\(\int (a+\frac {b}{x}) x^6 \, dx\) [1546]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {b x^6}{6}+\frac {a x^7}{7} \]

[Out]

1/6*b*x^6+1/7*a*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {a x^7}{7}+\frac {b x^6}{6} \]

[In]

Int[(a + b/x)*x^6,x]

[Out]

(b*x^6)/6 + (a*x^7)/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 (b x^5+a x^6\right ) \, dx \\ & = \frac {b x^6}{6}+\frac {a x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {b x^6}{6}+\frac {a x^7}{7} \]

[In]

Integrate[(a + b/x)*x^6,x]

[Out]

(b*x^6)/6 + (a*x^7)/7

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
gosper	\(\frac {x^{6} \left (6 a x +7 b \right )}{42}\)	\(14\)
default	\(\frac {1}{6} b \,x^{6}+\frac {1}{7} a \,x^{7}\)	\(14\)
norman	\(\frac {1}{6} b \,x^{6}+\frac {1}{7} a \,x^{7}\)	\(14\)
risch	\(\frac {1}{6} b \,x^{6}+\frac {1}{7} a \,x^{7}\)	\(14\)
parallelrisch	\(\frac {1}{6} b \,x^{6}+\frac {1}{7} a \,x^{7}\)	\(14\)

[In]

int((a+b/x)*x^6,x,method=_RETURNVERBOSE)

[Out]

1/42*x^6*(6*a*x+7*b)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {1}{7} \, a x^{7} + \frac {1}{6} \, b x^{6} \]

[In]

integrate((a+b/x)*x^6,x, algorithm="fricas")

[Out]

1/7*a*x^7 + 1/6*b*x^6

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {a x^{7}}{7} + \frac {b x^{6}}{6} \]

[In]

integrate((a+b/x)*x**6,x)

[Out]

a*x**7/7 + b*x**6/6

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {1}{7} \, a x^{7} + \frac {1}{6} \, b x^{6} \]

[In]

integrate((a+b/x)*x^6,x, algorithm="maxima")

[Out]

1/7*a*x^7 + 1/6*b*x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {1}{7} \, a x^{7} + \frac {1}{6} \, b x^{6} \]

[In]

integrate((a+b/x)*x^6,x, algorithm="giac")

[Out]

1/7*a*x^7 + 1/6*b*x^6

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \left (a+\frac {b}{x}\right ) x^6 \, dx=\frac {x^6\,\left (7\,b+6\,a\,x\right )}{42} \]

[In]

int(x^6*(a + b/x),x)

[Out]

(x^6*(7*b + 6*a*x))/42