3.689 \(\int \frac{(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=336 \[ -\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}} \]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(5/2))/(15015*c^5*d^5*e*(d + e*x)^(5/2)) + (128*g*(c*d*f - a*e
*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c^4*d^4*e*(d + e*x)^(
3/2)) + (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
)^(5/2))/(429*c^3*d^3*(d + e*x)^(5/2)) + (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)^(5/2))/(143*c^2*d^2*(d + e*x)^(5/2)) + (2*(f +
g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d*(d + e*x)^(5/2))

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Rubi [A]  time = 1.61184, 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 \[ -\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^3}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{429 c^3 d^3 (d+e x)^{5/2}}+\frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{143 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}} \]

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)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(-128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(5/2))/(15015*c^5*d^5*e*(d + e*x)^(5/2)) + (128*g*(c*d*f - a*e
*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c^4*d^4*e*(d + e*x)^(
3/2)) + (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
)^(5/2))/(429*c^3*d^3*(d + e*x)^(5/2)) + (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)^(5/2))/(143*c^2*d^2*(d + e*x)^(5/2)) + (2*(f +
g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d*(d + e*x)^(5/2))

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Rubi in Sympy [A]  time = 129.65, size = 330, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{13 c d \left (d + e x\right )^{\frac{5}{2}}} - \frac{16 \left (f + g x\right )^{3} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{143 c^{2} d^{2} \left (d + e x\right )^{\frac{5}{2}}} + \frac{32 \left (f + g x\right )^{2} \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{429 c^{3} d^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{128 g \left (a e g - c d f\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{3003 c^{4} d^{4} e \left (d + e x\right )^{\frac{3}{2}}} + \frac{128 \left (a e g - c d f\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} \left (2 a e^{2} g + 5 c d^{2} g - 7 c d e f\right )}{15015 c^{5} d^{5} e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(3/2)/(e*x+d)**(3/2),x)

[Out]

2*(f + g*x)**4*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(13*c*d*(d + e*
x)**(5/2)) - 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))**(5/2)/(143*c**2*d**2*(d + e*x)**(5/2)) + 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))**(5/2)/(429*c**3*d**3*(d + e*x)*
*(5/2)) - 128*g*(a*e*g - c*d*f)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(
5/2)/(3003*c**4*d**4*e*(d + e*x)**(3/2)) + 128*(a*e*g - c*d*f)**3*(a*d*e + c*d*e
*x**2 + x*(a*e**2 + c*d**2))**(5/2)*(2*a*e**2*g + 5*c*d**2*g - 7*c*d*e*f)/(15015
*c**5*d**5*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.334669, size = 195, normalized size = 0.58 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (13 f+5 g x)+16 a^2 c^2 d^2 e^2 g^2 \left (143 f^2+130 f g x+35 g^2 x^2\right )-8 a c^3 d^3 e g \left (429 f^3+715 f^2 g x+455 f g^2 x^2+105 g^3 x^3\right )+c^4 d^4 \left (3003 f^4+8580 f^3 g x+10010 f^2 g^2 x^2+5460 f g^3 x^3+1155 g^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \]

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)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*(13*f +
5*g*x) + 16*a^2*c^2*d^2*e^2*g^2*(143*f^2 + 130*f*g*x + 35*g^2*x^2) - 8*a*c^3*d^
3*e*g*(429*f^3 + 715*f^2*g*x + 455*f*g^2*x^2 + 105*g^3*x^3) + c^4*d^4*(3003*f^4
+ 8580*f^3*g*x + 10010*f^2*g^2*x^2 + 5460*f*g^3*x^3 + 1155*g^4*x^4)))/(15015*c^5
*d^5*(d + e*x)^(5/2))

