3.1931 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^7 \left (c d^2-a e^2\right )} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7
)

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Rubi [A]  time = 0.0843936, 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 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^7 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^7
)

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Rubi in Sympy [A]  time = 17.4113, size = 49, normalized size = 0.91 \[ - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{7 \left (d + e x\right )^{7} \left (a e^{2} - c d^{2}\right )} \]

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)**(5/2)/(e*x+d)**7,x)

[Out]

-2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(7/2)/(7*(d + e*x)**7*(a*e**2 - c
*d**2))

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Mathematica [A]  time = 0.154774, size = 53, normalized size = 0.98 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)}}{7 (d+e x)^4 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])/(7*(c*d^2 - a*e^2)*(d + e*x)^4
)

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Maple [A]  time = 0.01, size = 58, normalized size = 1.1 \[ -{\frac{2\,cdx+2\,ae}{7\, \left ( ex+d \right ) ^{6} \left ( a{e}^{2}-c{d}^{2} \right ) } \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2)/(e*x+d)^7,x)

[Out]

-2/7*(c*d*x+a*e)/(e*x+d)^6/(a*e^2-c*d^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. \[ \text{Exception raised: ValueError} \]

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)^(5/2)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2697, size = 230, normalized size = 4.26 \[ \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{7 \,{\left (c d^{6} - a d^{4} e^{2} +{\left (c d^{2} e^{4} - a e^{6}\right )} x^{4} + 4 \,{\left (c d^{3} e^{3} - a d e^{5}\right )} x^{3} + 6 \,{\left (c d^{4} e^{2} - a d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c d^{5} e - a d^{3} e^{3}\right )} x\right )}} \]

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)^(5/2)/(e*x + d)^7,x, algorithm="fricas")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)/(c*d^6 - a*d^4*e^2 + (c*d^2*e^4 - a*e^6)*x^4 + 4*(
c*d^3*e^3 - a*d*e^5)*x^3 + 6*(c*d^4*e^2 - a*d^2*e^4)*x^2 + 4*(c*d^5*e - a*d^3*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)**(5/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.933258, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

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)^(5/2)/(e*x + d)^7,x, algorithm="giac")

[Out]

Done