\(\int x^m (a+b x^7) \, dx\) [1435]

Optimal result

Integrand size = 11, antiderivative size = 25 \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {a x^{1+m}}{1+m}+\frac {b x^{8+m}}{8+m} \]

[Out]

a*x^(1+m)/(1+m)+b*x^(8+m)/(8+m)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, 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} \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {a x^{m+1}}{m+1}+\frac {b x^{m+8}}{m+8} \]

[In]

Int[x^m*(a + b*x^7),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(8 + m))/(8 + m)

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 (a x^m+b x^{7+m}\right ) \, dx \\ & = \frac {a x^{1+m}}{1+m}+\frac {b x^{8+m}}{8+m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {a x^{1+m}}{1+m}+\frac {b x^{8+m}}{8+m} \]

[In]

Integrate[x^m*(a + b*x^7),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(8 + m))/(8 + m)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20

method	result	size
norman	\(\frac {a x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b \,x^{8} {\mathrm e}^{m \ln \left (x \right )}}{8+m}\)	\(30\)
risch	\(\frac {x \left (b m \,x^{7}+b \,x^{7}+a m +8 a \right ) x^{m}}{\left (8+m \right ) \left (1+m \right )}\)	\(34\)
gosper	\(\frac {x^{1+m} \left (b m \,x^{7}+b \,x^{7}+a m +8 a \right )}{\left (1+m \right ) \left (8+m \right )}\)	\(35\)
parallelrisch	\(\frac {x^{8} x^{m} b m +x^{8} x^{m} b +x \,x^{m} a m +8 x \,x^{m} a}{\left (8+m \right ) \left (1+m \right )}\)	\(44\)

[In]

int(x^m*(b*x^7+a),x,method=_RETURNVERBOSE)

[Out]

a/(1+m)*x*exp(m*ln(x))+b/(8+m)*x^8*exp(m*ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {{\left ({\left (b m + b\right )} x^{8} + {\left (a m + 8 \, a\right )} x\right )} x^{m}}{m^{2} + 9 \, m + 8} \]

[In]

integrate(x^m*(b*x^7+a),x, algorithm="fricas")

[Out]

((b*m + b)*x^8 + (a*m + 8*a)*x)*x^m/(m^2 + 9*m + 8)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (19) = 38\).

Time = 0.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.76 \[ \int x^m \left (a+b x^7\right ) \, dx=\begin {cases} - \frac {a}{7 x^{7}} + b \log {\left (x \right )} & \text {for}\: m = -8 \\a \log {\left (x \right )} + \frac {b x^{7}}{7} & \text {for}\: m = -1 \\\frac {a m x x^{m}}{m^{2} + 9 m + 8} + \frac {8 a x x^{m}}{m^{2} + 9 m + 8} + \frac {b m x^{8} x^{m}}{m^{2} + 9 m + 8} + \frac {b x^{8} x^{m}}{m^{2} + 9 m + 8} & \text {otherwise} \end {cases} \]

[In]

integrate(x**m*(b*x**7+a),x)

[Out]

Piecewise((-a/(7*x**7) + b*log(x), Eq(m, -8)), (a*log(x) + b*x**7/7, Eq(m, -1)), (a*m*x*x**m/(m**2 + 9*m + 8)
+ 8*a*x*x**m/(m**2 + 9*m + 8) + b*m*x**8*x**m/(m**2 + 9*m + 8) + b*x**8*x**m/(m**2 + 9*m + 8), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {b x^{m + 8}}{m + 8} + \frac {a x^{m + 1}}{m + 1} \]

[In]

integrate(x^m*(b*x^7+a),x, algorithm="maxima")

[Out]

b*x^(m + 8)/(m + 8) + a*x^(m + 1)/(m + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {b m x^{8} x^{m} + b x^{8} x^{m} + a m x x^{m} + 8 \, a x x^{m}}{m^{2} + 9 \, m + 8} \]

[In]

integrate(x^m*(b*x^7+a),x, algorithm="giac")

[Out]

(b*m*x^8*x^m + b*x^8*x^m + a*m*x*x^m + 8*a*x*x^m)/(m^2 + 9*m + 8)

Mupad [B] (verification not implemented)

Time = 6.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int x^m \left (a+b x^7\right ) \, dx=\frac {x^{m+1}\,\left (8\,a+a\,m+b\,x^7+b\,m\,x^7\right )}{m^2+9\,m+8} \]

[In]

int(x^m*(a + b*x^7),x)

[Out]

(x^(m + 1)*(8*a + a*m + b*x^7 + b*m*x^7))/(9*m + m^2 + 8)