**3*e**3*(-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) + 6*a**2*c*d**4*e*(-d/sqrt(d + e*x) - sqrt(
d + e*x)) + 18*a**2*c*d**3*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)
**(3/2)/3) + 18*a**2*c*d**2*e*(-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) + 6*a**2*c*d*e*(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) + 6*a*c**2*d**5*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/
2)/3)/e + 18*a*c**2*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 + 18*a*c**2*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 + 6*a*c**2*d**2*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(
d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)
**(9/2)/9)/e + 2*c**3*d**6*(-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 + 6*c**3*d**5*(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 + 6*c**3*d**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**
3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e
*x)**(9/2)/9)/e**3 + 2*c**3*d**3*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*
d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2
*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**3)/e, Ne(e, 0)), (c**3*d**(9/2)
*x**4/4, True))

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GIAC/XCAS [A]  time = 0.220614, size = 250, normalized size = 2.1 \[ \frac{2}{1155} \,{\left (105 \,{\left (x e + d\right )}^{\frac{11}{2}} c^{3} d^{3} e^{40} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} d^{4} e^{40} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{5} e^{40} - 231 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{6} e^{40} + 385 \,{\left (x e + d\right )}^{\frac{9}{2}} a c^{2} d^{2} e^{42} - 990 \,{\left (x e + d\right )}^{\frac{7}{2}} a c^{2} d^{3} e^{42} + 693 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} d^{4} e^{42} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} c d e^{44} - 693 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} c d^{2} e^{44} + 231 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} e^{46}\right )} e^{\left (-44\right )} \]

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)^(3/2),x, algorithm="giac")

[Out]

2/1155*(105*(x*e + d)^(11/2)*c^3*d^3*e^40 - 385*(x*e + d)^(9/2)*c^3*d^4*e^40 + 4
95*(x*e + d)^(7/2)*c^3*d^5*e^40 - 231*(x*e + d)^(5/2)*c^3*d^6*e^40 + 385*(x*e +
d)^(9/2)*a*c^2*d^2*e^42 - 990*(x*e + d)^(7/2)*a*c^2*d^3*e^42 + 693*(x*e + d)^(5/
2)*a*c^2*d^4*e^42 + 495*(x*e + d)^(7/2)*a^2*c*d*e^44 - 693*(x*e + d)^(5/2)*a^2*c
*d^2*e^44 + 231*(x*e + d)^(5/2)*a^3*e^46)*e^(-44)