3.73 \(\int x^{5/2} \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]

[Out]

(-32*b^3*(b*x + c*x^2)^(3/2))/(315*c^4*x^(3/2)) + (16*b^2*(b*x + c*x^2)^(3/2))/(
105*c^3*Sqrt[x]) - (4*b*Sqrt[x]*(b*x + c*x^2)^(3/2))/(21*c^2) + (2*x^(3/2)*(b*x
+ c*x^2)^(3/2))/(9*c)

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Rubi [A]  time = 0.125796, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*Sqrt[b*x + c*x^2],x]

[Out]

(-32*b^3*(b*x + c*x^2)^(3/2))/(315*c^4*x^(3/2)) + (16*b^2*(b*x + c*x^2)^(3/2))/(
105*c^3*Sqrt[x]) - (4*b*Sqrt[x]*(b*x + c*x^2)^(3/2))/(21*c^2) + (2*x^(3/2)*(b*x
+ c*x^2)^(3/2))/(9*c)

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Rubi in Sympy [A]  time = 13.5387, size = 100, normalized size = 0.93 \[ - \frac{32 b^{3} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{315 c^{4} x^{\frac{3}{2}}} + \frac{16 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 c^{3} \sqrt{x}} - \frac{4 b \sqrt{x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{21 c^{2}} + \frac{2 x^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{9 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(c*x**2+b*x)**(1/2),x)

[Out]

-32*b**3*(b*x + c*x**2)**(3/2)/(315*c**4*x**(3/2)) + 16*b**2*(b*x + c*x**2)**(3/
2)/(105*c**3*sqrt(x)) - 4*b*sqrt(x)*(b*x + c*x**2)**(3/2)/(21*c**2) + 2*x**(3/2)
*(b*x + c*x**2)**(3/2)/(9*c)

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Mathematica [A]  time = 0.0271275, size = 64, normalized size = 0.59 \[ \frac{2 \sqrt{x (b+c x)} \left (-16 b^4+8 b^3 c x-6 b^2 c^2 x^2+5 b c^3 x^3+35 c^4 x^4\right )}{315 c^4 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[x*(b + c*x)]*(-16*b^4 + 8*b^3*c*x - 6*b^2*c^2*x^2 + 5*b*c^3*x^3 + 35*c^4
*x^4))/(315*c^4*Sqrt[x])

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Maple [A]  time = 0.008, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -35\,{x}^{3}{c}^{3}+30\,b{x}^{2}{c}^{2}-24\,{b}^{2}xc+16\,{b}^{3} \right ) }{315\,{c}^{4}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(c*x^2+b*x)^(1/2),x)

[Out]

-2/315*(c*x+b)*(-35*c^3*x^3+30*b*c^2*x^2-24*b^2*c*x+16*b^3)*(c*x^2+b*x)^(1/2)/c^
4/x^(1/2)

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Maxima [A]  time = 0.731618, size = 72, normalized size = 0.67 \[ \frac{2 \,{\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt{c x + b}}{315 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*x^(5/2),x, algorithm="maxima")

[Out]

2/315*(35*c^4*x^4 + 5*b*c^3*x^3 - 6*b^2*c^2*x^2 + 8*b^3*c*x - 16*b^4)*sqrt(c*x +
b)/c^4

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Fricas [A]  time = 0.226521, size = 100, normalized size = 0.93 \[ \frac{2 \,{\left (35 \, c^{5} x^{6} + 40 \, b c^{4} x^{5} - b^{2} c^{3} x^{4} + 2 \, b^{3} c^{2} x^{3} - 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )}}{315 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*c^5*x^6 + 40*b*c^4*x^5 - b^2*c^3*x^4 + 2*b^3*c^2*x^3 - 8*b^4*c*x^2 - 1
6*b^5*x)/(sqrt(c*x^2 + b*x)*c^4*sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**(5/2)*sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.213307, size = 78, normalized size = 0.72 \[ \frac{32 \, b^{\frac{9}{2}}}{315 \, c^{4}} + \frac{2 \,{\left (35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}\right )}}{315 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/315*b^(9/2)/c^4 + 2/315*(35*(c*x + b)^(9/2) - 135*(c*x + b)^(7/2)*b + 189*(c*
x + b)^(5/2)*b^2 - 105*(c*x + b)^(3/2)*b^3)/c^4