3.679 \(\int \frac{(f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=336 \[ -\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt{d+e x}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2))/(3465*c^5*d^5*e*(d + e*x)^(3/2)) + (128*g*(c*d*f - a*e*
g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^4*d^4*e*Sqrt[d + e*x
]) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(
3/2))/(231*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)
^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + e*x)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.63289, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt{d+e x}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2))/(3465*c^5*d^5*e*(d + e*x)^(3/2)) + (128*g*(c*d*f - a*e*
g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^4*d^4*e*Sqrt[d + e*x
]) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(
3/2))/(231*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)
^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + e*x)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 132.075, size = 330, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{11 c d \left (d + e x\right )^{\frac{3}{2}}} - \frac{16 \left (f + g x\right )^{3} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{99 c^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{32 \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{231 c^{3} d^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{128 g \left (a e g - c d f\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{1155 c^{4} d^{4} e \sqrt{d + e x}} + \frac{128 \left (a e g - c d f\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}} \left (2 a e^{2} g + 3 c d^{2} g - 5 c d e f\right )}{3465 c^{5} d^{5} e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)

[Out]

2*(f + g*x)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(11*c*d*(d + e*
x)**(3/2)) - 16*(f + g*x)**3*(a*e*g - c*d*f)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c
*d**2))**(3/2)/(99*c**2*d**2*(d + e*x)**(3/2)) + 32*(f + g*x)**2*(a*e*g - c*d*f)
**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(231*c**3*d**3*(d + e*x)**
(3/2)) - 128*g*(a*e*g - c*d*f)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3
/2)/(1155*c**4*d**4*e*sqrt(d + e*x)) + 128*(a*e*g - c*d*f)**3*(a*d*e + c*d*e*x**
2 + x*(a*e**2 + c*d**2))**(3/2)*(2*a*e**2*g + 3*c*d**2*g - 5*c*d*e*f)/(3465*c**5
*d**5*e*(d + e*x)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.255427, size = 195, normalized size = 0.58 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (11 f+3 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (33 f^2+22 f g x+5 g^2 x^2\right )-8 a c^3 d^3 e g \left (231 f^3+297 f^2 g x+165 f g^2 x^2+35 g^3 x^3\right )+c^4 d^4 \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(11*f +
3*g*x) + 48*a^2*c^2*d^2*e^2*g^2*(33*f^2 + 22*f*g*x + 5*g^2*x^2) - 8*a*c^3*d^3*e
*g*(231*f^3 + 297*f^2*g*x + 165*f*g^2*x^2 + 35*g^3*x^3) + c^4*d^4*(1155*f^4 + 27
72*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4*x^4)))/(3465*c^5*d^5*(d
+ e*x)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 283, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 315\,{g}^{4}{x}^{4}{c}^{4}{d}^{4}-280\,a{c}^{3}{d}^{3}e{g}^{4}{x}^{3}+1540\,{c}^{4}{d}^{4}f{g}^{3}{x}^{3}+240\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{g}^{4}{x}^{2}-1320\,a{c}^{3}{d}^{3}ef{g}^{3}{x}^{2}+2970\,{c}^{4}{d}^{4}{f}^{2}{g}^{2}{x}^{2}-192\,{a}^{3}cd{e}^{3}{g}^{4}x+1056\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}f{g}^{3}x-2376\,a{c}^{3}{d}^{3}e{f}^{2}{g}^{2}x+2772\,{c}^{4}{d}^{4}{f}^{3}gx+128\,{a}^{4}{e}^{4}{g}^{4}-704\,{a}^{3}cd{e}^{3}f{g}^{3}+1584\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{f}^{2}{g}^{2}-1848\,a{c}^{3}{d}^{3}e{f}^{3}g+1155\,{f}^{4}{c}^{4}{d}^{4} \right ) }{3465\,{c}^{5}{d}^{5}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)

[Out]