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Maple [A]  time = 0.013, size = 283, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 1155\,{g}^{4}{x}^{4}{c}^{4}{d}^{4}-840\,a{c}^{3}{d}^{3}e{g}^{4}{x}^{3}+5460\,{c}^{4}{d}^{4}f{g}^{3}{x}^{3}+560\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{g}^{4}{x}^{2}-3640\,a{c}^{3}{d}^{3}ef{g}^{3}{x}^{2}+10010\,{c}^{4}{d}^{4}{f}^{2}{g}^{2}{x}^{2}-320\,{a}^{3}cd{e}^{3}{g}^{4}x+2080\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}f{g}^{3}x-5720\,a{c}^{3}{d}^{3}e{f}^{2}{g}^{2}x+8580\,{c}^{4}{d}^{4}{f}^{3}gx+128\,{a}^{4}{e}^{4}{g}^{4}-832\,{a}^{3}cd{e}^{3}f{g}^{3}+2288\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{f}^{2}{g}^{2}-3432\,a{c}^{3}{d}^{3}e{f}^{3}g+3003\,{f}^{4}{c}^{4}{d}^{4} \right ) }{15015\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^(3/2),x)

[Out]

2/15015*(c*d*x+a*e)*(1155*c^4*d^4*g^4*x^4-840*a*c^3*d^3*e*g^4*x^3+5460*c^4*d^4*f
*g^3*x^3+560*a^2*c^2*d^2*e^2*g^4*x^2-3640*a*c^3*d^3*e*f*g^3*x^2+10010*c^4*d^4*f^
2*g^2*x^2-320*a^3*c*d*e^3*g^4*x+2080*a^2*c^2*d^2*e^2*f*g^3*x-5720*a*c^3*d^3*e*f^
2*g^2*x+8580*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-832*a^3*c*d*e^3*f*g^3+2288*a^2*c^2*
d^2*e^2*f^2*g^2-3432*a*c^3*d^3*e*f^3*g+3003*c^4*d^4*f^4)*(c*d*e*x^2+a*e^2*x+c*d^
2*x+a*d*e)^(3/2)/c^5/d^5/(e*x+d)^(3/2)

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Maxima [A]  time = 0.767313, size = 558, normalized size = 1.66 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{4}}{5 \, c d} + \frac{8 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{3} g}{35 \, c^{2} d^{2}} + \frac{4 \,{\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f^{2} g^{2}}{105 \, c^{3} d^{3}} + \frac{8 \,{\left (105 \, c^{5} d^{5} x^{5} + 140 \, a c^{4} d^{4} e x^{4} + 5 \, a^{2} c^{3} d^{3} e^{2} x^{3} - 6 \, a^{3} c^{2} d^{2} e^{3} x^{2} + 8 \, a^{4} c d e^{4} x - 16 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} f g^{3}}{1155 \, c^{4} d^{4}} + \frac{2 \,{\left (1155 \, c^{6} d^{6} x^{6} + 1470 \, a c^{5} d^{5} e x^{5} + 35 \, a^{2} c^{4} d^{4} e^{2} x^{4} - 40 \, a^{3} c^{3} d^{3} e^{3} x^{3} + 48 \, a^{4} c^{2} d^{2} e^{4} x^{2} - 64 \, a^{5} c d e^{5} x + 128 \, a^{6} e^{6}\right )} \sqrt{c d x + a e} g^{4}}{15015 \, c^{5} d^{5}} \]

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)^(3/2)*(g*x + f)^4/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*x + a*e)*f^4/(c*d) + 8/35*(5*
c^3*d^3*x^3 + 8*a*c^2*d^2*e*x^2 + a^2*c*d*e^2*x - 2*a^3*e^3)*sqrt(c*d*x + a*e)*f
^3*g/(c^2*d^2) + 4/105*(35*c^4*d^4*x^4 + 50*a*c^3*d^3*e*x^3 + 3*a^2*c^2*d^2*e^2*
x^2 - 4*a^3*c*d*e^3*x + 8*a^4*e^4)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*d^3) + 8/1155*
(105*c^5*d^5*x^5 + 140*a*c^4*d^4*e*x^4 + 5*a^2*c^3*d^3*e^2*x^3 - 6*a^3*c^2*d^2*e
^3*x^2 + 8*a^4*c*d*e^4*x - 16*a^5*e^5)*sqrt(c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/150
15*(1155*c^6*d^6*x^6 + 1470*a*c^5*d^5*e*x^5 + 35*a^2*c^4*d^4*e^2*x^4 - 40*a^3*c^
3*d^3*e^3*x^3 + 48*a^4*c^2*d^2*e^4*x^2 - 64*a^5*c*d*e^5*x + 128*a^6*e^6)*sqrt(c*
d*x + a*e)*g^4/(c^5*d^5)

