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

Optimal. Leaf size=129 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(
3/2)*(f + g*x)^(5/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1
5*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(3/2))

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Rubi [A]  time = 0.528434, 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 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)} \]

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)^(7/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(
3/2)*(f + g*x)^(5/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1
5*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(3/2))

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Rubi in Sympy [A]  time = 43.6312, size = 122, normalized size = 0.95 \[ \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{15 \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{5 \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c*d*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(15*(d + e*x)**(3/2)*(f
+ g*x)**(3/2)*(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)/(5*(d + e*x)**(3/2)*(f + g*x)**(5/2)*(a*e*g - c*d*f))

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Mathematica [A]  time = 0.145187, size = 69, normalized size = 0.53 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+2 g x)-3 a e g)}{15 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2} \]

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)^(7/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-3*a*e*g + c*d*(5*f + 2*g*x)))/(15*(c*d*f -
a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2))

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Maple [A]  time = 0.011, size = 99, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,xcdg+3\,aeg-5\,cdf \right ) }{15\,{a}^{2}{e}^{2}{g}^{2}-30\,acdefg+15\,{c}^{2}{d}^{2}{f}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( gx+f \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

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)^(7/2)/(e*x+d)^(1/2),x)

[Out]

-2/15*(c*d*x+a*e)*(-2*c*d*g*x+3*a*e*g-5*c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(1/2)/(g*x+f)^(5/2)/(a^2*e^2*g^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(e*x+d)^(1/2)

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

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)^(7/2)),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)^(
7/2)), x)

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Fricas [A]  time = 0.289128, size = 543, normalized size = 4.21 \[ \frac{2 \,{\left (2 \, c^{2} d^{2} g x^{2} + 5 \, a c d e f - 3 \, a^{2} e^{2} g +{\left (5 \, c^{2} d^{2} f - a c d e g\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{15 \,{\left (c^{2} d^{3} f^{5} - 2 \, a c d^{2} e f^{4} g + a^{2} d e^{2} f^{3} g^{2} +{\left (c^{2} d^{2} e f^{2} g^{3} - 2 \, a c d e^{2} f g^{4} + a^{2} e^{3} g^{5}\right )} x^{4} +{\left (3 \, c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} +{\left (c^{2} d^{3} - 6 \, a c d e^{2}\right )} f^{2} g^{3} -{\left (2 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} f g^{4}\right )} x^{3} + 3 \,{\left (c^{2} d^{2} e f^{4} g + a^{2} d e^{2} f g^{4} +{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{3} g^{2} -{\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{3}\right )} x^{2} +{\left (c^{2} d^{2} e f^{5} + 3 \, a^{2} d e^{2} f^{2} g^{3} +{\left (3 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g -{\left (6 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{2}\right )} x\right )}} \]

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)^(7/2)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

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)^(7/2)),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)^(
7/2)), x)