3.1940 \(\int \frac{1}{(d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x) \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^2 \left (c d^2-a e^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.178206, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{4 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x) \left (c d^2-a e^2\right )^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^2 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^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])/(3*(c*d^2 - a*e^2)*(d + e*x)^2)
+ (4*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d^2 - a*e^2)^2*(d +
e*x))

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Rubi in Sympy [A]  time = 35.0201, size = 100, normalized size = 0.9 \[ \frac{4 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.113885, size = 61, normalized size = 0.55 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (c d (3 d+2 e x)-a e^2\right )}{3 (d+e x)^2 \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

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

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Maple [A]  time = 0.012, size = 89, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,cdex+a{e}^{2}-3\,c{d}^{2} \right ) }{ \left ( 3\,ex+3\,d \right ) \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*d*x+a*e)*(-2*c*d*e*x+a*e^2-3*c*d^2)/(e*x+d)/(a^2*e^4-2*a*c*d^2*e^2+c^2*d
^4)/(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(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302138, size = 189, normalized size = 1.7 \[ \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + 3 \, c d^{2} - a e^{2}\right )}}{3 \,{\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^2),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*e*x + 3*c*d^2 - a*e^2)/(c
^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x
^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError