3.1068 \(\int \frac{(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]

[Out]

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

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Rubi [A]  time = 0.0739574, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.6076, size = 34, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{7}}{2 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0252697, size = 33, normalized size = 0.85 \[ \frac{x (d+e x) (2 d+e x)}{2 c^2 \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2),x]

[Out]

(x*(d + e*x)*(2*d + e*x))/(2*c^2*Sqrt[c*(d + e*x)^2])

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Maple [A]  time = 0.003, size = 40, normalized size = 1. \[{\frac{x \left ( ex+2\,d \right ) \left ( ex+d \right ) ^{5}}{2} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x)

[Out]

1/2*x*(e*x+2*d)*(e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)

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Maxima [A]  time = 0.70213, size = 313, normalized size = 8.03 \[ \frac{e^{4} x^{5}}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{5 \, d e^{3} x^{4}}{2 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} - \frac{25 \, c^{2} d^{6} e^{4}}{4 \, \left (c e^{2}\right )^{\frac{9}{2}}{\left (x + \frac{d}{e}\right )}^{4}} - \frac{10 \, d^{3} e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c} + \frac{50 \, c d^{5} e^{3}}{3 \, \left (c e^{2}\right )^{\frac{7}{2}}{\left (x + \frac{d}{e}\right )}^{3}} - \frac{26 \, d^{5}}{3 \,{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}} c e} - \frac{25 \, d^{4} e^{2}}{2 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{2}} + \frac{25 \, d^{6}}{4 \, \left (c e^{2}\right )^{\frac{5}{2}}{\left (x + \frac{d}{e}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="maxima")

[Out]

1/2*e^4*x^5/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c) + 5/2*d*e^3*x^4/((c*e^2*x^
2 + 2*c*d*e*x + c*d^2)^(3/2)*c) - 25/4*c^2*d^6*e^4/((c*e^2)^(9/2)*(x + d/e)^4) -
10*d^3*e*x^2/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c) + 50/3*c*d^5*e^3/((c*e^2
)^(7/2)*(x + d/e)^3) - 26/3*d^5/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c*e) - 25
/2*d^4*e^2/((c*e^2)^(5/2)*(x + d/e)^2) + 25/4*d^6/((c*e^2)^(5/2)*(x + d/e)^4)

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Fricas [A]  time = 0.218377, size = 65, normalized size = 1.67 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (c^{3} e x + c^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="fricas")

[Out]

1/2*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x^2 + 2*d*x)/(c^3*e*x + c^3*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{6}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Integral((d + e*x)**6/(c*(d + e*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.278614, size = 146, normalized size = 3.74 \[ -\frac{\frac{9 \, d^{5} e^{\left (-1\right )}}{c} - 4 \, C_{0} d^{3} e^{\left (-3\right )} -{\left (12 \, C_{0} d^{2} e^{\left (-2\right )} - \frac{25 \, d^{4}}{c} -{\left (\frac{20 \, d^{3} e}{c} - 12 \, C_{0} d e^{\left (-1\right )} -{\left (x{\left (\frac{x e^{4}}{c} + \frac{5 \, d e^{3}}{c}\right )} + 4 \, C_{0}\right )} x\right )} x\right )} x}{2 \,{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2),x, algorithm="giac")

[Out]

-1/2*(9*d^5*e^(-1)/c - 4*C_0*d^3*e^(-3) - (12*C_0*d^2*e^(-2) - 25*d^4/c - (20*d^
3*e/c - 12*C_0*d*e^(-1) - (x*(x*e^4/c + 5*d*e^3/c) + 4*C_0)*x)*x)*x)/(c*x^2*e^2
+ 2*c*d*x*e + c*d^2)^(3/2)