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

Optimal. Leaf size=209 \[ -\frac{4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (5 e f-d g)\right )}{15 c^3 d^3 e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (5 e f-d g)\right )}{15 c^2 d^2 e}+\frac{2 g (d+e x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d e} \]

[Out]

(-4*(c*d^2 - a*e^2)*(4*a*e^2*g - c*d*(5*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(15*c^3*d^3*e*Sqrt[d + e*x]) - (2*(4*a*e^2*g - c*d*(5*e*f - d*g
))*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^2*d^2*e) + (
2*g*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.543832, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (5 e f-d g)\right )}{15 c^3 d^3 e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (5 e f-d g)\right )}{15 c^2 d^2 e}+\frac{2 g (d+e x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d e} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(c*d^2 - a*e^2)*(4*a*e^2*g - c*d*(5*e*f - d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(15*c^3*d^3*e*Sqrt[d + e*x]) - (2*(4*a*e^2*g - c*d*(5*e*f - d*g
))*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^2*d^2*e) + (
2*g*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 54.7406, size = 202, normalized size = 0.97 \[ \frac{2 g \left (d + e x\right )^{\frac{3}{2}} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 c d e} - \frac{2 \sqrt{d + e x} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (4 a e^{2} g + c d^{2} g - 5 c d e f\right )}{15 c^{2} d^{2} e} + \frac{4 \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (4 a e^{2} g + c d^{2} g - 5 c d e f\right )}{15 c^{3} d^{3} e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*g*(d + e*x)**(3/2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(5*c*d*e) -
2*sqrt(d + e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(4*a*e**2*g + c*d
**2*g - 5*c*d*e*f)/(15*c**2*d**2*e) + 4*(a*e**2 - c*d**2)*sqrt(a*d*e + c*d*e*x**
2 + x*(a*e**2 + c*d**2))*(4*a*e**2*g + c*d**2*g - 5*c*d*e*f)/(15*c**3*d**3*e*sqr
t(d + e*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.130275, size = 96, normalized size = 0.46 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 a^2 e^3 g-2 a c d e (5 d g+5 e f+2 e g x)+c^2 d^2 (5 d (3 f+g x)+e x (5 f+3 g x))\right )}{15 c^3 d^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(3/2)*(f + g*x))/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)]*(8*a^2*e^3*g - 2*a*c*d*e*(5*e*f + 5*d*g + 2*e*g
*x) + c^2*d^2*(5*d*(3*f + g*x) + e*x*(5*f + 3*g*x))))/(15*c^3*d^3*Sqrt[d + e*x])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 131, normalized size = 0.6 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,eg{x}^{2}{c}^{2}{d}^{2}-4\,acd{e}^{2}gx+5\,{c}^{2}{d}^{3}gx+5\,{c}^{2}{d}^{2}efx+8\,{a}^{2}{e}^{3}g-10\,ac{d}^{2}eg-10\,acd{e}^{2}f+15\,{d}^{3}f{c}^{2} \right ) }{15\,{c}^{3}{d}^{3}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(c*d*x+a*e)*(3*c^2*d^2*e*g*x^2-4*a*c*d*e^2*g*x+5*c^2*d^3*g*x+5*c^2*d^2*e*f*
x+8*a^2*e^3*g-10*a*c*d^2*e*g-10*a*c*d*e^2*f+15*c^2*d^3*f)*(e*x+d)^(1/2)/c^3/d^3/
(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.72474, size = 227, normalized size = 1.09 \[ \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{2 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} g}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)/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/(sqr
t(c*d*x + a*e)*c^2*d^2) + 2/15*(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)*g/(sqrt(c*
d*x + a*e)*c^3*d^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.269422, size = 383, normalized size = 1.83 \[ \frac{2 \,{\left (3 \, c^{3} d^{3} e^{2} g x^{4} +{\left (5 \, c^{3} d^{3} e^{2} f +{\left (8 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} g\right )} x^{3} +{\left (5 \,{\left (4 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} f +{\left (5 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} + 4 \, a^{2} c d e^{4}\right )} g\right )} x^{2} + 5 \,{\left (3 \, a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3}\right )} f - 2 \,{\left (5 \, a^{2} c d^{3} e^{2} - 4 \, a^{3} d e^{4}\right )} g +{\left (5 \,{\left (3 \, c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} - 2 \, a^{2} c d e^{4}\right )} f -{\left (5 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} - 8 \, a^{3} e^{5}\right )} g\right )} x\right )}}{15 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*c^3*d^3*e^2*g*x^4 + (5*c^3*d^3*e^2*f + (8*c^3*d^4*e - a*c^2*d^2*e^3)*g)*
x^3 + (5*(4*c^3*d^4*e - a*c^2*d^2*e^3)*f + (5*c^3*d^5 - 6*a*c^2*d^3*e^2 + 4*a^2*
c*d*e^4)*g)*x^2 + 5*(3*a*c^2*d^4*e - 2*a^2*c*d^2*e^3)*f - 2*(5*a^2*c*d^3*e^2 - 4
*a^3*d*e^4)*g + (5*(3*c^3*d^5 + 2*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4)*f - (5*a*c^2*d^
4*e + 6*a^2*c*d^2*e^3 - 8*a^3*e^5)*g)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)*sqrt(e*x + d)*c^3*d^3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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