3.298 \(\int \frac{b x^2+c x^4}{x^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ 2 b \sqrt{x}+\frac{2}{5} c x^{5/2} \]

[Out]

2*b*Sqrt[x] + (2*c*x^(5/2))/5

________________________________________________________________________________________

Rubi [A]  time = 0.0047181, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {14} \[ 2 b \sqrt{x}+\frac{2}{5} c x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^(5/2),x]

[Out]

2*b*Sqrt[x] + (2*c*x^(5/2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0047264, size = 19, normalized size = 1. \[ 2 b \sqrt{x}+\frac{2}{5} c x^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^(5/2),x]

[Out]

2*b*Sqrt[x] + (2*c*x^(5/2))/5

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 15, normalized size = 0.8 \begin{align*}{\frac{2\,c{x}^{2}+10\,b}{5}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^(5/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.960773, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{5} \, c x^{\frac{5}{2}} + 2 \, b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(5/2),x, algorithm="maxima")

[Out]

2/5*c*x^(5/2) + 2*b*sqrt(x)

________________________________________________________________________________________

Fricas [A]  time = 1.20248, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{5} \,{\left (c x^{2} + 5 \, b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(5/2),x, algorithm="fricas")

[Out]

2/5*(c*x^2 + 5*b)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 1.1772, size = 17, normalized size = 0.89 \begin{align*} 2 b \sqrt{x} + \frac{2 c x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)/x**(5/2),x)

[Out]

2*b*sqrt(x) + 2*c*x**(5/2)/5

________________________________________________________________________________________

Giac [A]  time = 1.11644, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{5} \, c x^{\frac{5}{2}} + 2 \, b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(5/2),x, algorithm="giac")

[Out]

2/5*c*x^(5/2) + 2*b*sqrt(x)