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

Optimal. Leaf size=321 \[ \frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}-\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}} \]

[Out]

(8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g
))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^4*d^4*e*g*Sqrt[d + e*x])
- (8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*Sqrt[d + e*x]*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^3*d^3*e) - (2*(6*a*e^2*g + c*d*(e*f - 7*d
*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2*g*Sqrt
[d + e*x]) + (2*e*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*
d*g*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi [A]  time = 1.19965, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{8 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{105 c^4 d^4 e g \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (6 a e^2 g+c d (e f-7 d g)\right )}{105 c^3 d^3 e}-\frac{2 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g+c d (e f-7 d g)\right )}{35 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d g \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g
))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^4*d^4*e*g*Sqrt[d + e*x])
- (8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*Sqrt[d + e*x]*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(105*c^3*d^3*e) - (2*(6*a*e^2*g + c*d*(e*f - 7*d
*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(35*c^2*d^2*g*Sqrt
[d + e*x]) + (2*e*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(7*c*
d*g*Sqrt[d + e*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 95.1423, size = 325, normalized size = 1.01 \[ \frac{2 e \left (f + g x\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{7 c d g \sqrt{d + e x}} - \frac{2 \left (f + g x\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (6 a e^{2} g - 7 c d^{2} g + c d e f\right )}{35 c^{2} d^{2} g \sqrt{d + e x}} + \frac{8 \sqrt{d + e x} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (6 a e^{2} g - 7 c d^{2} g + c d e f\right )}{105 c^{3} d^{3} e} - \frac{8 \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (2 a e^{2} g + c d^{2} g - 3 c d e f\right ) \left (6 a e^{2} g - 7 c d^{2} g + c d e f\right )}{105 c^{4} d^{4} e g \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(f + g*x)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(7*c*d*g*sqrt(d
+ e*x)) - 2*(f + g*x)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(6*a*e**
2*g - 7*c*d**2*g + c*d*e*f)/(35*c**2*d**2*g*sqrt(d + e*x)) + 8*sqrt(d + e*x)*(a*
e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(6*a*e**2*g - 7*c*d*
*2*g + c*d*e*f)/(105*c**3*d**3*e) - 8*(a*e*g - c*d*f)*sqrt(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2))*(2*a*e**2*g + c*d**2*g - 3*c*d*e*f)*(6*a*e**2*g - 7*c*d**2*
g + c*d*e*f)/(105*c**4*d**4*e*g*sqrt(d + e*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.270622, size = 169, normalized size = 0.53 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-48 a^3 e^4 g^2+8 a^2 c d e^2 g (7 d g+14 e f+3 e g x)-2 a c^2 d^2 e \left (14 d g (5 f+g x)+e \left (35 f^2+28 f g x+9 g^2 x^2\right )\right )+c^3 d^3 \left (7 d \left (15 f^2+10 f g x+3 g^2 x^2\right )+e x \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )\right )}{105 c^4 d^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-48*a^3*e^4*g^2 + 8*a^2*c*d*e^2*g*(14*e*f + 7*
d*g + 3*e*g*x) - 2*a*c^2*d^2*e*(14*d*g*(5*f + g*x) + e*(35*f^2 + 28*f*g*x + 9*g^
2*x^2)) + c^3*d^3*(7*d*(15*f^2 + 10*f*g*x + 3*g^2*x^2) + e*x*(35*f^2 + 42*f*g*x
+ 15*g^2*x^2))))/(105*c^4*d^4*Sqrt[d + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 255, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -15\,e{g}^{2}{x}^{3}{c}^{3}{d}^{3}+18\,a{c}^{2}{d}^{2}{e}^{2}{g}^{2}{x}^{2}-21\,{c}^{3}{d}^{4}{g}^{2}{x}^{2}-42\,{c}^{3}{d}^{3}efg{x}^{2}-24\,{a}^{2}cd{e}^{3}{g}^{2}x+28\,a{c}^{2}{d}^{3}e{g}^{2}x+56\,a{c}^{2}{d}^{2}{e}^{2}fgx-70\,{c}^{3}{d}^{4}fgx-35\,{c}^{3}{d}^{3}e{f}^{2}x+48\,{a}^{3}{e}^{4}{g}^{2}-56\,{a}^{2}c{d}^{2}{e}^{2}{g}^{2}-112\,{a}^{2}cd{e}^{3}fg+140\,a{c}^{2}{d}^{3}efg+70\,a{c}^{2}{d}^{2}{e}^{2}{f}^{2}-105\,{d}^{4}{f}^{2}{c}^{3} \right ) }{105\,{c}^{4}{d}^{4}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(c*d*x+a*e)*(-15*c^3*d^3*e*g^2*x^3+18*a*c^2*d^2*e^2*g^2*x^2-21*c^3*d^4*g^
2*x^2-42*c^3*d^3*e*f*g*x^2-24*a^2*c*d*e^3*g^2*x+28*a*c^2*d^3*e*g^2*x+56*a*c^2*d^
2*e^2*f*g*x-70*c^3*d^4*f*g*x-35*c^3*d^3*e*f^2*x+48*a^3*e^4*g^2-56*a^2*c*d^2*e^2*
g^2-112*a^2*c*d*e^3*f*g+140*a*c^2*d^3*e*f*g+70*a*c^2*d^2*e^2*f^2-105*c^3*d^4*f^2
)*(e*x+d)^(1/2)/c^4/d^4/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.736747, size = 417, normalized size = 1.3 \[ \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{2}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{4 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f g}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{2 \,{\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \,{\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} -{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} g^{2}}{105 \, \sqrt{c d x + a e} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(c^2*d^2*e*x^2 + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)*f^2/(s
qrt(c*d*x + a*e)*c^2*d^2) + 4/15*(3*c^3*d^3*e*x^3 - 10*a^2*c*d^2*e^2 + 8*a^3*e^4
+ (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*c*d*e^3)*x)*f*g/(sqr
t(c*d*x + a*e)*c^3*d^3) + 2/105*(15*c^4*d^4*e*x^4 + 56*a^3*c*d^2*e^3 - 48*a^4*e^
5 + 3*(7*c^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*x^2
+ 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)*x)*g^2/(sqrt(c*d*x + a*e)*c^4*d^4)