2/3465*(c*d*x+a*e)*(315*c^4*d^4*g^4*x^4-280*a*c^3*d^3*e*g^4*x^3+1540*c^4*d^4*f*g
^3*x^3+240*a^2*c^2*d^2*e^2*g^4*x^2-1320*a*c^3*d^3*e*f*g^3*x^2+2970*c^4*d^4*f^2*g
^2*x^2-192*a^3*c*d*e^3*g^4*x+1056*a^2*c^2*d^2*e^2*f*g^3*x-2376*a*c^3*d^3*e*f^2*g
^2*x+2772*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-704*a^3*c*d*e^3*f*g^3+1584*a^2*c^2*d^2
*e^2*f^2*g^2-1848*a*c^3*d^3*e*f^3*g+1155*c^4*d^4*f^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x
+a*d*e)^(1/2)/c^5/d^5/(e*x+d)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.759534, size = 432, normalized size = 1.29 \[ \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f^{4}}{3 \, c d} + \frac{8 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{3} g}{15 \, c^{2} d^{2}} + \frac{4 \,{\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{2} g^{2}}{35 \, c^{3} d^{3}} + \frac{8 \,{\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f g^{3}}{315 \, c^{4} d^{4}} + \frac{2 \,{\left (315 \, c^{5} d^{5} x^{5} + 35 \, a c^{4} d^{4} e x^{4} - 40 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 48 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 64 \, a^{4} c d e^{4} x + 128 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} g^{4}}{3465 \, c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/3*(c*d*x + a*e)^(3/2)*f^4/(c*d) + 8/15*(3*c^2*d^2*x^2 + a*c*d*e*x - 2*a^2*e^2)
*sqrt(c*d*x + a*e)*f^3*g/(c^2*d^2) + 4/35*(15*c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 -
4*a^2*c*d*e^2*x + 8*a^3*e^3)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*d^3) + 8/315*(35*c^4
*d^4*x^4 + 5*a*c^3*d^3*e*x^3 - 6*a^2*c^2*d^2*e^2*x^2 + 8*a^3*c*d*e^3*x - 16*a^4*
e^4)*sqrt(c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/3465*(315*c^5*d^5*x^5 + 35*a*c^4*d^4*
e*x^4 - 40*a^2*c^3*d^3*e^2*x^3 + 48*a^3*c^2*d^2*e^3*x^2 - 64*a^4*c*d*e^4*x + 128
*a^5*e^5)*sqrt(c*d*x + a*e)*g^4/(c^5*d^5)

_______________________________________________________________________________________

Fricas [A]  time = 0.274664, size = 1127, normalized size = 3.35 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/3465*(315*c^6*d^6*e*g^4*x^7 + 1155*a^2*c^4*d^5*e^2*f^4 - 1848*a^3*c^3*d^4*e^3*
f^3*g + 1584*a^4*c^2*d^3*e^4*f^2*g^2 - 704*a^5*c*d^2*e^5*f*g^3 + 128*a^6*d*e^6*g
^4 + 35*(44*c^6*d^6*e*f*g^3 + (9*c^6*d^7 + 10*a*c^5*d^5*e^2)*g^4)*x^6 + 5*(594*c
^6*d^6*e*f^2*g^2 + 44*(7*c^6*d^7 + 8*a*c^5*d^5*e^2)*f*g^3 + (70*a*c^5*d^6*e - a^
2*c^4*d^4*e^3)*g^4)*x^5 + (2772*c^6*d^6*e*f^3*g + 594*(5*c^6*d^7 + 6*a*c^5*d^5*e
^2)*f^2*g^2 + 44*(40*a*c^5*d^6*e - a^2*c^4*d^4*e^3)*f*g^3 - (5*a^2*c^4*d^5*e^2 -
8*a^3*c^3*d^3*e^4)*g^4)*x^4 + (1155*c^6*d^6*e*f^4 + 924*(3*c^6*d^7 + 4*a*c^5*d^
5*e^2)*f^3*g + 198*(18*a*c^5*d^6*e - a^2*c^4*d^4*e^3)*f^2*g^2 - 44*(a^2*c^4*d^5*
e^2 - 2*a^3*c^3*d^3*e^4)*f*g^3 + 8*(a^3*c^3*d^4*e^3 - 2*a^4*c^2*d^2*e^5)*g^4)*x^
3 + (1155*(c^6*d^7 + 2*a*c^5*d^5*e^2)*f^4 + 924*(4*a*c^5*d^6*e - a^2*c^4*d^4*e^3
)*f^3*g - 198*(a^2*c^4*d^5*e^2 - 4*a^3*c^3*d^3*e^4)*f^2*g^2 + 88*(a^3*c^3*d^4*e^
3 - 4*a^4*c^2*d^2*e^5)*f*g^3 - 16*(a^4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*g^4)*x^2 + (
1155*(2*a*c^5*d^6*e + a^2*c^4*d^4*e^3)*f^4 - 924*(a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^
3*e^4)*f^3*g + 792*(a^3*c^3*d^4*e^3 + 2*a^4*c^2*d^2*e^5)*f^2*g^2 - 352*(a^4*c^2*
d^3*e^4 + 2*a^5*c*d*e^6)*f*g^3 + 64*(a^5*c*d^2*e^5 + 2*a^6*e^7)*g^4)*x)/(sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^5*d^5)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4/sqrt(e*x + d),x, algorithm="giac")

[Out]

Timed out