3.700 \(\int \frac{(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=336 \[ -\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(7/2))/(45045*c^5*d^5*e*(d + e*x)^(7/2)) + (128*g*(c*d*f - a*e
*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c^4*d^4*e*(d + e*x)^(
5/2)) + (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
)^(7/2))/(715*c^3*d^3*(d + e*x)^(7/2)) + (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)^(7/2))/(195*c^2*d^2*(d + e*x)^(7/2)) + (2*(f +
g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d*(d + e*x)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.62424, 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 \[ -\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(7/2))/(45045*c^5*d^5*e*(d + e*x)^(7/2)) + (128*g*(c*d*f - a*e
*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c^4*d^4*e*(d + e*x)^(
5/2)) + (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
)^(7/2))/(715*c^3*d^3*(d + e*x)^(7/2)) + (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)^(7/2))/(195*c^2*d^2*(d + e*x)^(7/2)) + (2*(f +
g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d*(d + e*x)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 129.653, size = 330, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{15 c d \left (d + e x\right )^{\frac{7}{2}}} - \frac{16 \left (f + g x\right )^{3} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{195 c^{2} d^{2} \left (d + e x\right )^{\frac{7}{2}}} + \frac{32 \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{715 c^{3} d^{3} \left (d + e x\right )^{\frac{7}{2}}} - \frac{128 g \left (a e g - c d f\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{6435 c^{4} d^{4} e \left (d + e x\right )^{\frac{5}{2}}} + \frac{128 \left (a e g - c d f\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}} \left (2 a e^{2} g + 7 c d^{2} g - 9 c d e f\right )}{45045 c^{5} d^{5} e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(5/2)/(e*x+d)**(5/2),x)

[Out]

2*(f + g*x)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(15*c*d*(d + e*
x)**(7/2)) - 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))**(7/2)/(195*c**2*d**2*(d + e*x)**(7/2)) + 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))**(7/2)/(715*c**3*d**3*(d + e*x)*
*(7/2)) - 128*g*(a*e*g - c*d*f)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(
7/2)/(6435*c**4*d**4*e*(d + e*x)**(5/2)) + 128*(a*e*g - c*d*f)**3*(a*d*e + c*d*e
*x**2 + x*(a*e**2 + c*d**2))**(7/2)*(2*a*e**2*g + 7*c*d**2*g - 9*c*d*e*f)/(45045
*c**5*d**5*e*(d + e*x)**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.353751, size = 205, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (15 f+7 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (65 f^2+70 f g x+21 g^2 x^2\right )-8 a c^3 d^3 e g \left (715 f^3+1365 f^2 g x+945 f g^2 x^2+231 g^3 x^3\right )+c^4 d^4 \left (6435 f^4+20020 f^3 g x+24570 f^2 g^2 x^2+13860 f g^3 x^3+3003 g^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^4*g^4 - 64*a^3*c*d*e
^3*g^3*(15*f + 7*g*x) + 48*a^2*c^2*d^2*e^2*g^2*(65*f^2 + 70*f*g*x + 21*g^2*x^2)
- 8*a*c^3*d^3*e*g*(715*f^3 + 1365*f^2*g*x + 945*f*g^2*x^2 + 231*g^3*x^3) + c^4*d
^4*(6435*f^4 + 20020*f^3*g*x + 24570*f^2*g^2*x^2 + 13860*f*g^3*x^3 + 3003*g^4*x^
4)))/(45045*c^5*d^5*Sqrt[d + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 283, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3003\,{g}^{4}{x}^{4}{c}^{4}{d}^{4}-1848\,a{c}^{3}{d}^{3}e{g}^{4}{x}^{3}+13860\,{c}^{4}{d}^{4}f{g}^{3}{x}^{3}+1008\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{g}^{4}{x}^{2}-7560\,a{c}^{3}{d}^{3}ef{g}^{3}{x}^{2}+24570\,{c}^{4}{d}^{4}{f}^{2}{g}^{2}{x}^{2}-448\,{a}^{3}cd{e}^{3}{g}^{4}x+3360\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}f{g}^{3}x-10920\,a{c}^{3}{d}^{3}e{f}^{2}{g}^{2}x+20020\,{c}^{4}{d}^{4}{f}^{3}gx+128\,{a}^{4}{e}^{4}{g}^{4}-960\,{a}^{3}cd{e}^{3}f{g}^{3}+3120\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{f}^{2}{g}^{2}-5720\,a{c}^{3}{d}^{3}e{f}^{3}g+6435\,{f}^{4}{c}^{4}{d}^{4} \right ) }{45045\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2)/(e*x+d)^(5/2),x)

