3.1984 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.167271, 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 \[ -\frac{6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac{6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac{2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]

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)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 46.3625, size = 110, normalized size = 0.92 \[ \frac{2 c^{3} d^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{6 c^{2} d^{2} \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - c d^{2}\right )}{7 e^{4}} + \frac{6 c d \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{3}}{3 e^{4}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.144757, size = 111, normalized size = 0.93 \[ \frac{2 (d+e x)^{3/2} \left (105 a^3 e^6-63 a^2 c d e^4 (2 d-3 e x)+9 a c^2 d^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \]

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)^(5/2),x]

[Out]

(2*(d + e*x)^(3/2)*(105*a^3*e^6 - 63*a^2*c*d*e^4*(2*d - 3*e*x) + 9*a*c^2*d^2*e^2
*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + c^3*d^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2
+ 35*e^3*x^3)))/(315*e^4)

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Maple [A]  time = 0.01, size = 131, normalized size = 1.1 \[{\frac{70\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+270\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-60\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+378\,x{a}^{2}cd{e}^{5}-216\,xa{c}^{2}{d}^{3}{e}^{3}+48\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-252\,{a}^{2}c{d}^{2}{e}^{4}+144\,{c}^{2}{d}^{4}a{e}^{2}-32\,{c}^{3}{d}^{6}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

2/315*(e*x+d)^(3/2)*(35*c^3*d^3*e^3*x^3+135*a*c^2*d^2*e^4*x^2-30*c^3*d^4*e^2*x^2
+189*a^2*c*d*e^5*x-108*a*c^2*d^3*e^3*x+24*c^3*d^5*e*x+105*a^3*e^6-126*a^2*c*d^2*
e^4+72*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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Maxima [A]  time = 0.737483, size = 185, normalized size = 1.55 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} d^{3} - 135 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/315*(35*(e*x + d)^(9/2)*c^3*d^3 - 135*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(7/2
) + 189*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(5/2) - 105*(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)^(3/2))/e^4

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Fricas [A]  time = 0.21622, size = 242, normalized size = 2.03 \[ \frac{2 \,{\left (35 \, c^{3} d^{3} e^{4} x^{4} - 16 \, c^{3} d^{7} + 72 \, a c^{2} d^{5} e^{2} - 126 \, a^{2} c d^{3} e^{4} + 105 \, a^{3} d e^{6} + 5 \,{\left (c^{3} d^{4} e^{3} + 27 \, a c^{2} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{5} e^{2} - 9 \, a c^{2} d^{3} e^{4} - 63 \, a^{2} c d e^{6}\right )} x^{2} +{\left (8 \, c^{3} d^{6} e - 36 \, a c^{2} d^{4} e^{3} + 63 \, a^{2} c d^{2} e^{5} + 105 \, a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*c^3*d^3*e^4*x^4 - 16*c^3*d^7 + 72*a*c^2*d^5*e^2 - 126*a^2*c*d^3*e^4 +
105*a^3*d*e^6 + 5*(c^3*d^4*e^3 + 27*a*c^2*d^2*e^5)*x^3 - 3*(2*c^3*d^5*e^2 - 9*a*
c^2*d^3*e^4 - 63*a^2*c*d*e^6)*x^2 + (8*c^3*d^6*e - 36*a*c^2*d^4*e^3 + 63*a^2*c*d
^2*e^5 + 105*a^3*e^7)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 38.095, size = 644, normalized size = 5.41 \[ \text{result too large to display} \]

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)**(5/2),x)

[Out]

Piecewise((-(2*a**3*d**2*e**3/sqrt(d + e*x) + 4*a**3*d*e**3*(-d/sqrt(d + e*x) -
sqrt(d + e*x)) + 2*a**3*e**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)
**(3/2)/3) + 6*a**2*c*d**3*e*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 12*a**2*c*d**2
*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) + 6*a**2*c*d*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*c**2*d**4*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2
)/3)/e + 12*a*c**2*d**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)/e + 6*a*c**2*d**2*(d**4/sqrt(d + e*x) + 4*d**3*sq
rt(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 + 2*c**3*d**5*(-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 + 4*c**3*d**4*(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 + 2*c**3*d**3*(-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)/e, Ne(e, 0)), (c**3*d**(7/2)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.221994, size = 250, normalized size = 2.1 \[ \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} d^{3} e^{32} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{4} e^{32} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{5} e^{32} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{6} e^{32} + 135 \,{\left (x e + d\right )}^{\frac{7}{2}} a c^{2} d^{2} e^{34} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} d^{3} e^{34} + 315 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d^{4} e^{34} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} c d e^{36} - 315 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} c d^{2} e^{36} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} e^{38}\right )} e^{\left (-36\right )} \]

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)^(5/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*d^3*e^32 - 135*(x*e + d)^(7/2)*c^3*d^4*e^32 + 189*
(x*e + d)^(5/2)*c^3*d^5*e^32 - 105*(x*e + d)^(3/2)*c^3*d^6*e^32 + 135*(x*e + d)^
(7/2)*a*c^2*d^2*e^34 - 378*(x*e + d)^(5/2)*a*c^2*d^3*e^34 + 315*(x*e + d)^(3/2)*
a*c^2*d^4*e^34 + 189*(x*e + d)^(5/2)*a^2*c*d*e^36 - 315*(x*e + d)^(3/2)*a^2*c*d^
2*e^36 + 105*(x*e + d)^(3/2)*a^3*e^38)*e^(-36)