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

Optimal. Leaf size=269 \[ -\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{1155 c^4 d^4 e (d+e x)^{5/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}} \]

[Out]

(-16*(c*d*f - a*e*g)^2*(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))/(1155*c^4*d^4*e*(d + e*x)^(5/2)) + (16*g*(c*d*f - a*e*g)
^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*c^3*d^3*e*(d + e*x)^(3/2)
) + (4*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)
)/(33*c^2*d^2*(d + e*x)^(5/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(5/2))/(11*c*d*(d + e*x)^(5/2))

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Rubi [A]  time = 1.10891, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{1155 c^4 d^4 e (d+e x)^{5/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)^2}{231 c^3 d^3 e (d+e x)^{3/2}}+\frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} (c d f-a e g)}{33 c^2 d^2 (d+e x)^{5/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(-16*(c*d*f - a*e*g)^2*(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))/(1155*c^4*d^4*e*(d + e*x)^(5/2)) + (16*g*(c*d*f - a*e*g)
^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*c^3*d^3*e*(d + e*x)^(3/2)
) + (4*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)
)/(33*c^2*d^2*(d + e*x)^(5/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d
*e*x^2)^(5/2))/(11*c*d*(d + e*x)^(5/2))

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Rubi in Sympy [A]  time = 93.7573, size = 264, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{11 c d \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 \left (f + g x\right )^{2} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{33 c^{2} d^{2} \left (d + e x\right )^{\frac{5}{2}}} + \frac{16 g \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{231 c^{3} d^{3} e \left (d + e x\right )^{\frac{3}{2}}} - \frac{16 \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} \left (2 a e^{2} g + 5 c d^{2} g - 7 c d e f\right )}{1155 c^{4} d^{4} e \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**3*(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)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(11*c*d*(d + e*
x)**(5/2)) - 4*(f + g*x)**2*(a*e*g - c*d*f)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*
d**2))**(5/2)/(33*c**2*d**2*(d + e*x)**(5/2)) + 16*g*(a*e*g - c*d*f)**2*(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(231*c**3*d**3*e*(d + e*x)**(3/2)) - 1
6*(a*e*g - c*d*f)**2*(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)/(1155*c**4*d**4*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.242918, size = 137, normalized size = 0.51 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (11 f+5 g x)-2 a c^2 d^2 e g \left (99 f^2+110 f g x+35 g^2 x^2\right )+c^3 d^3 \left (231 f^3+495 f^2 g x+385 f g^2 x^2+105 g^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^3*(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)*(-16*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(11*f +
5*g*x) - 2*a*c^2*d^2*e*g*(99*f^2 + 110*f*g*x + 35*g^2*x^2) + c^3*d^3*(231*f^3 +
495*f^2*g*x + 385*f*g^2*x^2 + 105*g^3*x^3)))/(1155*c^4*d^4*(d + e*x)^(5/2))

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Maple [A]  time = 0.012, size = 188, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -105\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+70\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-385\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-40\,{a}^{2}cd{e}^{2}{g}^{3}x+220\,a{c}^{2}{d}^{2}ef{g}^{2}x-495\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-88\,{a}^{2}cd{e}^{2}f{g}^{2}+198\,a{c}^{2}{d}^{2}e{f}^{2}g-231\,{f}^{3}{c}^{3}{d}^{3} \right ) }{1155\,{c}^{4}{d}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x)

[Out]

-2/1155*(c*d*x+a*e)*(-105*c^3*d^3*g^3*x^3+70*a*c^2*d^2*e*g^3*x^2-385*c^3*d^3*f*g
^2*x^2-40*a^2*c*d*e^2*g^3*x+220*a*c^2*d^2*e*f*g^2*x-495*c^3*d^3*f^2*g*x+16*a^3*e
^3*g^3-88*a^2*c*d*e^2*f*g^2+198*a*c^2*d^2*e*f^2*g-231*c^3*d^3*f^3)*(c*d*e*x^2+a*
e^2*x+c*d^2*x+a*d*e)^(3/2)/c^4/d^4/(e*x+d)^(3/2)

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Maxima [A]  time = 0.743763, size = 397, normalized size = 1.48 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{3}}{5 \, c d} + \frac{6 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{2} g}{35 \, c^{2} d^{2}} + \frac{2 \,{\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f g^{2}}{105 \, c^{3} d^{3}} + \frac{2 \,{\left (105 \, c^{5} d^{5} x^{5} + 140 \, a c^{4} d^{4} e x^{4} + 5 \, a^{2} c^{3} d^{3} e^{2} x^{3} - 6 \, a^{3} c^{2} d^{2} e^{3} x^{2} + 8 \, a^{4} c d e^{4} x - 16 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} g^{3}}{1155 \, c^{4} d^{4}} \]

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)^(3/2)*(g*x + f)^3/(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^3/(c*d) + 6/35*(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
^2*g/(c^2*d^2) + 2/105*(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*g^2/(c^3*d^3) + 2/1155*(1
05*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)*g^3/(c^4*d^4)

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Fricas [A]  time = 0.276334, size = 990, normalized size = 3.68 \[ \frac{2 \,{\left (105 \, c^{6} d^{6} e g^{3} x^{7} + 231 \, a^{3} c^{3} d^{4} e^{3} f^{3} - 198 \, a^{4} c^{2} d^{3} e^{4} f^{2} g + 88 \, a^{5} c d^{2} e^{5} f g^{2} - 16 \, a^{6} d e^{6} g^{3} + 35 \,{\left (11 \, c^{6} d^{6} e f g^{2} +{\left (3 \, c^{6} d^{7} + 7 \, a c^{5} d^{5} e^{2}\right )} g^{3}\right )} x^{6} + 5 \,{\left (99 \, c^{6} d^{6} e f^{2} g + 11 \,{\left (7 \, c^{6} d^{7} + 17 \, a c^{5} d^{5} e^{2}\right )} f g^{2} +{\left (49 \, a c^{5} d^{6} e + 29 \, a^{2} c^{4} d^{4} e^{3}\right )} g^{3}\right )} x^{5} +{\left (231 \, c^{6} d^{6} e f^{3} + 99 \,{\left (5 \, c^{6} d^{7} + 13 \, a c^{5} d^{5} e^{2}\right )} f^{2} g + 11 \,{\left (85 \, a c^{5} d^{6} e + 53 \, a^{2} c^{4} d^{4} e^{3}\right )} f g^{2} +{\left (145 \, a^{2} c^{4} d^{5} e^{2} - a^{3} c^{3} d^{3} e^{4}\right )} g^{3}\right )} x^{4} +{\left (231 \,{\left (c^{6} d^{7} + 3 \, a c^{5} d^{5} e^{2}\right )} f^{3} + 99 \,{\left (13 \, a c^{5} d^{6} e + 9 \, a^{2} c^{4} d^{4} e^{3}\right )} f^{2} g + 11 \,{\left (53 \, a^{2} c^{4} d^{5} e^{2} - a^{3} c^{3} d^{3} e^{4}\right )} f g^{2} -{\left (a^{3} c^{3} d^{4} e^{3} - 2 \, a^{4} c^{2} d^{2} e^{5}\right )} g^{3}\right )} x^{3} +{\left (693 \,{\left (a c^{5} d^{6} e + a^{2} c^{4} d^{4} e^{3}\right )} f^{3} + 99 \,{\left (9 \, a^{2} c^{4} d^{5} e^{2} - a^{3} c^{3} d^{3} e^{4}\right )} f^{2} g - 11 \,{\left (a^{3} c^{3} d^{4} e^{3} - 4 \, a^{4} c^{2} d^{2} e^{5}\right )} f g^{2} + 2 \,{\left (a^{4} c^{2} d^{3} e^{4} - 4 \, a^{5} c d e^{6}\right )} g^{3}\right )} x^{2} +{\left (231 \,{\left (3 \, a^{2} c^{4} d^{5} e^{2} + a^{3} c^{3} d^{3} e^{4}\right )} f^{3} - 99 \,{\left (a^{3} c^{3} d^{4} e^{3} + 2 \, a^{4} c^{2} d^{2} e^{5}\right )} f^{2} g + 44 \,{\left (a^{4} c^{2} d^{3} e^{4} + 2 \, a^{5} c d e^{6}\right )} f g^{2} - 8 \,{\left (a^{5} c d^{2} e^{5} + 2 \, a^{6} e^{7}\right )} g^{3}\right )} x\right )}}{1155 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

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

[Out]

2/1155*(105*c^6*d^6*e*g^3*x^7 + 231*a^3*c^3*d^4*e^3*f^3 - 198*a^4*c^2*d^3*e^4*f^
2*g + 88*a^5*c*d^2*e^5*f*g^2 - 16*a^6*d*e^6*g^3 + 35*(11*c^6*d^6*e*f*g^2 + (3*c^
6*d^7 + 7*a*c^5*d^5*e^2)*g^3)*x^6 + 5*(99*c^6*d^6*e*f^2*g + 11*(7*c^6*d^7 + 17*a
*c^5*d^5*e^2)*f*g^2 + (49*a*c^5*d^6*e + 29*a^2*c^4*d^4*e^3)*g^3)*x^5 + (231*c^6*
d^6*e*f^3 + 99*(5*c^6*d^7 + 13*a*c^5*d^5*e^2)*f^2*g + 11*(85*a*c^5*d^6*e + 53*a^
2*c^4*d^4*e^3)*f*g^2 + (145*a^2*c^4*d^5*e^2 - a^3*c^3*d^3*e^4)*g^3)*x^4 + (231*(
c^6*d^7 + 3*a*c^5*d^5*e^2)*f^3 + 99*(13*a*c^5*d^6*e + 9*a^2*c^4*d^4*e^3)*f^2*g +
11*(53*a^2*c^4*d^5*e^2 - a^3*c^3*d^3*e^4)*f*g^2 - (a^3*c^3*d^4*e^3 - 2*a^4*c^2*
d^2*e^5)*g^3)*x^3 + (693*(a*c^5*d^6*e + a^2*c^4*d^4*e^3)*f^3 + 99*(9*a^2*c^4*d^5
*e^2 - a^3*c^3*d^3*e^4)*f^2*g - 11*(a^3*c^3*d^4*e^3 - 4*a^4*c^2*d^2*e^5)*f*g^2 +
2*(a^4*c^2*d^3*e^4 - 4*a^5*c*d*e^6)*g^3)*x^2 + (231*(3*a^2*c^4*d^5*e^2 + a^3*c^
3*d^3*e^4)*f^3 - 99*(a^3*c^3*d^4*e^3 + 2*a^4*c^2*d^2*e^5)*f^2*g + 44*(a^4*c^2*d^
3*e^4 + 2*a^5*c*d*e^6)*f*g^2 - 8*(a^5*c*d^2*e^5 + 2*a^6*e^7)*g^3)*x)/(sqrt(c*d*e
*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*c^4*d^4)

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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)**3*(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)^3/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Timed out