[Out]

2/45045*(c*d*x+a*e)*(3003*c^4*d^4*g^4*x^4-1848*a*c^3*d^3*e*g^4*x^3+13860*c^4*d^4
*f*g^3*x^3+1008*a^2*c^2*d^2*e^2*g^4*x^2-7560*a*c^3*d^3*e*f*g^3*x^2+24570*c^4*d^4
*f^2*g^2*x^2-448*a^3*c*d*e^3*g^4*x+3360*a^2*c^2*d^2*e^2*f*g^3*x-10920*a*c^3*d^3*
e*f^2*g^2*x+20020*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-960*a^3*c*d*e^3*f*g^3+3120*a^2
*c^2*d^2*e^2*f^2*g^2-5720*a*c^3*d^3*e*f^3*g+6435*c^4*d^4*f^4)*(c*d*e*x^2+a*e^2*x
+c*d^2*x+a*d*e)^(5/2)/c^5/d^5/(e*x+d)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.767596, size = 672, normalized size = 2. \[ \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{4}}{7 \, c d} + \frac{8 \,{\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f^{3} g}{63 \, c^{2} d^{2}} + \frac{4 \,{\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} f^{2} g^{2}}{231 \, c^{3} d^{3}} + \frac{8 \,{\left (231 \, c^{6} d^{6} x^{6} + 567 \, a c^{5} d^{5} e x^{5} + 371 \, a^{2} c^{4} d^{4} e^{2} x^{4} + 5 \, a^{3} c^{3} d^{3} e^{3} x^{3} - 6 \, a^{4} c^{2} d^{2} e^{4} x^{2} + 8 \, a^{5} c d e^{5} x - 16 \, a^{6} e^{6}\right )} \sqrt{c d x + a e} f g^{3}}{3003 \, c^{4} d^{4}} + \frac{2 \,{\left (3003 \, c^{7} d^{7} x^{7} + 7161 \, a c^{6} d^{6} e x^{6} + 4473 \, a^{2} c^{5} d^{5} e^{2} x^{5} + 35 \, a^{3} c^{4} d^{4} e^{3} x^{4} - 40 \, a^{4} c^{3} d^{3} e^{4} x^{3} + 48 \, a^{5} c^{2} d^{2} e^{5} x^{2} - 64 \, a^{6} c d e^{6} x + 128 \, a^{7} e^{7}\right )} \sqrt{c d x + a e} g^{4}}{45045 \, c^{5} d^{5}} \]

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)^(5/2)*(g*x + f)^4/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*x + a
*e)*f^4/(c*d) + 8/63*(7*c^4*d^4*x^4 + 19*a*c^3*d^3*e*x^3 + 15*a^2*c^2*d^2*e^2*x^
2 + a^3*c*d*e^3*x - 2*a^4*e^4)*sqrt(c*d*x + a*e)*f^3*g/(c^2*d^2) + 4/231*(63*c^5
*d^5*x^5 + 161*a*c^4*d^4*e*x^4 + 113*a^2*c^3*d^3*e^2*x^3 + 3*a^3*c^2*d^2*e^3*x^2
- 4*a^4*c*d*e^4*x + 8*a^5*e^5)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*d^3) + 8/3003*(23
1*c^6*d^6*x^6 + 567*a*c^5*d^5*e*x^5 + 371*a^2*c^4*d^4*e^2*x^4 + 5*a^3*c^3*d^3*e^
3*x^3 - 6*a^4*c^2*d^2*e^4*x^2 + 8*a^5*c*d*e^5*x - 16*a^6*e^6)*sqrt(c*d*x + a*e)*
f*g^3/(c^4*d^4) + 2/45045*(3003*c^7*d^7*x^7 + 7161*a*c^6*d^6*e*x^6 + 4473*a^2*c^
5*d^5*e^2*x^5 + 35*a^3*c^4*d^4*e^3*x^4 - 40*a^4*c^3*d^3*e^4*x^3 + 48*a^5*c^2*d^2
*e^5*x^2 - 64*a^6*c*d*e^6*x + 128*a^7*e^7)*sqrt(c*d*x + a*e)*g^4/(c^5*d^5)

_______________________________________________________________________________________

Fricas [A]  time = 0.287211, size = 1615, normalized size = 4.81 \[ \text{result too large to display} \]

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)^(5/2)*(g*x + f)^4/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*c^8*d^8*e*g^4*x^9 + 6435*a^4*c^4*d^5*e^4*f^4 - 5720*a^5*c^3*d^4*e^
5*f^3*g + 3120*a^6*c^2*d^3*e^6*f^2*g^2 - 960*a^7*c*d^2*e^7*f*g^3 + 128*a^8*d*e^8
*g^4 + 231*(60*c^8*d^8*e*f*g^3 + (13*c^8*d^9 + 44*a*c^7*d^7*e^2)*g^4)*x^8 + 42*(
585*c^8*d^8*e*f^2*g^2 + 30*(11*c^8*d^9 + 38*a*c^7*d^7*e^2)*f*g^3 + (242*a*c^7*d^
8*e + 277*a^2*c^6*d^6*e^3)*g^4)*x^7 + 14*(1430*c^8*d^8*e*f^3*g + 195*(9*c^8*d^9
+ 32*a*c^7*d^7*e^2)*f^2*g^2 + 60*(57*a*c^7*d^8*e + 67*a^2*c^6*d^6*e^3)*f*g^3 + (
831*a^2*c^6*d^7*e^2 + 322*a^3*c^5*d^5*e^4)*g^4)*x^6 + (6435*c^8*d^8*e*f^4 + 2860
*(7*c^8*d^9 + 26*a*c^7*d^7*e^2)*f^3*g + 780*(112*a*c^7*d^8*e + 137*a^2*c^6*d^6*e
^3)*f^2*g^2 + 120*(469*a^2*c^6*d^7*e^2 + 188*a^3*c^5*d^5*e^4)*f*g^3 + (4508*a^3*
c^5*d^6*e^3 - 5*a^4*c^4*d^4*e^5)*g^4)*x^5 + (6435*(c^8*d^9 + 4*a*c^7*d^7*e^2)*f^
4 + 5720*(13*a*c^7*d^8*e + 17*a^2*c^6*d^6*e^3)*f^3*g + 780*(137*a^2*c^6*d^7*e^2
+ 58*a^3*c^5*d^5*e^4)*f^2*g^2 + 60*(376*a^3*c^5*d^6*e^3 - a^4*c^4*d^4*e^5)*f*g^3
- (5*a^4*c^4*d^5*e^4 - 8*a^5*c^3*d^3*e^6)*g^4)*x^4 + 2*(6435*(2*a*c^7*d^8*e + 3
*a^2*c^6*d^6*e^3)*f^4 + 2860*(17*a^2*c^6*d^7*e^2 + 8*a^3*c^5*d^5*e^4)*f^3*g + 19
5*(116*a^3*c^5*d^6*e^3 - a^4*c^4*d^4*e^5)*f^2*g^2 - 30*(a^4*c^4*d^5*e^4 - 2*a^5*
c^3*d^3*e^6)*f*g^3 + 4*(a^5*c^3*d^4*e^5 - 2*a^6*c^2*d^2*e^7)*g^4)*x^3 + 2*(6435*
(3*a^2*c^6*d^7*e^2 + 2*a^3*c^5*d^5*e^4)*f^4 + 1430*(16*a^3*c^5*d^6*e^3 - a^4*c^4
*d^4*e^5)*f^3*g - 195*(a^4*c^4*d^5*e^4 - 4*a^5*c^3*d^3*e^6)*f^2*g^2 + 60*(a^5*c^
3*d^4*e^5 - 4*a^6*c^2*d^2*e^7)*f*g^3 - 8*(a^6*c^2*d^3*e^6 - 4*a^7*c*d*e^8)*g^4)*
x^2 + (6435*(4*a^3*c^5*d^6*e^3 + a^4*c^4*d^4*e^5)*f^4 - 2860*(a^4*c^4*d^5*e^4 +
2*a^5*c^3*d^3*e^6)*f^3*g + 1560*(a^5*c^3*d^4*e^5 + 2*a^6*c^2*d^2*e^7)*f^2*g^2 -
480*(a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*f*g^3 + 64*(a^7*c*d^2*e^7 + 2*a^8*e^9)*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)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

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)^(5/2)*(g*x + f)^4/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError