\(\int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx\) [1047]

Optimal result

Integrand size = 18, antiderivative size = 29 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=-\frac {2 a}{5 x^{5/2}}-\frac {2 b}{\sqrt {x}}+\frac {2}{3} c x^{3/2} \]

[Out]

-2/5*a/x^(5/2)+2/3*c*x^(3/2)-2*b/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} \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=-\frac {2 a}{5 x^{5/2}}-\frac {2 b}{\sqrt {x}}+\frac {2}{3} c x^{3/2} \]

[In]

Int[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x] + (2*c*x^(3/2))/3

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=\frac {2 \left (-3 a-15 b x^2+5 c x^4\right )}{15 x^{5/2}} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(7/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(-\frac {2 a}{5 x^{\frac {5}{2}}}+\frac {2 c \,x^{\frac {3}{2}}}{3}-\frac {2 b}{\sqrt {x}}\)	\(20\)
default	\(-\frac {2 a}{5 x^{\frac {5}{2}}}+\frac {2 c \,x^{\frac {3}{2}}}{3}-\frac {2 b}{\sqrt {x}}\)	\(20\)
gosper	\(-\frac {2 \left (-5 c \,x^{4}+15 b \,x^{2}+3 a \right )}{15 x^{\frac {5}{2}}}\)	\(22\)
trager	\(-\frac {2 \left (-5 c \,x^{4}+15 b \,x^{2}+3 a \right )}{15 x^{\frac {5}{2}}}\)	\(22\)
risch	\(-\frac {2 \left (-5 c \,x^{4}+15 b \,x^{2}+3 a \right )}{15 x^{\frac {5}{2}}}\)	\(22\)

[In]

int((c*x^4+b*x^2+a)/x^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*a/x^(5/2)+2/3*c*x^(3/2)-2*b/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=\frac {2 \, {\left (5 \, c x^{4} - 15 \, b x^{2} - 3 \, a\right )}}{15 \, x^{\frac {5}{2}}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^(7/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=- \frac {2 a}{5 x^{\frac {5}{2}}} - \frac {2 b}{\sqrt {x}} + \frac {2 c x^{\frac {3}{2}}}{3} \]

[In]

integrate((c*x**4+b*x**2+a)/x**(7/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=\frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac {5}{2}}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^(7/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=\frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac {5}{2}}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^(7/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x^2+c x^4}{x^{7/2}} \, dx=-\frac {-10\,c\,x^4+30\,b\,x^2+6\,a}{15\,x^{5/2}} \]

[In]

int((a + b*x^2 + c*x^4)/x^(7/2),x)

[Out]

-(6*a + 30*b*x^2 - 10*c*x^4)/(15*x^(5/2))