3.74 \(\int x^{3/2} \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=80 \[ \frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac{8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c} \]

[Out]

(16*b^2*(b*x + c*x^2)^(3/2))/(105*c^3*x^(3/2)) - (8*b*(b*x + c*x^2)^(3/2))/(35*c
^2*Sqrt[x]) + (2*Sqrt[x]*(b*x + c*x^2)^(3/2))/(7*c)

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Rubi [A]  time = 0.0880756, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac{8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*Sqrt[b*x + c*x^2],x]

[Out]

(16*b^2*(b*x + c*x^2)^(3/2))/(105*c^3*x^(3/2)) - (8*b*(b*x + c*x^2)^(3/2))/(35*c
^2*Sqrt[x]) + (2*Sqrt[x]*(b*x + c*x^2)^(3/2))/(7*c)

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Rubi in Sympy [A]  time = 9.20138, size = 73, normalized size = 0.91 \[ \frac{16 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 c^{3} x^{\frac{3}{2}}} - \frac{8 b \left (b x + c x^{2}\right )^{\frac{3}{2}}}{35 c^{2} \sqrt{x}} + \frac{2 \sqrt{x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(c*x**2+b*x)**(1/2),x)

[Out]

16*b**2*(b*x + c*x**2)**(3/2)/(105*c**3*x**(3/2)) - 8*b*(b*x + c*x**2)**(3/2)/(3
5*c**2*sqrt(x)) + 2*sqrt(x)*(b*x + c*x**2)**(3/2)/(7*c)

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Mathematica [A]  time = 0.0237818, size = 53, normalized size = 0.66 \[ \frac{2 \sqrt{x (b+c x)} \left (8 b^3-4 b^2 c x+3 b c^2 x^2+15 c^3 x^3\right )}{105 c^3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[x*(b + c*x)]*(8*b^3 - 4*b^2*c*x + 3*b*c^2*x^2 + 15*c^3*x^3))/(105*c^3*Sq
rt[x])

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 15\,{c}^{2}{x}^{2}-12\,bcx+8\,{b}^{2} \right ) }{105\,{c}^{3}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(c*x^2+b*x)^(1/2),x)

[Out]

2/105*(c*x+b)*(15*c^2*x^2-12*b*c*x+8*b^2)*(c*x^2+b*x)^(1/2)/c^3/x^(1/2)

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Maxima [A]  time = 0.741628, size = 57, normalized size = 0.71 \[ \frac{2 \,{\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x + b}}{105 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*x^(3/2),x, algorithm="maxima")

[Out]

2/105*(15*c^3*x^3 + 3*b*c^2*x^2 - 4*b^2*c*x + 8*b^3)*sqrt(c*x + b)/c^3

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Fricas [A]  time = 0.233082, size = 85, normalized size = 1.06 \[ \frac{2 \,{\left (15 \, c^{4} x^{5} + 18 \, b c^{3} x^{4} - b^{2} c^{2} x^{3} + 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )}}{105 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*x^(3/2),x, algorithm="fricas")

[Out]

2/105*(15*c^4*x^5 + 18*b*c^3*x^4 - b^2*c^2*x^3 + 4*b^3*c*x^2 + 8*b^4*x)/(sqrt(c*
x^2 + b*x)*c^3*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{\frac{3}{2}} \sqrt{x \left (b + c x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(x**(3/2)*sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.212627, size = 62, normalized size = 0.78 \[ -\frac{16 \, b^{\frac{7}{2}}}{105 \, c^{3}} + \frac{2 \,{\left (15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}\right )}}{105 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*x^(3/2),x, algorithm="giac")

[Out]

-16/105*b^(7/2)/c^3 + 2/105*(15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x
+ b)^(3/2)*b^2)/c^3