\(\int \frac {a+c x^4}{\sqrt {x}} \, dx\) [720]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2}{9} c x^{9/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2}{9} c x^{9/2} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=\frac {2}{9} \left (9 a \sqrt {x}+c x^{9/2}\right ) \]

[In]

Integrate[(a + c*x^4)/Sqrt[x],x]

[Out]

(2*(9*a*Sqrt[x] + c*x^(9/2)))/9

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {2 c \,x^{\frac {9}{2}}}{9}+2 a \sqrt {x}\)	\(14\)
default	\(\frac {2 c \,x^{\frac {9}{2}}}{9}+2 a \sqrt {x}\)	\(14\)
gosper	\(\frac {2 \sqrt {x}\, \left (x^{4} c +9 a \right )}{9}\)	\(15\)
trager	\(\left (\frac {2 x^{4} c}{9}+2 a \right ) \sqrt {x}\)	\(15\)
risch	\(\frac {2 \sqrt {x}\, \left (x^{4} c +9 a \right )}{9}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=\frac {2}{9} \, {\left (c x^{4} + 9 \, a\right )} \sqrt {x} \]

[In]

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

[Out]

2/9*(c*x^4 + 9*a)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=2 a \sqrt {x} + \frac {2 c x^{\frac {9}{2}}}{9} \]

[In]

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

[Out]

2*a*sqrt(x) + 2*c*x**(9/2)/9

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=\frac {2}{9} \, c x^{\frac {9}{2}} + 2 \, a \sqrt {x} \]

[In]

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

[Out]

2/9*c*x^(9/2) + 2*a*sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=\frac {2}{9} \, c x^{\frac {9}{2}} + 2 \, a \sqrt {x} \]

[In]

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

[Out]

2/9*c*x^(9/2) + 2*a*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {a+c x^4}{\sqrt {x}} \, dx=\frac {2\,\sqrt {x}\,\left (c\,x^4+9\,a\right )}{9} \]

[In]

int((a + c*x^4)/x^(1/2),x)

[Out]

(2*x^(1/2)*(9*a + c*x^4))/9