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

Optimal. Leaf size=80 \[ \frac{16 b^2 \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{8 b \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c} \]

[Out]

(16*b^2*Sqrt[b*x + c*x^2])/(15*c^3*Sqrt[x]) - (8*b*Sqrt[x]*Sqrt[b*x + c*x^2])/(1
5*c^2) + (2*x^(3/2)*Sqrt[b*x + c*x^2])/(5*c)

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Rubi [A]  time = 0.09171, 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 \[ \frac{16 b^2 \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{8 b \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(16*b^2*Sqrt[b*x + c*x^2])/(15*c^3*Sqrt[x]) - (8*b*Sqrt[x]*Sqrt[b*x + c*x^2])/(1
5*c^2) + (2*x^(3/2)*Sqrt[b*x + c*x^2])/(5*c)

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Rubi in Sympy [A]  time = 9.4552, size = 73, normalized size = 0.91 \[ \frac{16 b^{2} \sqrt{b x + c x^{2}}}{15 c^{3} \sqrt{x}} - \frac{8 b \sqrt{x} \sqrt{b x + c x^{2}}}{15 c^{2}} + \frac{2 x^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{5 c} \]

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]

16*b**2*sqrt(b*x + c*x**2)/(15*c**3*sqrt(x)) - 8*b*sqrt(x)*sqrt(b*x + c*x**2)/(1
5*c**2) + 2*x**(3/2)*sqrt(b*x + c*x**2)/(5*c)

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Mathematica [A]  time = 0.0310636, size = 42, normalized size = 0.52 \[ \frac{2 \sqrt{x (b+c x)} \left (8 b^2-4 b c x+3 c^2 x^2\right )}{15 c^3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3\,{c}^{2}{x}^{2}-4\,bcx+8\,{b}^{2} \right ) }{15\,{c}^{3}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(c*x+b)*(3*c^2*x^2-4*b*c*x+8*b^2)*x^(1/2)/c^3/(c*x^2+b*x)^(1/2)

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Maxima [A]  time = 0.707365, size = 57, normalized size = 0.71 \[ \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )}}{15 \, \sqrt{c x + b} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*c^3*x^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 = 0.220536, size = 70, normalized size = 0.88 \[ \frac{2 \,{\left (3 \, c^{3} x^{4} - b c^{2} x^{3} + 4 \, b^{2} c x^{2} + 8 \, b^{3} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 = 0.210165, size = 62, normalized size = 0.78 \[ -\frac{16 \, b^{\frac{5}{2}}}{15 \, c^{3}} + \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 10 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{c x + b} b^{2}\right )}}{15 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/15*b^(5/2)/c^3 + 2/15*(3*(c*x + b)^(5/2) - 10*(c*x + b)^(3/2)*b + 15*sqrt(c*
x + b)*b^2)/c^3