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

Optimal. Leaf size=125 \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{35 c^2 d^2 e (d+e x)^{5/2}} \]

[Out]

(-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))/(35*c^2*d^2*e*(d + e*x)^(5/2)) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2)^(5/2))/(7*c*d*e*(d + e*x)^(3/2))

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Rubi [A]  time = 0.327352, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{35 c^2 d^2 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(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*(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))/(35*c^2*d^2*e*(d + e*x)^(5/2)) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2)^(5/2))/(7*c*d*e*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 34.9547, size = 121, normalized size = 0.97 \[ \frac{2 g \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 c d e \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} \left (2 a e^{2} g + 5 c d^{2} g - 7 c d e f\right )}{35 c^{2} d^{2} e \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(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*g*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/(7*c*d*e*(d + e*x)**(3/2))
- 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)/(35*c**2*d**2*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.0991003, size = 54, normalized size = 0.43 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} (c d (7 f+5 g x)-2 a e g)}{35 c^2 d^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(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)*(-2*a*e*g + c*d*(7*f + 5*g*x)))/(35*c^2*d^2*(
d + e*x)^(5/2))

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Maple [A]  time = 0.007, size = 67, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,xcdg+2\,aeg-7\,cdf \right ) }{35\,{c}^{2}{d}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(c*d*x+a*e)*(-5*c*d*g*x+2*a*e*g-7*c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(3/2)/c^2/d^2/(e*x+d)^(3/2)

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Maxima [A]  time = 0.72322, size = 144, normalized size = 1.15 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f}{5 \, c d} + \frac{2 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} g}{35 \, c^{2} d^{2}} \]

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)/(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/(c*d) + 2/35*(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)*g/(
c^2*d^2)

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Fricas [A]  time = 0.267671, size = 382, normalized size = 3.06 \[ \frac{2 \,{\left (5 \, c^{4} d^{4} e g x^{5} + 7 \, a^{3} c d^{2} e^{3} f - 2 \, a^{4} d e^{4} g +{\left (7 \, c^{4} d^{4} e f +{\left (5 \, c^{4} d^{5} + 13 \, a c^{3} d^{3} e^{2}\right )} g\right )} x^{4} +{\left (7 \,{\left (c^{4} d^{5} + 3 \, a c^{3} d^{3} e^{2}\right )} f +{\left (13 \, a c^{3} d^{4} e + 9 \, a^{2} c^{2} d^{2} e^{3}\right )} g\right )} x^{3} +{\left (21 \,{\left (a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f +{\left (9 \, a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} g\right )} x^{2} +{\left (7 \,{\left (3 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f -{\left (a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5}\right )} g\right )} x\right )}}{35 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]

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

[Out]

2/35*(5*c^4*d^4*e*g*x^5 + 7*a^3*c*d^2*e^3*f - 2*a^4*d*e^4*g + (7*c^4*d^4*e*f + (
5*c^4*d^5 + 13*a*c^3*d^3*e^2)*g)*x^4 + (7*(c^4*d^5 + 3*a*c^3*d^3*e^2)*f + (13*a*
c^3*d^4*e + 9*a^2*c^2*d^2*e^3)*g)*x^3 + (21*(a*c^3*d^4*e + a^2*c^2*d^2*e^3)*f +
(9*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*g)*x^2 + (7*(3*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*
f - (a^3*c*d^2*e^3 + 2*a^4*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^2*d^2)

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

[Out]

Timed out