3.748 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)} \]

[Out]

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

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Rubi [A]  time = 0.525762, 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 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(9/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 43.1757, size = 122, normalized size = 0.95 \[ \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{35 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{7}{2}} \left (a e g - c d f\right )} \]

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)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.206252, size = 69, normalized size = 0.53 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} (c d (7 f+2 g x)-5 a e g)}{35 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(9/2)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 99, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,xcdg+5\,aeg-7\,cdf \right ) }{35\,{a}^{2}{e}^{2}{g}^{2}-70\,acdefg+35\,{c}^{2}{d}^{2}{f}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(9/2),x)

[Out]

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

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Maxima [A]  time = 1.20462, size = 707, normalized size = 5.48 \[ \frac{2 \,{\left (2 \, c^{3} d^{3} g x^{3} + 7 \, a^{2} c d e^{2} f - 5 \, a^{3} e^{3} g +{\left (7 \, c^{3} d^{3} f - a c^{2} d^{2} e g\right )} x^{2} + 2 \,{\left (7 \, a c^{2} d^{2} e f - 4 \, a^{2} c d e^{2} g\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )} \sqrt{g x + f}}{35 \,{\left (c^{2} d^{3} f^{6} - 2 \, a c d^{2} e f^{5} g + a^{2} d e^{2} f^{4} g^{2} +{\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{5} +{\left ({\left (4 \, e^{3} f g^{5} + d e^{2} g^{6}\right )} a^{2} - 2 \,{\left (4 \, d e^{2} f^{2} g^{4} + d^{2} e f g^{5}\right )} a c +{\left (4 \, d^{2} e f^{3} g^{3} + d^{3} f^{2} g^{4}\right )} c^{2}\right )} x^{4} + 2 \,{\left ({\left (3 \, e^{3} f^{2} g^{4} + 2 \, d e^{2} f g^{5}\right )} a^{2} - 2 \,{\left (3 \, d e^{2} f^{3} g^{3} + 2 \, d^{2} e f^{2} g^{4}\right )} a c +{\left (3 \, d^{2} e f^{4} g^{2} + 2 \, d^{3} f^{3} g^{3}\right )} c^{2}\right )} x^{3} + 2 \,{\left ({\left (2 \, e^{3} f^{3} g^{3} + 3 \, d e^{2} f^{2} g^{4}\right )} a^{2} - 2 \,{\left (2 \, d e^{2} f^{4} g^{2} + 3 \, d^{2} e f^{3} g^{3}\right )} a c +{\left (2 \, d^{2} e f^{5} g + 3 \, d^{3} f^{4} g^{2}\right )} c^{2}\right )} x^{2} +{\left ({\left (e^{3} f^{4} g^{2} + 4 \, d e^{2} f^{3} g^{3}\right )} a^{2} - 2 \,{\left (d e^{2} f^{5} g + 4 \, d^{2} e f^{4} g^{2}\right )} a c +{\left (d^{2} e f^{6} + 4 \, d^{3} f^{5} g\right )} c^{2}\right )} x\right )}} \]

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)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(9/2)),x, algorithm="maxima")

[Out]

2/35*(2*c^3*d^3*g*x^3 + 7*a^2*c*d*e^2*f - 5*a^3*e^3*g + (7*c^3*d^3*f - a*c^2*d^2
*e*g)*x^2 + 2*(7*a*c^2*d^2*e*f - 4*a^2*c*d*e^2*g)*x)*sqrt(c*d*x + a*e)*(e*x + d)
*sqrt(g*x + f)/(c^2*d^3*f^6 - 2*a*c*d^2*e*f^5*g + a^2*d*e^2*f^4*g^2 + (c^2*d^2*e
*f^2*g^4 - 2*a*c*d*e^2*f*g^5 + a^2*e^3*g^6)*x^5 + ((4*e^3*f*g^5 + d*e^2*g^6)*a^2
- 2*(4*d*e^2*f^2*g^4 + d^2*e*f*g^5)*a*c + (4*d^2*e*f^3*g^3 + d^3*f^2*g^4)*c^2)*
x^4 + 2*((3*e^3*f^2*g^4 + 2*d*e^2*f*g^5)*a^2 - 2*(3*d*e^2*f^3*g^3 + 2*d^2*e*f^2*
g^4)*a*c + (3*d^2*e*f^4*g^2 + 2*d^3*f^3*g^3)*c^2)*x^3 + 2*((2*e^3*f^3*g^3 + 3*d*
e^2*f^2*g^4)*a^2 - 2*(2*d*e^2*f^4*g^2 + 3*d^2*e*f^3*g^3)*a*c + (2*d^2*e*f^5*g +
3*d^3*f^4*g^2)*c^2)*x^2 + ((e^3*f^4*g^2 + 4*d*e^2*f^3*g^3)*a^2 - 2*(d*e^2*f^5*g
+ 4*d^2*e*f^4*g^2)*a*c + (d^2*e*f^6 + 4*d^3*f^5*g)*c^2)*x)