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Fricas [A]  time = 0.275302, size = 1368, normalized size = 4.07 \[ \text{result too large to display} \]

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)^(3/2)*(g*x + f)^4/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*c^7*d^7*e*g^4*x^8 + 3003*a^3*c^4*d^5*e^3*f^4 - 3432*a^4*c^3*d^4*e^
4*f^3*g + 2288*a^5*c^2*d^3*e^5*f^2*g^2 - 832*a^6*c*d^2*e^6*f*g^3 + 128*a^7*d*e^7
*g^4 + 105*(52*c^7*d^7*e*f*g^3 + (11*c^7*d^8 + 25*a*c^6*d^6*e^2)*g^4)*x^7 + 35*(
286*c^7*d^7*e*f^2*g^2 + 52*(3*c^7*d^8 + 7*a*c^6*d^6*e^2)*f*g^3 + (75*a*c^6*d^7*e
+ 43*a^2*c^5*d^5*e^3)*g^4)*x^6 + 5*(1716*c^7*d^7*e*f^3*g + 286*(7*c^7*d^8 + 17*
a*c^6*d^6*e^2)*f^2*g^2 + 52*(49*a*c^6*d^7*e + 29*a^2*c^5*d^5*e^3)*f*g^3 + (301*a
^2*c^5*d^6*e^2 - a^3*c^4*d^4*e^4)*g^4)*x^5 + (3003*c^7*d^7*e*f^4 + 1716*(5*c^7*d
^8 + 13*a*c^6*d^6*e^2)*f^3*g + 286*(85*a*c^6*d^7*e + 53*a^2*c^5*d^5*e^3)*f^2*g^2
+ 52*(145*a^2*c^5*d^6*e^2 - a^3*c^4*d^4*e^4)*f*g^3 - (5*a^3*c^4*d^5*e^3 - 8*a^4
*c^3*d^3*e^5)*g^4)*x^4 + (3003*(c^7*d^8 + 3*a*c^6*d^6*e^2)*f^4 + 1716*(13*a*c^6*
d^7*e + 9*a^2*c^5*d^5*e^3)*f^3*g + 286*(53*a^2*c^5*d^6*e^2 - a^3*c^4*d^4*e^4)*f^
2*g^2 - 52*(a^3*c^4*d^5*e^3 - 2*a^4*c^3*d^3*e^5)*f*g^3 + 8*(a^4*c^3*d^4*e^4 - 2*
a^5*c^2*d^2*e^6)*g^4)*x^3 + (9009*(a*c^6*d^7*e + a^2*c^5*d^5*e^3)*f^4 + 1716*(9*
a^2*c^5*d^6*e^2 - a^3*c^4*d^4*e^4)*f^3*g - 286*(a^3*c^4*d^5*e^3 - 4*a^4*c^3*d^3*
e^5)*f^2*g^2 + 104*(a^4*c^3*d^4*e^4 - 4*a^5*c^2*d^2*e^6)*f*g^3 - 16*(a^5*c^2*d^3
*e^5 - 4*a^6*c*d*e^7)*g^4)*x^2 + (3003*(3*a^2*c^5*d^6*e^2 + a^3*c^4*d^4*e^4)*f^4
- 1716*(a^3*c^4*d^5*e^3 + 2*a^4*c^3*d^3*e^5)*f^3*g + 1144*(a^4*c^3*d^4*e^4 + 2*
a^5*c^2*d^2*e^6)*f^2*g^2 - 416*(a^5*c^2*d^3*e^5 + 2*a^6*c*d*e^7)*f*g^3 + 64*(a^6
*c*d^2*e^6 + 2*a^7*e^8)*g^4)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr
t(e*x + d)*c^5*d^5)

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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)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^4/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Timed out