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

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Rubi [A]  time = 0.0265036, 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, Rules used = {656, 648} \[ \frac{16 b^2 \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{8 b \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c} \]

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

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c}-\frac{(4 b) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{5 c}\\ &=-\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}+\frac{\left (8 b^2\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{15 c^2}\\ &=\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}\\ \end{align*}

Mathematica [A]  time = 0.0227216, 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 = 0.049, size = 44, normalized size = 0.6 \begin{align*}{\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}}}} \end{align*}

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 = 1.15306, size = 57, normalized size = 0.71 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(c*x^2+b*x)^(1/2),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 = 2.0872, size = 92, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (3 \, c^{2} x^{2} - 4 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x}}{15 \, c^{3} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

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

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 [A]  time = 1.28111, size = 62, normalized size = 0.78 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(c*x^2+b*x)^(1/2),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