_______________________________________________________________________________________

Fricas [A]  time = 0.270736, size = 689, normalized size = 2.15 \[ \frac{2 \,{\left (15 \, c^{4} d^{4} e^{2} g^{2} x^{5} + 3 \,{\left (14 \, c^{4} d^{4} e^{2} f g +{\left (12 \, c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} g^{2}\right )} x^{4} +{\left (35 \, c^{4} d^{4} e^{2} f^{2} + 14 \,{\left (8 \, c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} f g +{\left (21 \, c^{4} d^{6} - 10 \, a c^{3} d^{4} e^{2} + 6 \, a^{2} c^{2} d^{2} e^{4}\right )} g^{2}\right )} x^{3} + 35 \,{\left (3 \, a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3}\right )} f^{2} - 28 \,{\left (5 \, a^{2} c^{2} d^{4} e^{2} - 4 \, a^{3} c d^{2} e^{4}\right )} f g + 8 \,{\left (7 \, a^{3} c d^{3} e^{3} - 6 \, a^{4} d e^{5}\right )} g^{2} +{\left (35 \,{\left (4 \, c^{4} d^{5} e - a c^{3} d^{3} e^{3}\right )} f^{2} + 14 \,{\left (5 \, c^{4} d^{6} - 6 \, a c^{3} d^{4} e^{2} + 4 \, a^{2} c^{2} d^{2} e^{4}\right )} f g -{\left (7 \, a c^{3} d^{5} e - 34 \, a^{2} c^{2} d^{3} e^{3} + 24 \, a^{3} c d e^{5}\right )} g^{2}\right )} x^{2} +{\left (35 \,{\left (3 \, c^{4} d^{6} + 2 \, a c^{3} d^{4} e^{2} - 2 \, a^{2} c^{2} d^{2} e^{4}\right )} f^{2} - 14 \,{\left (5 \, a c^{3} d^{5} e + 6 \, a^{2} c^{2} d^{3} e^{3} - 8 \, a^{3} c d e^{5}\right )} f g + 4 \,{\left (7 \, a^{2} c^{2} d^{4} e^{2} + 8 \, a^{3} c d^{2} e^{4} - 12 \, a^{4} e^{6}\right )} g^{2}\right )} x\right )}}{105 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*c^4*d^4*e^2*g^2*x^5 + 3*(14*c^4*d^4*e^2*f*g + (12*c^4*d^5*e - a*c^3*d^
3*e^3)*g^2)*x^4 + (35*c^4*d^4*e^2*f^2 + 14*(8*c^4*d^5*e - a*c^3*d^3*e^3)*f*g + (
21*c^4*d^6 - 10*a*c^3*d^4*e^2 + 6*a^2*c^2*d^2*e^4)*g^2)*x^3 + 35*(3*a*c^3*d^5*e
- 2*a^2*c^2*d^3*e^3)*f^2 - 28*(5*a^2*c^2*d^4*e^2 - 4*a^3*c*d^2*e^4)*f*g + 8*(7*a
^3*c*d^3*e^3 - 6*a^4*d*e^5)*g^2 + (35*(4*c^4*d^5*e - a*c^3*d^3*e^3)*f^2 + 14*(5*
c^4*d^6 - 6*a*c^3*d^4*e^2 + 4*a^2*c^2*d^2*e^4)*f*g - (7*a*c^3*d^5*e - 34*a^2*c^2
*d^3*e^3 + 24*a^3*c*d*e^5)*g^2)*x^2 + (35*(3*c^4*d^6 + 2*a*c^3*d^4*e^2 - 2*a^2*c
^2*d^2*e^4)*f^2 - 14*(5*a*c^3*d^5*e + 6*a^2*c^2*d^3*e^3 - 8*a^3*c*d*e^5)*f*g + 4
*(7*a^2*c^2*d^4*e^2 + 8*a^3*c*d^2*e^4 - 12*a^4*e^6)*g^2)*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)

_______________________________________________________________________________________

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((e*x+d)**(3/2)*(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{2}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^(3/2)*(g*x + f)^2/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
), x)