3.758 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(
7/2)*(f + g*x)^(9/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6
3*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.528807, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(11/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(
7/2)*(f + g*x)^(9/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6
3*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.1642, size = 122, normalized size = 0.95 \[ \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{63 \left (d + e x\right )^{\frac{7}{2}} \left (f + g x\right )^{\frac{7}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{9 \left (d + e x\right )^{\frac{7}{2}} \left (f + g x\right )^{\frac{9}{2}} \left (a e g - c d f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(11/2),x)

[Out]

4*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(63*(d + e*x)**(7/2)*(f
+ g*x)**(7/2)*(a*e*g - c*d*f)**2) - 2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
**(7/2)/(9*(d + e*x)**(7/2)*(f + g*x)**(9/2)*(a*e*g - c*d*f))

_______________________________________________________________________________________

Mathematica [A]  time = 0.265054, size = 79, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} (c d (9 f+2 g x)-7 a e g)}{63 \sqrt{d+e x} (f+g x)^{9/2} (c d f-a e g)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(11/2)),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-7*a*e*g + c*d*(9*f + 2*g*x)))
/(63*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 99, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,xcdg+7\,aeg-9\,cdf \right ) }{63\,{a}^{2}{e}^{2}{g}^{2}-126\,acdefg+63\,{c}^{2}{d}^{2}{f}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( gx+f \right ) ^{-{\frac{9}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(11/2),x)

[Out]

-2/63*(c*d*x+a*e)*(-2*c*d*g*x+7*a*e*g-9*c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(5/2)/(g*x+f)^(9/2)/(a^2*e^2*g^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(e*x+d)^(5/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{11}{2}}}\,{d x} \]

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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(11/2)),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x +
f)^(11/2)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.303034, size = 863, normalized size = 6.69 \[ \frac{2 \,{\left (2 \, c^{4} d^{4} g x^{4} + 9 \, a^{3} c d e^{3} f - 7 \, a^{4} e^{4} g +{\left (9 \, c^{4} d^{4} f - a c^{3} d^{3} e g\right )} x^{3} + 3 \,{\left (9 \, a c^{3} d^{3} e f - 5 \, a^{2} c^{2} d^{2} e^{2} g\right )} x^{2} +{\left (27 \, a^{2} c^{2} d^{2} e^{2} f - 19 \, a^{3} c d e^{3} g\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{63 \,{\left (c^{2} d^{3} f^{7} - 2 \, a c d^{2} e f^{6} g + a^{2} d e^{2} f^{5} g^{2} +{\left (c^{2} d^{2} e f^{2} g^{5} - 2 \, a c d e^{2} f g^{6} + a^{2} e^{3} g^{7}\right )} x^{6} +{\left (5 \, c^{2} d^{2} e f^{3} g^{4} + a^{2} d e^{2} g^{7} +{\left (c^{2} d^{3} - 10 \, a c d e^{2}\right )} f^{2} g^{5} -{\left (2 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} f g^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{2} d^{2} e f^{4} g^{3} + a^{2} d e^{2} f g^{6} +{\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{3} g^{4} - 2 \,{\left (a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{5}\right )} x^{4} + 10 \,{\left (c^{2} d^{2} e f^{5} g^{2} + a^{2} d e^{2} f^{2} g^{5} +{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g^{3} -{\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{4}\right )} x^{3} + 5 \,{\left (c^{2} d^{2} e f^{6} g + 2 \, a^{2} d e^{2} f^{3} g^{4} + 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} f^{5} g^{2} -{\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{3}\right )} x^{2} +{\left (c^{2} d^{2} e f^{7} + 5 \, a^{2} d e^{2} f^{4} g^{3} +{\left (5 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{6} g -{\left (10 \, a c d^{2} e - a^{2} e^{3}\right )} f^{5} g^{2}\right )} x\right )}} \]

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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(11/2)),x, algorithm="fricas")

[Out]

2/63*(2*c^4*d^4*g*x^4 + 9*a^3*c*d*e^3*f - 7*a^4*e^4*g + (9*c^4*d^4*f - a*c^3*d^3
*e*g)*x^3 + 3*(9*a*c^3*d^3*e*f - 5*a^2*c^2*d^2*e^2*g)*x^2 + (27*a^2*c^2*d^2*e^2*
f - 19*a^3*c*d*e^3*g)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x +
d)*sqrt(g*x + f)/(c^2*d^3*f^7 - 2*a*c*d^2*e*f^6*g + a^2*d*e^2*f^5*g^2 + (c^2*d^2
*e*f^2*g^5 - 2*a*c*d*e^2*f*g^6 + a^2*e^3*g^7)*x^6 + (5*c^2*d^2*e*f^3*g^4 + a^2*d
*e^2*g^7 + (c^2*d^3 - 10*a*c*d*e^2)*f^2*g^5 - (2*a*c*d^2*e - 5*a^2*e^3)*f*g^6)*x
^5 + 5*(2*c^2*d^2*e*f^4*g^3 + a^2*d*e^2*f*g^6 + (c^2*d^3 - 4*a*c*d*e^2)*f^3*g^4
- 2*(a*c*d^2*e - a^2*e^3)*f^2*g^5)*x^4 + 10*(c^2*d^2*e*f^5*g^2 + a^2*d*e^2*f^2*g
^5 + (c^2*d^3 - 2*a*c*d*e^2)*f^4*g^3 - (2*a*c*d^2*e - a^2*e^3)*f^3*g^4)*x^3 + 5*
(c^2*d^2*e*f^6*g + 2*a^2*d*e^2*f^3*g^4 + 2*(c^2*d^3 - a*c*d*e^2)*f^5*g^2 - (4*a*
c*d^2*e - a^2*e^3)*f^4*g^3)*x^2 + (c^2*d^2*e*f^7 + 5*a^2*d*e^2*f^4*g^3 + (5*c^2*
d^3 - 2*a*c*d*e^2)*f^6*g - (10*a*c*d^2*e - a^2*e^3)*f^5*g^2)*x)

_______________________________________________________________________________________

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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(11/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(11/2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError