3.1916 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0820094, 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 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.3723, size = 49, normalized size = 0.91 \[ - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{5 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )} \]

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)**(3/2)/(e*x+d)**5,x)

[Out]

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

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Mathematica [A]  time = 0.124726, size = 43, normalized size = 0.8 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2}}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

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

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Maple [A]  time = 0.01, size = 58, normalized size = 1.1 \[ -{\frac{2\,cdx+2\,ae}{5\, \left ( ex+d \right ) ^{4} \left ( a{e}^{2}-c{d}^{2} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^5,x)

[Out]

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

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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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.432042, size = 176, normalized size = 3.26 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{5 \,{\left (c d^{5} - a d^{3} e^{2} +{\left (c d^{2} e^{3} - a e^{5}\right )} x^{3} + 3 \,{\left (c d^{3} e^{2} - a d e^{4}\right )} x^{2} + 3 \,{\left (c d^{4} e - a d^{2} e^{3}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^5,x, algorithm="fricas")

[Out]

2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)/(c*d^5 - a*d^3*e^2 + (c*d^2*e^3 - a*e^5)*x^3 + 3*(c*d^3*e^2 - a*d*e^4)*x^2
+ 3*(c*d^4*e - a*d^2*e^3)*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)**(3/2)/(e*x+d)**5,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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^5,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError