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

Optimal. Leaf size=48 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d (d+e x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0723978, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d (d+e x)^{7/2}} \]

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)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.404, size = 42, normalized size = 0.88 \[ \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{7}{2}}}{7 c d \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0651428, size = 37, normalized size = 0.77 \[ \frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 c d (d+e x)^{7/2}} \]

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)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 50, normalized size = 1. \[{\frac{2\,cdx+2\,ae}{7\,cd} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.710899, size = 81, normalized size = 1.69 \[ \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d x + a e}}{7 \, c d} \]

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)^(5/2),x, algorithm="maxima")

[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*x + a
*e)/(c*d)

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Fricas [A]  time = 0.269069, size = 227, normalized size = 4.73 \[ \frac{2 \,{\left (c^{4} d^{4} e x^{5} + a^{4} d e^{4} +{\left (c^{4} d^{5} + 4 \, a c^{3} d^{3} e^{2}\right )} x^{4} + 2 \,{\left (2 \, a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{3} + 2 \,{\left (3 \, a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} x^{2} +{\left (4 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x\right )}}{7 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c d} \]

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)^(5/2),x, algorithm="fricas")

[Out]

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

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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)**(5/2),x)

[Out]

Timed out

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

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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError