3.15.36 \(\int x^8 (a+b x^7) \, dx\) [1436]

Optimal. Leaf size=17 \[ \frac {a x^9}{9}+\frac {b x^{16}}{16} \]

[Out]

1/9*a*x^9+1/16*b*x^16

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \begin {gather*} \frac {a x^9}{9}+\frac {b x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^7),x]

[Out]

(a*x^9)/9 + (b*x^16)/16

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 x^8 \left (a+b x^7\right ) \, dx &=\int \left (a x^8+b x^{15}\right ) \, dx\\ &=\frac {a x^9}{9}+\frac {b x^{16}}{16}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^9}{9}+\frac {b x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^7),x]

[Out]

(a*x^9)/9 + (b*x^16)/16

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 14, normalized size = 0.82

method	result	size
gosper	\(\frac {1}{9} a \,x^{9}+\frac {1}{16} b \,x^{16}\)	\(14\)
default	\(\frac {1}{9} a \,x^{9}+\frac {1}{16} b \,x^{16}\)	\(14\)
norman	\(\frac {1}{9} a \,x^{9}+\frac {1}{16} b \,x^{16}\)	\(14\)
risch	\(\frac {1}{9} a \,x^{9}+\frac {1}{16} b \,x^{16}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^7+a),x,method=_RETURNVERBOSE)

[Out]

1/9*a*x^9+1/16*b*x^16

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{16} \, b x^{16} + \frac {1}{9} \, a x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a),x, algorithm="maxima")

[Out]

1/16*b*x^16 + 1/9*a*x^9

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{16} \, b x^{16} + \frac {1}{9} \, a x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a),x, algorithm="fricas")

[Out]

1/16*b*x^16 + 1/9*a*x^9

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{9}}{9} + \frac {b x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**7+a),x)

[Out]

a*x**9/9 + b*x**16/16

________________________________________________________________________________________

Giac [A]
time = 1.12, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{16} \, b x^{16} + \frac {1}{9} \, a x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^7+a),x, algorithm="giac")

[Out]

1/16*b*x^16 + 1/9*a*x^9

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {b\,x^{16}}{16}+\frac {a\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^7),x)

[Out]

(a*x^9)/9 + (b*x^16)/16

________________________________________________________________________________________