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

Optimal. Leaf size=128 \[ \frac{4 g \sqrt{d+e x} \sqrt{f+g x}}{3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2} \sqrt{f+g x}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[Out]

(-2*(d + e*x)^(3/2)*Sqrt[f + g*x])/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2)) + (4*g*Sqrt[d + e*x]*Sqrt[f + g*x])/(3*(c*d*f - a*e*g)^2*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.532058, antiderivative size = 128, 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 \[ \frac{4 g \sqrt{d+e x} \sqrt{f+g x}}{3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2} \sqrt{f+g x}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d + e*x)^(3/2)*Sqrt[f + g*x])/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2)) + (4*g*Sqrt[d + e*x]*Sqrt[f + g*x])/(3*(c*d*f - a*e*g)^2*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi in Sympy [A]  time = 44.197, size = 121, normalized size = 0.95 \[ \frac{4 g \sqrt{d + e x} \sqrt{f + g x}}{3 \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x}}{3 \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(g*x+f)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.144077, size = 68, normalized size = 0.53 \[ \frac{2 (d+e x)^{3/2} \sqrt{f+g x} (3 a e g-c d (f-2 g x))}{3 ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 99, normalized size = 0.8 \[{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 2\,xcdg+3\,aeg-cdf \right ) }{3\,{a}^{2}{e}^{2}{g}^{2}-6\,acdefg+3\,{c}^{2}{d}^{2}{f}^{2}}\sqrt{gx+f} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(g*x+f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.292028, size = 429, normalized size = 3.35 \[ \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d g x - c d f + 3 \, a e g\right )} \sqrt{e x + d} \sqrt{g x + f}}{3 \,{\left (a^{2} c^{2} d^{3} e^{2} f^{2} - 2 \, a^{3} c d^{2} e^{3} f g + a^{4} d e^{4} g^{2} +{\left (c^{4} d^{4} e f^{2} - 2 \, a c^{3} d^{3} e^{2} f g + a^{2} c^{2} d^{2} e^{3} g^{2}\right )} x^{3} +{\left ({\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} f^{2} - 2 \,{\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3}\right )} f g +{\left (a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} g^{2}\right )} x^{2} +{\left ({\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f^{2} - 2 \,{\left (2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g +{\left (2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{2}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*sqrt(g*x + f)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError