3.1904 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=111 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.173627, 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 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )} \]

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)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 34.2739, size = 102, normalized size = 0.92 \[ \frac{4 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{15 \left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{5 \left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right )} \]

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)**4,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*(a*e**2
 - c*d**2)**2) - 2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(5*(d + e*x
)**4*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.0860105, size = 81, normalized size = 0.73 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-3 a^2 e^3+a c d e (5 d-e x)+c^2 d^2 x (5 d+2 e x)\right )}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2} \]

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)^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 90, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,cdex+3\,a{e}^{2}-5\,c{d}^{2} \right ) }{15\, \left ( ex+d \right ) ^{3} \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}} \]

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)^4,x)

[Out]

-2/15*(c*d*x+a*e)*(-2*c*d*e*x+3*a*e^2-5*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)
^(1/2)/(e*x+d)^3/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.426278, size = 277, normalized size = 2.5 \[ \frac{2 \,{\left (2 \, c^{2} d^{2} e x^{2} + 5 \, a c d^{2} e - 3 \, a^{2} e^{3} +{\left (5 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{15 \,{\left (c^{2} d^{7} - 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} +{\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

2/15*(2*c^2*d^2*e*x^2 + 5*a*c*d^2*e - 3*a^2*e^3 + (5*c^2*d^3 - a*c*d*e^2)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^2*d^7 - 2*a*c*d^5*e^2 + a^2*d^3*e^4
+ (c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*x^3 + 3*(c^2*d^5*e^2 - 2*a*c*d^3*e^4 +
 a^2*d*e^6)*x^2 + 3*(c^2*d^6*e - 2*a*c*d^4*e^3 + a^2*d^2*e^5)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{4}}\, dx \]

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)**4,x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.421494, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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)^4,x, algorithm="giac")

[Out]

Done