3.106 \(\int \frac{x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.059282, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.06826, size = 42, normalized size = 0.88 \[ \frac{4 b \sqrt{x}}{c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{3}{2}}}{c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

4*b*sqrt(x)/(c**2*sqrt(b*x + c*x**2)) + 2*x**(3/2)/(c*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 0.0213057, size = 28, normalized size = 0.58 \[ \frac{2 \sqrt{x} (2 b+c x)}{c^2 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(2*Sqrt[x]*(2*b + c*x))/(c^2*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.004, size = 32, normalized size = 0.7 \[ 2\,{\frac{ \left ( cx+b \right ) \left ( cx+2\,b \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2*(c*x+b)*(c*x+2*b)*x^(3/2)/c^2/(c*x^2+b*x)^(3/2)

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Maxima [A]  time = 0.735935, size = 47, normalized size = 0.98 \[ \frac{2 \, \sqrt{c x + b} x}{c^{2} x + b c} + \frac{4 \, b}{\sqrt{c x + b} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

2*sqrt(c*x + b)*x/(c^2*x + b*c) + 4*b/(sqrt(c*x + b)*c^2)

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Fricas [A]  time = 0.225112, size = 39, normalized size = 0.81 \[ \frac{2 \,{\left (c x^{2} + 2 \, b x\right )}}{\sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

2*(c*x^2 + 2*b*x)/(sqrt(c*x^2 + b*x)*c^2*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(x**(5/2)/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.20994, size = 42, normalized size = 0.88 \[ \frac{2 \,{\left (\sqrt{c x + b} + \frac{b}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{4 \, \sqrt{b}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

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