3.720 \(\int \frac{a+c x^4}{\sqrt{x}} \, dx\)

Optimal. Leaf size=19 \[ 2 a \sqrt{x}+\frac{2}{9} c x^{9/2} \]

[Out]

2*a*Sqrt[x] + (2*c*x^(9/2))/9

________________________________________________________________________________________

Rubi [A]  time = 0.0040556, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ 2 a \sqrt{x}+\frac{2}{9} c x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*c*x^(9/2))/9

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{a+c x^4}{\sqrt{x}} \, dx &=\int \left (\frac{a}{\sqrt{x}}+c x^{7/2}\right ) \, dx\\ &=2 a \sqrt{x}+\frac{2}{9} c x^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0046182, size = 19, normalized size = 1. \[ 2 a \sqrt{x}+\frac{2}{9} c x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*c*x^(9/2))/9

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{2\,c{x}^{4}+18\,a}{9}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)/x^(1/2),x)

[Out]

2/9*x^(1/2)*(c*x^4+9*a)

________________________________________________________________________________________

Maxima [A]  time = 0.978482, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{9} \, c x^{\frac{9}{2}} + 2 \, a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)/x^(1/2),x, algorithm="maxima")

[Out]

2/9*c*x^(9/2) + 2*a*sqrt(x)

________________________________________________________________________________________

Fricas [A]  time = 1.39485, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{9} \,{\left (c x^{4} + 9 \, a\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)/x^(1/2),x, algorithm="fricas")

[Out]

2/9*(c*x^4 + 9*a)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 0.982763, size = 17, normalized size = 0.89 \begin{align*} 2 a \sqrt{x} + \frac{2 c x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)/x**(1/2),x)

[Out]

2*a*sqrt(x) + 2*c*x**(9/2)/9

________________________________________________________________________________________

Giac [A]  time = 1.12666, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{9} \, c x^{\frac{9}{2}} + 2 \, a \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)/x^(1/2),x, algorithm="giac")

[Out]

2/9*c*x^(9/2) + 2*a*sqrt(x)