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

Optimal. Leaf size=198 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(
3/2)*(f + g*x)^(7/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3
5*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c^2*d^2*(a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*
x)^(3/2))

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Rubi [A]  time = 0.816242, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(
3/2)*(f + g*x)^(7/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3
5*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c^2*d^2*(a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*
x)^(3/2))

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Rubi in Sympy [A]  time = 69.2863, size = 190, normalized size = 0.96 \[ - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{105 \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{3}} + \frac{8 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{35 \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{7 \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{7}{2}} \left (a e g - c d f\right )} \]

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)**(1/2)/(g*x+f)**(9/2)/(e*x+d)**(1/2),x)

[Out]

-16*c**2*d**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(105*(d + e*x)**
(3/2)*(f + g*x)**(3/2)*(a*e*g - c*d*f)**3) + 8*c*d*(a*d*e + c*d*e*x**2 + x*(a*e*
*2 + c*d**2))**(3/2)/(35*(d + e*x)**(3/2)*(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))**(3/2)/(7*(d + e*x)**(3/2)*(f + g*
x)**(7/2)*(a*e*g - c*d*f))

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Mathematica [A]  time = 0.212353, size = 105, normalized size = 0.53 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (15 a^2 e^2 g^2-6 a c d e g (7 f+2 g x)+c^2 d^2 \left (35 f^2+28 f g x+8 g^2 x^2\right )\right )}{105 (d+e x)^{3/2} (f+g x)^{7/2} (c d f-a e g)^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(15*a^2*e^2*g^2 - 6*a*c*d*e*g*(7*f + 2*g*x) +
c^2*d^2*(35*f^2 + 28*f*g*x + 8*g^2*x^2)))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2
)*(f + g*x)^(7/2))

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Maple [A]  time = 0.013, size = 169, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}-12\,acde{g}^{2}x+28\,{c}^{2}{d}^{2}fgx+15\,{a}^{2}{e}^{2}{g}^{2}-42\,acdefg+35\,{c}^{2}{d}^{2}{f}^{2} \right ) }{105\,{a}^{3}{e}^{3}{g}^{3}-315\,{a}^{2}cd{e}^{2}f{g}^{2}+315\,a{c}^{2}{d}^{2}e{f}^{2}g-105\,{c}^{3}{d}^{3}{f}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( gx+f \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(1/2)/(g*x+f)^(9/2)/(e*x+d)^(1/2),x)

[Out]

-2/105*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-12*a*c*d*e*g^2*x+28*c^2*d^2*f*g*x+15*a^2*e
^2*g^2-42*a*c*d*e*f*g+35*c^2*d^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(g
*x+f)^(7/2)/(a^3*e^3*g^3-3*a^2*c*d*e^2*f*g^2+3*a*c^2*d^2*e*f^2*g-c^3*d^3*f^3)/(e
*x+d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(9/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.297231, size = 1010, normalized size = 5.1 \[ \frac{2 \,{\left (8 \, c^{3} d^{3} g^{2} x^{3} + 35 \, a c^{2} d^{2} e f^{2} - 42 \, a^{2} c d e^{2} f g + 15 \, a^{3} e^{3} g^{2} + 4 \,{\left (7 \, c^{3} d^{3} f g - a c^{2} d^{2} e g^{2}\right )} x^{2} +{\left (35 \, c^{3} d^{3} f^{2} - 14 \, a c^{2} d^{2} e f g + 3 \, a^{2} c d e^{2} g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{105 \,{\left (c^{3} d^{4} f^{7} - 3 \, a c^{2} d^{3} e f^{6} g + 3 \, a^{2} c d^{2} e^{2} f^{5} g^{2} - a^{3} d e^{3} f^{4} g^{3} +{\left (c^{3} d^{3} e f^{3} g^{4} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{5} + 3 \, a^{2} c d e^{3} f g^{6} - a^{3} e^{4} g^{7}\right )} x^{5} +{\left (4 \, c^{3} d^{3} e f^{4} g^{3} - a^{3} d e^{3} g^{7} +{\left (c^{3} d^{4} - 12 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{4} - 3 \,{\left (a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} f^{2} g^{5} +{\left (3 \, a^{2} c d^{2} e^{2} - 4 \, a^{3} e^{4}\right )} f g^{6}\right )} x^{4} + 2 \,{\left (3 \, c^{3} d^{3} e f^{5} g^{2} - 2 \, a^{3} d e^{3} f g^{6} +{\left (2 \, c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{3} - 3 \,{\left (2 \, a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{3} g^{4} + 3 \,{\left (2 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{5}\right )} x^{3} + 2 \,{\left (2 \, c^{3} d^{3} e f^{6} g - 3 \, a^{3} d e^{3} f^{2} g^{5} + 3 \,{\left (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2}\right )} f^{5} g^{2} - 3 \,{\left (3 \, a c^{2} d^{3} e - 2 \, a^{2} c d e^{3}\right )} f^{4} g^{3} +{\left (9 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4}\right )} f^{3} g^{4}\right )} x^{2} +{\left (c^{3} d^{3} e f^{7} - 4 \, a^{3} d e^{3} f^{3} g^{4} +{\left (4 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{6} g - 3 \,{\left (4 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{5} g^{2} +{\left (12 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{4} g^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(9/2)),x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*g^2*x^3 + 35*a*c^2*d^2*e*f^2 - 42*a^2*c*d*e^2*f*g + 15*a^3*e^3*
g^2 + 4*(7*c^3*d^3*f*g - a*c^2*d^2*e*g^2)*x^2 + (35*c^3*d^3*f^2 - 14*a*c^2*d^2*e
*f*g + 3*a^2*c*d*e^2*g^2)*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^3*d^4*f^7 - 3*a*c^2*d^3*e*f^6*g + 3*a^2*c*d^2*e^2*f^5*g^
2 - a^3*d*e^3*f^4*g^3 + (c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d
*e^3*f*g^6 - a^3*e^4*g^7)*x^5 + (4*c^3*d^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4
- 12*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 4*a^2*c*d*e^3)*f^2*g^5 + (3*a^2*c
*d^2*e^2 - 4*a^3*e^4)*f*g^6)*x^4 + 2*(3*c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 +
(2*c^3*d^4 - 9*a*c^2*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^3*g^
4 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x^3 + 2*(2*c^3*d^3*e*f^6*g - 3*a^3*d*
e^3*f^2*g^5 + 3*(c^3*d^4 - 2*a*c^2*d^2*e^2)*f^5*g^2 - 3*(3*a*c^2*d^3*e - 2*a^2*c
*d*e^3)*f^4*g^3 + (9*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^3*g^4)*x^2 + (c^3*d^3*e*f^7 -
4*a^3*d*e^3*f^3*g^4 + (4*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^6*g - 3*(4*a*c^2*d^3*e - a
^2*c*d*e^3)*f^5*g^2 + (12*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^3)*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)**(1/2)/(g*x+f)**(9/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(e*x + d)*(g*x + f)^(9/2)),x, algorithm="giac")

[Out]

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