\(\int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx\) [1045]

Optimal result

Integrand size = 18, antiderivative size = 29 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=-\frac {2 a}{\sqrt {x}}+\frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

[Out]

2/3*b*x^(3/2)+2/7*c*x^(7/2)-2*a/x^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=-\frac {2 a}{\sqrt {x}}+\frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

[In]

Int[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3 + (2*c*x^(7/2))/7

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}{x^{3/2}}+b \sqrt {x}+c x^{5/2}\right ) \, dx \\ & = -\frac {2 a}{\sqrt {x}}+\frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=-\frac {2 \left (21 a-7 b x^2-3 c x^4\right )}{21 \sqrt {x}} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(-2*(21*a - 7*b*x^2 - 3*c*x^4))/(21*Sqrt[x])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 c \,x^{\frac {7}{2}}}{7}-\frac {2 a}{\sqrt {x}}\)	\(20\)
default	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 c \,x^{\frac {7}{2}}}{7}-\frac {2 a}{\sqrt {x}}\)	\(20\)
gosper	\(-\frac {2 \left (-3 c \,x^{4}-7 b \,x^{2}+21 a \right )}{21 \sqrt {x}}\)	\(22\)
trager	\(-\frac {2 \left (-3 c \,x^{4}-7 b \,x^{2}+21 a \right )}{21 \sqrt {x}}\)	\(22\)
risch	\(-\frac {2 \left (-3 c \,x^{4}-7 b \,x^{2}+21 a \right )}{21 \sqrt {x}}\)	\(22\)

[In]

int((c*x^4+b*x^2+a)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3*b*x^(3/2)+2/7*c*x^(7/2)-2*a/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=\frac {2 \, {\left (3 \, c x^{4} + 7 \, b x^{2} - 21 \, a\right )}}{21 \, \sqrt {x}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*c*x^4 + 7*b*x^2 - 21*a)/sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=- \frac {2 a}{\sqrt {x}} + \frac {2 b x^{\frac {3}{2}}}{3} + \frac {2 c x^{\frac {7}{2}}}{7} \]

[In]

integrate((c*x**4+b*x**2+a)/x**(3/2),x)

[Out]

-2*a/sqrt(x) + 2*b*x**(3/2)/3 + 2*c*x**(7/2)/7

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=\frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} - \frac {2 \, a}{\sqrt {x}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="maxima")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2) - 2*a/sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=\frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} - \frac {2 \, a}{\sqrt {x}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="giac")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2) - 2*a/sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx=\frac {6\,c\,x^4+14\,b\,x^2-42\,a}{21\,\sqrt {x}} \]

[In]

int((a + b*x^2 + c*x^4)/x^(3/2),x)

[Out]

(14*b*x^2 - 42*a + 6*c*x^4)/(21*x^(1/2))