3.718 \(\int \frac{\sqrt{d+e x}}{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=198 \[ \frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}+\frac{8 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)} \]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)*Sqrt[d + e*x]
*(f + g*x)^(5/2)) + (8*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*(c*d
*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*Sqrt[f + g*x])

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Rubi [A]  time = 0.795988, 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 \[ \frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}+\frac{8 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/((f + g*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)*Sqrt[d + e*x]
*(f + g*x)^(5/2)) + (8*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*(c*d
*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (16*c^2*d^2*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*Sqrt[f + g*x])

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Rubi in Sympy [A]  time = 70.0242, size = 190, normalized size = 0.96 \[ - \frac{16 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 \sqrt{d + e x} \sqrt{f + g x} \left (a e g - c d f\right )^{3}} + \frac{8 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{15 \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 \sqrt{d + e x} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(1/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

-16*c**2*d**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(15*sqrt(d + e*x)*s
qrt(f + g*x)*(a*e*g - c*d*f)**3) + 8*c*d*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c
*d**2))/(15*sqrt(d + e*x)*(f + g*x)**(3/2)*(a*e*g - c*d*f)**2) - 2*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2))/(5*sqrt(d + e*x)*(f + g*x)**(5/2)*(a*e*g - c*d
*f))

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Mathematica [A]  time = 0.254817, size = 105, normalized size = 0.53 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (3 a^2 e^2 g^2-2 a c d e g (5 f+2 g x)+c^2 d^2 \left (15 f^2+20 f g x+8 g^2 x^2\right )\right )}{15 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/((f + g*x)^(7/2)*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)]*(3*a^2*e^2*g^2 - 2*a*c*d*e*g*(5*f + 2*g*x) + c^
2*d^2*(15*f^2 + 20*f*g*x + 8*g^2*x^2)))/(15*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f +
g*x)^(5/2))

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Maple [A]  time = 0.013, size = 169, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}-4\,acde{g}^{2}x+20\,{c}^{2}{d}^{2}fgx+3\,{a}^{2}{e}^{2}{g}^{2}-10\,acdefg+15\,{c}^{2}{d}^{2}{f}^{2} \right ) }{15\,{a}^{3}{e}^{3}{g}^{3}-45\,{a}^{2}cd{e}^{2}f{g}^{2}+45\,a{c}^{2}{d}^{2}e{f}^{2}g-15\,{c}^{3}{d}^{3}{f}^{3}}\sqrt{ex+d} \left ( gx+f \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(g*x+f)^(7/2)/(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)*(8*c^2*d^2*g^2*x^2-4*a*c*d*e*g^2*x+20*c^2*d^2*f*g*x+3*a^2*e^2*
g^2-10*a*c*d*e*f*g+15*c^2*d^2*f^2)*(e*x+d)^(1/2)/(g*x+f)^(5/2)/(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)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*
e)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(7/2)),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(
7/2)), x)

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Fricas [A]  time = 0.289814, size = 772, normalized size = 3.9 \[ \frac{2 \,{\left (8 \, c^{2} d^{2} g^{2} x^{2} + 15 \, c^{2} d^{2} f^{2} - 10 \, a c d e f g + 3 \, a^{2} e^{2} g^{2} + 4 \,{\left (5 \, c^{2} d^{2} f g - a c d e g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{15 \,{\left (c^{3} d^{4} f^{6} - 3 \, a c^{2} d^{3} e f^{5} g + 3 \, a^{2} c d^{2} e^{2} f^{4} g^{2} - a^{3} d e^{3} f^{3} g^{3} +{\left (c^{3} d^{3} e f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{4} + 3 \, a^{2} c d e^{3} f g^{5} - a^{3} e^{4} g^{6}\right )} x^{4} +{\left (3 \, c^{3} d^{3} e f^{4} g^{2} - a^{3} d e^{3} g^{6} +{\left (c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{3} - 3 \,{\left (a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{2} g^{4} + 3 \,{\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{5}\right )} x^{3} + 3 \,{\left (c^{3} d^{3} e f^{5} g - a^{3} d e^{3} f g^{5} +{\left (c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{2} - 3 \,{\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{3} g^{3} +{\left (3 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{4}\right )} x^{2} +{\left (c^{3} d^{3} e f^{6} - 3 \, a^{3} d e^{3} f^{2} g^{4} + 3 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{5} g - 3 \,{\left (3 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{4} g^{2} +{\left (9 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{3} g^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(7/2)),x, algorithm="fricas")

[Out]

2/15*(8*c^2*d^2*g^2*x^2 + 15*c^2*d^2*f^2 - 10*a*c*d*e*f*g + 3*a^2*e^2*g^2 + 4*(5
*c^2*d^2*f*g - a*c*d*e*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^6 - 3*a*c^2*d^3*e*f^5*g + 3*a^2*c*d^2*e^2*f^4*
g^2 - a^3*d*e^3*f^3*g^3 + (c^3*d^3*e*f^3*g^3 - 3*a*c^2*d^2*e^2*f^2*g^4 + 3*a^2*c
*d*e^3*f*g^5 - a^3*e^4*g^6)*x^4 + (3*c^3*d^3*e*f^4*g^2 - a^3*d*e^3*g^6 + (c^3*d^
4 - 9*a*c^2*d^2*e^2)*f^3*g^3 - 3*(a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^2*g^4 + 3*(a^2*
c*d^2*e^2 - a^3*e^4)*f*g^5)*x^3 + 3*(c^3*d^3*e*f^5*g - a^3*d*e^3*f*g^5 + (c^3*d^
4 - 3*a*c^2*d^2*e^2)*f^4*g^2 - 3*(a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^3 + (3*a^2*c*
d^2*e^2 - a^3*e^4)*f^2*g^4)*x^2 + (c^3*d^3*e*f^6 - 3*a^3*d*e^3*f^2*g^4 + 3*(c^3*
d^4 - a*c^2*d^2*e^2)*f^5*g - 3*(3*a*c^2*d^3*e - a^2*c*d*e^3)*f^4*g^2 + (9*a^2*c*
d^2*e^2 - a^3*e^4)*f^3*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((e*x+d)**(1/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(7/2)),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(
7/2)), x)