3.1985 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{7/2}} \, dx\)

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 = 0.163917, 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 \[ -\frac{6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac{2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4} \]

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)

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Rubi in Sympy [A]  time = 46.2808, size = 107, normalized size = 0.93 \[ \frac{2 c^{3} d^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{4}} + \frac{6 c^{2} d^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e^{4}} + \frac{2 c d \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{e^{4}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3}}{e^{4}} \]

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)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.100084, size = 110, normalized size = 0.96 \[ \frac{2 \sqrt{d+e x} \left (35 a^3 e^6+35 a^2 c d e^4 (e x-2 d)+7 a c^2 d^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )}{35 e^4} \]

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 = 0.009, size = 131, normalized size = 1.1 \[{\frac{10\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+42\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-12\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+70\,x{a}^{2}cd{e}^{5}-56\,xa{c}^{2}{d}^{3}{e}^{3}+16\,{c}^{3}{d}^{5}ex+70\,{a}^{3}{e}^{6}-140\,{a}^{2}c{d}^{2}{e}^{4}+112\,{c}^{2}{d}^{4}a{e}^{2}-32\,{c}^{3}{d}^{6}}{35\,{e}^{4}}\sqrt{ex+d}} \]

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)^(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 = 0.747452, size = 185, normalized size = 1.61 \[ \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} d^{3} - 21 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt{e x + d}\right )}}{35 \, e^{4}} \]

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)^(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 = 0.233994, size = 176, normalized size = 1.53 \[ \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 56 \, a c^{2} d^{4} e^{2} - 70 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} - 3 \,{\left (2 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (8 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{4}} \]

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)^(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 = 23.032, size = 376, normalized size = 3.27 \[ \begin{cases} - \frac{\frac{2 a^{3} d e^{3}}{\sqrt{d + e x}} + 2 a^{3} e^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} c d^{2} e \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} c d e \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right ) + \frac{6 a c^{2} d^{3} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{6 a c^{2} d^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 c^{3} d^{4} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 c^{3} d^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{3} d^{\frac{5}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.222477, size = 250, normalized size = 2.17 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{3} e^{24} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{4} e^{24} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{5} e^{24} - 35 \, \sqrt{x e + d} c^{3} d^{6} e^{24} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} d^{2} e^{26} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d^{3} e^{26} + 105 \, \sqrt{x e + d} a c^{2} d^{4} e^{26} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} c d e^{28} - 105 \, \sqrt{x e + d} a^{2} c d^{2} e^{28} + 35 \, \sqrt{x e + d} a^{3} e^{30}\right )} e^{\left (-28\right )} \]

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)^(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*sq
rt(x*e + d)*a^3*e^30)*e^(-28)