3.1040 \(\int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=39 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \]

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))/(8*c*e)

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Rubi [A]  time = 0.0734767, 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 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{8 c e} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2))/(8*c*e)

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Rubi in Sympy [A]  time = 18.2494, size = 34, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{3} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}}{8 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**3*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(5/2)/(8*e)

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Mathematica [A]  time = 0.0443816, size = 28, normalized size = 0.72 \[ \frac{(d+e x) \left (c (d+e x)^2\right )^{7/2}}{8 c e} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^(7/2))/(8*c*e)

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Maple [B]  time = 0.006, size = 106, normalized size = 2.7 \[{\frac{x \left ({e}^{7}{x}^{7}+8\,d{e}^{6}{x}^{6}+28\,{d}^{2}{e}^{5}{x}^{5}+56\,{d}^{3}{e}^{4}{x}^{4}+70\,{d}^{4}{e}^{3}{x}^{3}+56\,{d}^{5}{e}^{2}{x}^{2}+28\,{d}^{6}ex+8\,{d}^{7} \right ) }{8\, \left ( ex+d \right ) ^{5}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x*(e^7*x^7+8*d*e^6*x^6+28*d^2*e^5*x^5+56*d^3*e^4*x^4+70*d^4*e^3*x^3+56*d^5*e
^2*x^2+28*d^6*e*x+8*d^7)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222657, size = 177, normalized size = 4.54 \[ \frac{{\left (c^{2} e^{7} x^{8} + 8 \, c^{2} d e^{6} x^{7} + 28 \, c^{2} d^{2} e^{5} x^{6} + 56 \, c^{2} d^{3} e^{4} x^{5} + 70 \, c^{2} d^{4} e^{3} x^{4} + 56 \, c^{2} d^{5} e^{2} x^{3} + 28 \, c^{2} d^{6} e x^{2} + 8 \, c^{2} d^{7} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{8 \,{\left (e x + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(c^2*e^7*x^8 + 8*c^2*d*e^6*x^7 + 28*c^2*d^2*e^5*x^6 + 56*c^2*d^3*e^4*x^5 + 7
0*c^2*d^4*e^3*x^4 + 56*c^2*d^5*e^2*x^3 + 28*c^2*d^6*e*x^2 + 8*c^2*d^7*x)*sqrt(c*
e^2*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223802, size = 155, normalized size = 3.97 \[ \frac{1}{8} \,{\left (c^{2} d^{7} e^{\left (-1\right )} +{\left (7 \, c^{2} d^{6} +{\left (21 \, c^{2} d^{5} e +{\left (35 \, c^{2} d^{4} e^{2} +{\left (35 \, c^{2} d^{3} e^{3} +{\left (21 \, c^{2} d^{2} e^{4} +{\left (c^{2} x e^{6} + 7 \, c^{2} d e^{5}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(c^2*d^7*e^(-1) + (7*c^2*d^6 + (21*c^2*d^5*e + (35*c^2*d^4*e^2 + (35*c^2*d^3
*e^3 + (21*c^2*d^2*e^4 + (c^2*x*e^6 + 7*c^2*d*e^5)*x)*x)*x)*x)*x)*x)*sqrt(c*x^2*
e^2 + 2*c*d*x*e + c*d^2)