3.1044 \(\int \frac {a+b x^2+c x^4}{\sqrt {x}} \, dx\)

Optimal. Leaf size=29 \[ 2 a \sqrt {x}+\frac {2}{5} b x^{5/2}+\frac {2}{9} c x^{9/2} \]

[Out]

2/5*b*x^(5/2)+2/9*c*x^(9/2)+2*a*x^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \[ 2 a \sqrt {x}+\frac {2}{5} b x^{5/2}+\frac {2}{9} c x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(5/2))/5 + (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+b x^2+c x^4}{\sqrt {x}} \, dx &=\int \left (\frac {a}{\sqrt {x}}+b x^{3/2}+c x^{7/2}\right ) \, dx\\ &=2 a \sqrt {x}+\frac {2}{5} b x^{5/2}+\frac {2}{9} c x^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.86 \[ \frac {2}{45} \sqrt {x} \left (45 a+9 b x^2+5 c x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(45*a + 9*b*x^2 + 5*c*x^4))/45

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fricas [A]  time = 0.50, size = 21, normalized size = 0.72 \[ \frac {2}{45} \, {\left (5 \, c x^{4} + 9 \, b x^{2} + 45 \, a\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(5*c*x^4 + 9*b*x^2 + 45*a)*sqrt(x)

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giac [A]  time = 0.15, size = 19, normalized size = 0.66 \[ \frac {2}{9} \, c x^{\frac {9}{2}} + \frac {2}{5} \, b x^{\frac {5}{2}} + 2 \, a \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*c*x^(9/2) + 2/5*b*x^(5/2) + 2*a*sqrt(x)

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maple [A]  time = 0.00, size = 22, normalized size = 0.76 \[ \frac {2 \left (5 c \,x^{4}+9 b \,x^{2}+45 a \right ) \sqrt {x}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*x^(1/2)*(5*c*x^4+9*b*x^2+45*a)

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maxima [A]  time = 1.03, size = 19, normalized size = 0.66 \[ \frac {2}{9} \, c x^{\frac {9}{2}} + \frac {2}{5} \, b x^{\frac {5}{2}} + 2 \, a \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*c*x^(9/2) + 2/5*b*x^(5/2) + 2*a*sqrt(x)

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mupad [B]  time = 0.03, size = 21, normalized size = 0.72 \[ \frac {2\,\sqrt {x}\,\left (5\,c\,x^4+9\,b\,x^2+45\,a\right )}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^(1/2),x)

[Out]

(2*x^(1/2)*(45*a + 9*b*x^2 + 5*c*x^4))/45

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sympy [A]  time = 0.82, size = 27, normalized size = 0.93 \[ 2 a \sqrt {x} + \frac {2 b x^{\frac {5}{2}}}{5} + \frac {2 c x^{\frac {9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) + 2*b*x**(5/2)/5 + 2*c*x**(9/2)/9

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