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

Optimal. Leaf size=269 \[ -\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt{d+e x}}+\frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]

[Out]

(-16*(c*d*f - a*e*g)^2*(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))/(315*c^4*d^4*e*(d + e*x)^(3/2)) + (16*g*(c*d*f - a*e*g)^
2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*c^3*d^3*e*Sqrt[d + e*x]) +
(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)^(3/2))/(
21*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2)^(3/2))/(9*c*d*(d + e*x)^(3/2))

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Rubi [A]  time = 1.10868, 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 \[ -\frac{16 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{315 c^4 d^4 e (d+e x)^{3/2}}+\frac{16 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{105 c^3 d^3 e \sqrt{d+e x}}+\frac{4 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{21 c^2 d^2 (d+e x)^{3/2}}+\frac{2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-16*(c*d*f - a*e*g)^2*(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))/(315*c^4*d^4*e*(d + e*x)^(3/2)) + (16*g*(c*d*f - a*e*g)^
2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*c^3*d^3*e*Sqrt[d + e*x]) +
(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)^(3/2))/(
21*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2)^(3/2))/(9*c*d*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 95.0646, size = 264, normalized size = 0.98 \[ \frac{2 \left (f + g x\right )^{3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{9 c d \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 \left (f + g x\right )^{2} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{21 c^{2} d^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{16 g \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{105 c^{3} d^{3} e \sqrt{d + e x}} - \frac{16 \left (a e g - c d f\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}} \left (2 a e^{2} g + 3 c d^{2} g - 5 c d e f\right )}{315 c^{4} d^{4} e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**3*(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)**3*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(9*c*d*(d + e*x
)**(3/2)) - 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))**(3/2)/(21*c**2*d**2*(d + e*x)**(3/2)) + 16*g*(a*e*g - c*d*f)**2*(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(105*c**3*d**3*e*sqrt(d + e*x)) - 16*(a
*e*g - c*d*f)**2*(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)/(315*c**4*d**4*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.214077, size = 136, normalized size = 0.51 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (3 f+g x)-6 a c^2 d^2 e g \left (21 f^2+18 f g x+5 g^2 x^2\right )+c^3 d^3 \left (105 f^3+189 f^2 g x+135 f g^2 x^2+35 g^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^3*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)*(-16*a^3*e^3*g^3 + 24*a^2*c*d*e^2*g^2*(3*f +
g*x) - 6*a*c^2*d^2*e*g*(21*f^2 + 18*f*g*x + 5*g^2*x^2) + c^3*d^3*(105*f^3 + 189*
f^2*g*x + 135*f*g^2*x^2 + 35*g^3*x^3)))/(315*c^4*d^4*(d + e*x)^(3/2))

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Maple [A]  time = 0.012, size = 188, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -35\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+30\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-135\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-24\,{a}^{2}cd{e}^{2}{g}^{3}x+108\,a{c}^{2}{d}^{2}ef{g}^{2}x-189\,{c}^{3}{d}^{3}{f}^{2}gx+16\,{a}^{3}{e}^{3}{g}^{3}-72\,{a}^{2}cd{e}^{2}f{g}^{2}+126\,a{c}^{2}{d}^{2}e{f}^{2}g-105\,{f}^{3}{c}^{3}{d}^{3} \right ) }{315\,{c}^{4}{d}^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)

[Out]

-2/315*(c*d*x+a*e)*(-35*c^3*d^3*g^3*x^3+30*a*c^2*d^2*e*g^3*x^2-135*c^3*d^3*f*g^2
*x^2-24*a^2*c*d*e^2*g^3*x+108*a*c^2*d^2*e*f*g^2*x-189*c^3*d^3*f^2*g*x+16*a^3*e^3
*g^3-72*a^2*c*d*e^2*f*g^2+126*a*c^2*d^2*e*f^2*g-105*c^3*d^3*f^3)*(c*d*e*x^2+a*e^
2*x+c*d^2*x+a*d*e)^(1/2)/c^4/d^4/(e*x+d)^(1/2)

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Maxima [A]  time = 0.747628, size = 294, normalized size = 1.09 \[ \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f^{3}}{3 \, c d} + \frac{2 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} f^{2} g}{5 \, c^{2} d^{2}} + \frac{2 \,{\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} f g^{2}}{35 \, c^{3} d^{3}} + \frac{2 \,{\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} g^{3}}{315 \, c^{4} d^{4}} \]

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)^3/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/3*(c*d*x + a*e)^(3/2)*f^3/(c*d) + 2/5*(3*c^2*d^2*x^2 + a*c*d*e*x - 2*a^2*e^2)*
sqrt(c*d*x + a*e)*f^2*g/(c^2*d^2) + 2/35*(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*g^2/(c^3*d^3) + 2/315*(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)*g^3/(c^4*d^4)