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Fricas [A]  time = 0.296334, size = 710, normalized size = 5.5 \[ \frac{2 \,{\left (2 \, c^{3} d^{3} g x^{3} + 7 \, a^{2} c d e^{2} f - 5 \, a^{3} e^{3} g +{\left (7 \, c^{3} d^{3} f - a c^{2} d^{2} e g\right )} x^{2} + 2 \,{\left (7 \, a c^{2} d^{2} e f - 4 \, a^{2} c d e^{2} g\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{35 \,{\left (c^{2} d^{3} f^{6} - 2 \, a c d^{2} e f^{5} g + a^{2} d e^{2} f^{4} g^{2} +{\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{5} +{\left (4 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} +{\left (c^{2} d^{3} - 8 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \,{\left (a c d^{2} e - 2 \, a^{2} e^{3}\right )} f g^{5}\right )} x^{4} + 2 \,{\left (3 \, c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \,{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} f^{3} g^{3} -{\left (4 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} f^{2} g^{4}\right )} x^{3} + 2 \,{\left (2 \, c^{2} d^{2} e f^{5} g + 3 \, a^{2} d e^{2} f^{2} g^{4} +{\left (3 \, c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{4} g^{2} - 2 \,{\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{3}\right )} x^{2} +{\left (c^{2} d^{2} e f^{6} + 4 \, a^{2} d e^{2} f^{3} g^{3} + 2 \,{\left (2 \, c^{2} d^{3} - a c d e^{2}\right )} f^{5} g -{\left (8 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{2}\right )} x\right )}} \]

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)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(9/2)),x, algorithm="fricas")

[Out]

2/35*(2*c^3*d^3*g*x^3 + 7*a^2*c*d*e^2*f - 5*a^3*e^3*g + (7*c^3*d^3*f - a*c^2*d^2
*e*g)*x^2 + 2*(7*a*c^2*d^2*e*f - 4*a^2*c*d*e^2*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^6 - 2*a*c*d^2*e*f^5*g +
a^2*d*e^2*f^4*g^2 + (c^2*d^2*e*f^2*g^4 - 2*a*c*d*e^2*f*g^5 + a^2*e^3*g^6)*x^5 +
(4*c^2*d^2*e*f^3*g^3 + a^2*d*e^2*g^6 + (c^2*d^3 - 8*a*c*d*e^2)*f^2*g^4 - 2*(a*c*
d^2*e - 2*a^2*e^3)*f*g^5)*x^4 + 2*(3*c^2*d^2*e*f^4*g^2 + 2*a^2*d*e^2*f*g^5 + 2*(
c^2*d^3 - 3*a*c*d*e^2)*f^3*g^3 - (4*a*c*d^2*e - 3*a^2*e^3)*f^2*g^4)*x^3 + 2*(2*c
^2*d^2*e*f^5*g + 3*a^2*d*e^2*f^2*g^4 + (3*c^2*d^3 - 4*a*c*d*e^2)*f^4*g^2 - 2*(3*
a*c*d^2*e - a^2*e^3)*f^3*g^3)*x^2 + (c^2*d^2*e*f^6 + 4*a^2*d*e^2*f^3*g^3 + 2*(2*
c^2*d^3 - a*c*d*e^2)*f^5*g - (8*a*c*d^2*e - a^2*e^3)*f^4*g^2)*x)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(9/2)),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x +
f)^(9/2)), x)