3.460 \(\int \left (\frac{b^2}{4 c}+b \sqrt{x}+c x\right )^2 \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (b+2 c \sqrt{x}\right )^6}{192 c^4}-\frac{b \left (b+2 c \sqrt{x}\right )^5}{160 c^4} \]

[Out]

-(b*(b + 2*c*Sqrt[x])^5)/(160*c^4) + (b + 2*c*Sqrt[x])^6/(192*c^4)

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Rubi [A]  time = 0.0464199, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\left (b+2 c \sqrt{x}\right )^6}{192 c^4}-\frac{b \left (b+2 c \sqrt{x}\right )^5}{160 c^4} \]

Antiderivative was successfully verified.

[In]  Int[(b^2/(4*c) + b*Sqrt[x] + c*x)^2,x]

[Out]

-(b*(b + 2*c*Sqrt[x])^5)/(160*c^4) + (b + 2*c*Sqrt[x])^6/(192*c^4)

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Rubi in Sympy [A]  time = 13.3516, size = 34, normalized size = 0.85 \[ - \frac{b \left (b + 2 c \sqrt{x}\right )^{5}}{160 c^{4}} + \frac{\left (b + 2 c \sqrt{x}\right )^{6}}{192 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1/4/c*b**2+c*x+b*x**(1/2))**2,x)

[Out]

-b*(b + 2*c*sqrt(x))**5/(160*c**4) + (b + 2*c*sqrt(x))**6/(192*c**4)

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Mathematica [A]  time = 0.0165751, size = 60, normalized size = 1.5 \[ \frac{b^4 x+\frac{16}{3} b^3 c x^{3/2}+12 b^2 c^2 x^2+\frac{64}{5} b c^3 x^{5/2}+\frac{16 c^4 x^3}{3}}{16 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b^2/(4*c) + b*Sqrt[x] + c*x)^2,x]

[Out]

(b^4*x + (16*b^3*c*x^(3/2))/3 + 12*b^2*c^2*x^2 + (64*b*c^3*x^(5/2))/5 + (16*c^4*
x^3)/3)/(16*c^2)

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Maple [A]  time = 0.003, size = 52, normalized size = 1.3 \[{\frac{{x}^{2}{b}^{2}}{2}}+{\frac{b}{2\,c} \left ({\frac{8\,{c}^{2}}{5}{x}^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}}{3}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{1}{3\,c} \left ({\frac{{b}^{2}}{4\,c}}+cx \right ) ^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/4*b^2/c+c*x+b*x^(1/2))^2,x)

[Out]

1/2*x^2*b^2+1/2*b/c*(8/5*x^(5/2)*c^2+2/3*x^(3/2)*b^2)+1/3*(1/4*b^2/c+c*x)^3/c

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Maxima [A]  time = 0.747215, size = 73, normalized size = 1.82 \[ \frac{1}{3} \, c^{2} x^{3} + \frac{4}{5} \, b c x^{\frac{5}{2}} + \frac{1}{2} \, b^{2} x^{2} + \frac{b^{4} x}{16 \, c^{2}} + \frac{{\left (3 \, c x^{2} + 4 \, b x^{\frac{3}{2}}\right )} b^{2}}{12 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/16*(4*c*x + 4*b*sqrt(x) + b^2/c)^2,x, algorithm="maxima")

[Out]

1/3*c^2*x^3 + 4/5*b*c*x^(5/2) + 1/2*b^2*x^2 + 1/16*b^4*x/c^2 + 1/12*(3*c*x^2 + 4
*b*x^(3/2))*b^2/c

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Fricas [A]  time = 0.274786, size = 72, normalized size = 1.8 \[ \frac{80 \, c^{4} x^{3} + 180 \, b^{2} c^{2} x^{2} + 15 \, b^{4} x + 16 \,{\left (12 \, b c^{3} x^{2} + 5 \, b^{3} c x\right )} \sqrt{x}}{240 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/16*(4*c*x + 4*b*sqrt(x) + b^2/c)^2,x, algorithm="fricas")

[Out]

1/240*(80*c^4*x^3 + 180*b^2*c^2*x^2 + 15*b^4*x + 16*(12*b*c^3*x^2 + 5*b^3*c*x)*s
qrt(x))/c^2

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Sympy [A]  time = 1.63181, size = 58, normalized size = 1.45 \[ \frac{b^{4} x + \frac{16 b^{3} c x^{\frac{3}{2}}}{3} + 12 b^{2} c^{2} x^{2} + \frac{64 b c^{3} x^{\frac{5}{2}}}{5} + \frac{16 c^{4} x^{3}}{3}}{16 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1/4/c*b**2+c*x+b*x**(1/2))**2,x)

[Out]

(b**4*x + 16*b**3*c*x**(3/2)/3 + 12*b**2*c**2*x**2 + 64*b*c**3*x**(5/2)/5 + 16*c
**4*x**3/3)/(16*c**2)

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GIAC/XCAS [A]  time = 0.295246, size = 66, normalized size = 1.65 \[ \frac{80 \, c^{4} x^{3} + 192 \, b c^{3} x^{\frac{5}{2}} + 180 \, b^{2} c^{2} x^{2} + 80 \, b^{3} c x^{\frac{3}{2}} + 15 \, b^{4} x}{240 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/16*(4*c*x + 4*b*sqrt(x) + b^2/c)^2,x, algorithm="giac")

[Out]

1/240*(80*c^4*x^3 + 192*b*c^3*x^(5/2) + 180*b^2*c^2*x^2 + 80*b^3*c*x^(3/2) + 15*
b^4*x)/c^2