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Fricas [A]  time = 0.276498, size = 799, normalized size = 2.97 \[ \frac{2 \,{\left (35 \, c^{5} d^{5} e g^{3} x^{6} + 105 \, a^{2} c^{3} d^{4} e^{2} f^{3} - 126 \, a^{3} c^{2} d^{3} e^{3} f^{2} g + 72 \, a^{4} c d^{2} e^{4} f g^{2} - 16 \, a^{5} d e^{5} g^{3} + 5 \,{\left (27 \, c^{5} d^{5} e f g^{2} +{\left (7 \, c^{5} d^{6} + 8 \, a c^{4} d^{4} e^{2}\right )} g^{3}\right )} x^{5} +{\left (189 \, c^{5} d^{5} e f^{2} g + 27 \,{\left (5 \, c^{5} d^{6} + 6 \, a c^{4} d^{4} e^{2}\right )} f g^{2} +{\left (40 \, a c^{4} d^{5} e - a^{2} c^{3} d^{3} e^{3}\right )} g^{3}\right )} x^{4} +{\left (105 \, c^{5} d^{5} e f^{3} + 63 \,{\left (3 \, c^{5} d^{6} + 4 \, a c^{4} d^{4} e^{2}\right )} f^{2} g + 9 \,{\left (18 \, a c^{4} d^{5} e - a^{2} c^{3} d^{3} e^{3}\right )} f g^{2} -{\left (a^{2} c^{3} d^{4} e^{2} - 2 \, a^{3} c^{2} d^{2} e^{4}\right )} g^{3}\right )} x^{3} +{\left (105 \,{\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} f^{3} + 63 \,{\left (4 \, a c^{4} d^{5} e - a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g - 9 \,{\left (a^{2} c^{3} d^{4} e^{2} - 4 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{2} + 2 \,{\left (a^{3} c^{2} d^{3} e^{3} - 4 \, a^{4} c d e^{5}\right )} g^{3}\right )} x^{2} +{\left (105 \,{\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{3} - 63 \,{\left (a^{2} c^{3} d^{4} e^{2} + 2 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g + 36 \,{\left (a^{3} c^{2} d^{3} e^{3} + 2 \, a^{4} c d e^{5}\right )} f g^{2} - 8 \,{\left (a^{4} c d^{2} e^{4} + 2 \, a^{5} e^{6}\right )} g^{3}\right )} x\right )}}{315 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^3/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/315*(35*c^5*d^5*e*g^3*x^6 + 105*a^2*c^3*d^4*e^2*f^3 - 126*a^3*c^2*d^3*e^3*f^2*
g + 72*a^4*c*d^2*e^4*f*g^2 - 16*a^5*d*e^5*g^3 + 5*(27*c^5*d^5*e*f*g^2 + (7*c^5*d
^6 + 8*a*c^4*d^4*e^2)*g^3)*x^5 + (189*c^5*d^5*e*f^2*g + 27*(5*c^5*d^6 + 6*a*c^4*
d^4*e^2)*f*g^2 + (40*a*c^4*d^5*e - a^2*c^3*d^3*e^3)*g^3)*x^4 + (105*c^5*d^5*e*f^
3 + 63*(3*c^5*d^6 + 4*a*c^4*d^4*e^2)*f^2*g + 9*(18*a*c^4*d^5*e - a^2*c^3*d^3*e^3
)*f*g^2 - (a^2*c^3*d^4*e^2 - 2*a^3*c^2*d^2*e^4)*g^3)*x^3 + (105*(c^5*d^6 + 2*a*c
^4*d^4*e^2)*f^3 + 63*(4*a*c^4*d^5*e - a^2*c^3*d^3*e^3)*f^2*g - 9*(a^2*c^3*d^4*e^
2 - 4*a^3*c^2*d^2*e^4)*f*g^2 + 2*(a^3*c^2*d^3*e^3 - 4*a^4*c*d*e^5)*g^3)*x^2 + (1
05*(2*a*c^4*d^5*e + a^2*c^3*d^3*e^3)*f^3 - 63*(a^2*c^3*d^4*e^2 + 2*a^3*c^2*d^2*e
^4)*f^2*g + 36*(a^3*c^2*d^3*e^3 + 2*a^4*c*d*e^5)*f*g^2 - 8*(a^4*c*d^2*e^4 + 2*a^
5*e^6)*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)**(1/2)/(e*x+d)**(1/2),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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^3/sqrt(e*x + d),x, algorithm="giac")

[Out]

Timed out