\(\int x (a+b x^3)^8 \, dx\) [310]

Optimal result

Integrand size = 11, antiderivative size = 106 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26} \]

[Out]

1/2*a^8*x^2+8/5*a^7*b*x^5+7/2*a^6*b^2*x^8+56/11*a^5*b^3*x^11+5*a^4*b^4*x^14+56/17*a^3*b^5*x^17+7/5*a^2*b^6*x^2
0+8/23*a*b^7*x^23+1/26*b^8*x^26

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, 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 = {276} \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26} \]

[In]

Int[x*(a + b*x^3)^8,x]

[Out]

(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/
17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^23)/23 + (b^8*x^26)/26

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^8 x+8 a^7 b x^4+28 a^6 b^2 x^7+56 a^5 b^3 x^{10}+70 a^4 b^4 x^{13}+56 a^3 b^5 x^{16}+28 a^2 b^6 x^{19}+8 a b^7 x^{22}+b^8 x^{25}\right ) \, dx \\ & = \frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^2}{2}+\frac {8}{5} a^7 b x^5+\frac {7}{2} a^6 b^2 x^8+\frac {56}{11} a^5 b^3 x^{11}+5 a^4 b^4 x^{14}+\frac {56}{17} a^3 b^5 x^{17}+\frac {7}{5} a^2 b^6 x^{20}+\frac {8}{23} a b^7 x^{23}+\frac {b^8 x^{26}}{26} \]

[In]

Integrate[x*(a + b*x^3)^8,x]

[Out]

(a^8*x^2)/2 + (8*a^7*b*x^5)/5 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 + 5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/
17 + (7*a^2*b^6*x^20)/5 + (8*a*b^7*x^23)/23 + (b^8*x^26)/26

Maple [A] (verified)

Time = 3.65 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} x^{5} b \,a^{7}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} x^{11} b^{3} a^{5}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\)	\(91\)
default	\(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} x^{5} b \,a^{7}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} x^{11} b^{3} a^{5}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\)	\(91\)
norman	\(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} x^{5} b \,a^{7}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} x^{11} b^{3} a^{5}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\)	\(91\)
risch	\(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} x^{5} b \,a^{7}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} x^{11} b^{3} a^{5}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\)	\(91\)
parallelrisch	\(\frac {1}{2} a^{8} x^{2}+\frac {8}{5} x^{5} b \,a^{7}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {56}{11} x^{11} b^{3} a^{5}+5 a^{4} b^{4} x^{14}+\frac {56}{17} a^{3} b^{5} x^{17}+\frac {7}{5} a^{2} b^{6} x^{20}+\frac {8}{23} a \,b^{7} x^{23}+\frac {1}{26} b^{8} x^{26}\)	\(91\)

[In]

int(x*(b*x^3+a)^8,x,method=_RETURNVERBOSE)

[Out]

1/2*a^8*x^2+8/5*x^5*b*a^7+7/2*a^6*b^2*x^8+56/11*x^11*b^3*a^5+5*a^4*b^4*x^14+56/17*a^3*b^5*x^17+7/5*a^2*b^6*x^2
0+8/23*a*b^7*x^23+1/26*b^8*x^26

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {1}{26} \, b^{8} x^{26} + \frac {8}{23} \, a b^{7} x^{23} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{2} \, a^{8} x^{2} \]

[In]

integrate(x*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11
+ 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/2*a^8*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{2}}{2} + \frac {8 a^{7} b x^{5}}{5} + \frac {7 a^{6} b^{2} x^{8}}{2} + \frac {56 a^{5} b^{3} x^{11}}{11} + 5 a^{4} b^{4} x^{14} + \frac {56 a^{3} b^{5} x^{17}}{17} + \frac {7 a^{2} b^{6} x^{20}}{5} + \frac {8 a b^{7} x^{23}}{23} + \frac {b^{8} x^{26}}{26} \]

[In]

integrate(x*(b*x**3+a)**8,x)

[Out]

a**8*x**2/2 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**8/2 + 56*a**5*b**3*x**11/11 + 5*a**4*b**4*x**14 + 56*a**3*b**5*
x**17/17 + 7*a**2*b**6*x**20/5 + 8*a*b**7*x**23/23 + b**8*x**26/26

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {1}{26} \, b^{8} x^{26} + \frac {8}{23} \, a b^{7} x^{23} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{2} \, a^{8} x^{2} \]

[In]

integrate(x*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11
+ 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/2*a^8*x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {1}{26} \, b^{8} x^{26} + \frac {8}{23} \, a b^{7} x^{23} + \frac {7}{5} \, a^{2} b^{6} x^{20} + \frac {56}{17} \, a^{3} b^{5} x^{17} + 5 \, a^{4} b^{4} x^{14} + \frac {56}{11} \, a^{5} b^{3} x^{11} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {8}{5} \, a^{7} b x^{5} + \frac {1}{2} \, a^{8} x^{2} \]

[In]

integrate(x*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/26*b^8*x^26 + 8/23*a*b^7*x^23 + 7/5*a^2*b^6*x^20 + 56/17*a^3*b^5*x^17 + 5*a^4*b^4*x^14 + 56/11*a^5*b^3*x^11
+ 7/2*a^6*b^2*x^8 + 8/5*a^7*b*x^5 + 1/2*a^8*x^2

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int x \left (a+b x^3\right )^8 \, dx=\frac {a^8\,x^2}{2}+\frac {8\,a^7\,b\,x^5}{5}+\frac {7\,a^6\,b^2\,x^8}{2}+\frac {56\,a^5\,b^3\,x^{11}}{11}+5\,a^4\,b^4\,x^{14}+\frac {56\,a^3\,b^5\,x^{17}}{17}+\frac {7\,a^2\,b^6\,x^{20}}{5}+\frac {8\,a\,b^7\,x^{23}}{23}+\frac {b^8\,x^{26}}{26} \]

[In]

int(x*(a + b*x^3)^8,x)

[Out]

(a^8*x^2)/2 + (b^8*x^26)/26 + (8*a^7*b*x^5)/5 + (8*a*b^7*x^23)/23 + (7*a^6*b^2*x^8)/2 + (56*a^5*b^3*x^11)/11 +
5*a^4*b^4*x^14 + (56*a^3*b^5*x^17)/17 + (7*a^2*b^6*x^20)/5