3.1046 \(\int \frac {a+b x^2+c x^4}{x^{5/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac {2 a}{3 x^{3/2}}+2 b \sqrt {x}+\frac {2}{5} c x^{5/2} \]

[Out]

-2/3*a/x^(3/2)+2/5*c*x^(5/2)+2*b*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} \[ -\frac {2 a}{3 x^{3/2}}+2 b \sqrt {x}+\frac {2}{5} c x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^(5/2),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.86 \[ \frac {2 \left (-5 a+15 b x^2+3 c x^4\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(5/2),x]

[Out]

(2*(-5*a + 15*b*x^2 + 3*c*x^4))/(15*x^(3/2))

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fricas [A]  time = 0.59, size = 21, normalized size = 0.72 \[ \frac {2 \, {\left (3 \, c x^{4} + 15 \, b x^{2} - 5 \, a\right )}}{15 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*c*x^4 + 15*b*x^2 - 5*a)/x^(3/2)

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giac [A]  time = 0.33, size = 19, normalized size = 0.66 \[ \frac {2}{5} \, c x^{\frac {5}{2}} + 2 \, b \sqrt {x} - \frac {2 \, a}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*c*x^(5/2) + 2*b*sqrt(x) - 2/3*a/x^(3/2)

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maple [A]  time = 0.00, size = 22, normalized size = 0.76 \[ -\frac {2 \left (-3 c \,x^{4}-15 b \,x^{2}+5 a \right )}{15 x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-3*c*x^4-15*b*x^2+5*a)/x^(3/2)

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maxima [A]  time = 1.02, size = 19, normalized size = 0.66 \[ \frac {2}{5} \, c x^{\frac {5}{2}} + 2 \, b \sqrt {x} - \frac {2 \, a}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*c*x^(5/2) + 2*b*sqrt(x) - 2/3*a/x^(3/2)

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mupad [B]  time = 0.03, size = 21, normalized size = 0.72 \[ \frac {6\,c\,x^4+30\,b\,x^2-10\,a}{15\,x^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^(5/2),x)

[Out]

(30*b*x^2 - 10*a + 6*c*x^4)/(15*x^(3/2))

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sympy [A]  time = 1.28, size = 27, normalized size = 0.93 \[ - \frac {2 a}{3 x^{\frac {3}{2}}} + 2 b \sqrt {x} + \frac {2 c x^{\frac {5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/(3*x**(3/2)) + 2*b*sqrt(x) + 2*c*x**(5/2)/5

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