3.1903 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*(c*d^2 - a*e^2)*(d + e*x)^3
)

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Rubi [A]  time = 0.0835962, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*(c*d^2 - a*e^2)*(d + e*x)^3
)

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Rubi in Sympy [A]  time = 17.662, size = 49, normalized size = 0.91 \[ - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )} \]

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)/(e*x+d)**3,x)

[Out]

-2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(3*(d + e*x)**3*(a*e**2 - c
*d**2))

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Mathematica [A]  time = 0.066823, size = 43, normalized size = 0.8 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2}}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2))/(3*(c*d^2 - a*e^2)*(d + e*x)^3)

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Maple [A]  time = 0.009, size = 58, normalized size = 1.1 \[ -{\frac{2\,cdx+2\,ae}{3\, \left ( ex+d \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^3,x)

[Out]

-2/3*(c*d*x+a*e)/(e*x+d)^2/(a*e^2-c*d^2)*(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. \[ \text{Exception raised: ValueError} \]

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)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263554, size = 122, normalized size = 2.26 \[ \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d x + a e\right )}}{3 \,{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{2} e^{2} - a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - a d e^{3}\right )} x\right )}} \]

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)/(e*x + d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{3}}\, dx \]

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)/(e*x+d)**3,x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.272428, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

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)/(e*x + d)^3,x, algorithm="giac")

[Out]

Done