\(\int x^3 (a+b x^3)^8 \, dx\) [309]

Optimal result

Integrand size = 13, antiderivative size = 108 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^4}{4}+\frac {8}{7} a^7 b x^7+\frac {14}{5} a^6 b^2 x^{10}+\frac {56}{13} a^5 b^3 x^{13}+\frac {35}{8} a^4 b^4 x^{16}+\frac {56}{19} a^3 b^5 x^{19}+\frac {14}{11} a^2 b^6 x^{22}+\frac {8}{25} a b^7 x^{25}+\frac {b^8 x^{28}}{28} \]

[Out]

1/4*a^8*x^4+8/7*a^7*b*x^7+14/5*a^6*b^2*x^10+56/13*a^5*b^3*x^13+35/8*a^4*b^4*x^16+56/19*a^3*b^5*x^19+14/11*a^2*
b^6*x^22+8/25*a*b^7*x^25+1/28*b^8*x^28

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^4}{4}+\frac {8}{7} a^7 b x^7+\frac {14}{5} a^6 b^2 x^{10}+\frac {56}{13} a^5 b^3 x^{13}+\frac {35}{8} a^4 b^4 x^{16}+\frac {56}{19} a^3 b^5 x^{19}+\frac {14}{11} a^2 b^6 x^{22}+\frac {8}{25} a b^7 x^{25}+\frac {b^8 x^{28}}{28} \]

[In]

Int[x^3*(a + b*x^3)^8,x]

[Out]

(a^8*x^4)/4 + (8*a^7*b*x^7)/7 + (14*a^6*b^2*x^10)/5 + (56*a^5*b^3*x^13)/13 + (35*a^4*b^4*x^16)/8 + (56*a^3*b^5
*x^19)/19 + (14*a^2*b^6*x^22)/11 + (8*a*b^7*x^25)/25 + (b^8*x^28)/28

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^3+8 a^7 b x^6+28 a^6 b^2 x^9+56 a^5 b^3 x^{12}+70 a^4 b^4 x^{15}+56 a^3 b^5 x^{18}+28 a^2 b^6 x^{21}+8 a b^7 x^{24}+b^8 x^{27}\right ) \, dx \\ & = \frac {a^8 x^4}{4}+\frac {8}{7} a^7 b x^7+\frac {14}{5} a^6 b^2 x^{10}+\frac {56}{13} a^5 b^3 x^{13}+\frac {35}{8} a^4 b^4 x^{16}+\frac {56}{19} a^3 b^5 x^{19}+\frac {14}{11} a^2 b^6 x^{22}+\frac {8}{25} a b^7 x^{25}+\frac {b^8 x^{28}}{28} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^4}{4}+\frac {8}{7} a^7 b x^7+\frac {14}{5} a^6 b^2 x^{10}+\frac {56}{13} a^5 b^3 x^{13}+\frac {35}{8} a^4 b^4 x^{16}+\frac {56}{19} a^3 b^5 x^{19}+\frac {14}{11} a^2 b^6 x^{22}+\frac {8}{25} a b^7 x^{25}+\frac {b^8 x^{28}}{28} \]

[In]

Integrate[x^3*(a + b*x^3)^8,x]

[Out]

(a^8*x^4)/4 + (8*a^7*b*x^7)/7 + (14*a^6*b^2*x^10)/5 + (56*a^5*b^3*x^13)/13 + (35*a^4*b^4*x^16)/8 + (56*a^3*b^5
*x^19)/19 + (14*a^2*b^6*x^22)/11 + (8*a*b^7*x^25)/25 + (b^8*x^28)/28

Maple [A] (verified)

Time = 3.55 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {1}{4} a^{8} x^{4}+\frac {8}{7} x^{7} b \,a^{7}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{13} x^{13} b^{3} a^{5}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {8}{25} a \,b^{7} x^{25}+\frac {1}{28} b^{8} x^{28}\)	\(91\)
default	\(\frac {1}{4} a^{8} x^{4}+\frac {8}{7} x^{7} b \,a^{7}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{13} x^{13} b^{3} a^{5}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {8}{25} a \,b^{7} x^{25}+\frac {1}{28} b^{8} x^{28}\)	\(91\)
norman	\(\frac {1}{4} a^{8} x^{4}+\frac {8}{7} x^{7} b \,a^{7}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{13} x^{13} b^{3} a^{5}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {8}{25} a \,b^{7} x^{25}+\frac {1}{28} b^{8} x^{28}\)	\(91\)
risch	\(\frac {1}{4} a^{8} x^{4}+\frac {8}{7} x^{7} b \,a^{7}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{13} x^{13} b^{3} a^{5}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {8}{25} a \,b^{7} x^{25}+\frac {1}{28} b^{8} x^{28}\)	\(91\)
parallelrisch	\(\frac {1}{4} a^{8} x^{4}+\frac {8}{7} x^{7} b \,a^{7}+\frac {14}{5} a^{6} b^{2} x^{10}+\frac {56}{13} x^{13} b^{3} a^{5}+\frac {35}{8} a^{4} b^{4} x^{16}+\frac {56}{19} a^{3} b^{5} x^{19}+\frac {14}{11} a^{2} b^{6} x^{22}+\frac {8}{25} a \,b^{7} x^{25}+\frac {1}{28} b^{8} x^{28}\)	\(91\)

[In]

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

[Out]

1/4*a^8*x^4+8/7*x^7*b*a^7+14/5*a^6*b^2*x^10+56/13*x^13*b^3*a^5+35/8*a^4*b^4*x^16+56/19*a^3*b^5*x^19+14/11*a^2*
b^6*x^22+8/25*a*b^7*x^25+1/28*b^8*x^28

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {1}{28} \, b^{8} x^{28} + \frac {8}{25} \, a b^{7} x^{25} + \frac {14}{11} \, a^{2} b^{6} x^{22} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {35}{8} \, a^{4} b^{4} x^{16} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{4} \, a^{8} x^{4} \]

[In]

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

[Out]

1/28*b^8*x^28 + 8/25*a*b^7*x^25 + 14/11*a^2*b^6*x^22 + 56/19*a^3*b^5*x^19 + 35/8*a^4*b^4*x^16 + 56/13*a^5*b^3*
x^13 + 14/5*a^6*b^2*x^10 + 8/7*a^7*b*x^7 + 1/4*a^8*x^4

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{4}}{4} + \frac {8 a^{7} b x^{7}}{7} + \frac {14 a^{6} b^{2} x^{10}}{5} + \frac {56 a^{5} b^{3} x^{13}}{13} + \frac {35 a^{4} b^{4} x^{16}}{8} + \frac {56 a^{3} b^{5} x^{19}}{19} + \frac {14 a^{2} b^{6} x^{22}}{11} + \frac {8 a b^{7} x^{25}}{25} + \frac {b^{8} x^{28}}{28} \]

[In]

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

[Out]

a**8*x**4/4 + 8*a**7*b*x**7/7 + 14*a**6*b**2*x**10/5 + 56*a**5*b**3*x**13/13 + 35*a**4*b**4*x**16/8 + 56*a**3*
b**5*x**19/19 + 14*a**2*b**6*x**22/11 + 8*a*b**7*x**25/25 + b**8*x**28/28

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {1}{28} \, b^{8} x^{28} + \frac {8}{25} \, a b^{7} x^{25} + \frac {14}{11} \, a^{2} b^{6} x^{22} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {35}{8} \, a^{4} b^{4} x^{16} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{4} \, a^{8} x^{4} \]

[In]

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

[Out]

1/28*b^8*x^28 + 8/25*a*b^7*x^25 + 14/11*a^2*b^6*x^22 + 56/19*a^3*b^5*x^19 + 35/8*a^4*b^4*x^16 + 56/13*a^5*b^3*
x^13 + 14/5*a^6*b^2*x^10 + 8/7*a^7*b*x^7 + 1/4*a^8*x^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {1}{28} \, b^{8} x^{28} + \frac {8}{25} \, a b^{7} x^{25} + \frac {14}{11} \, a^{2} b^{6} x^{22} + \frac {56}{19} \, a^{3} b^{5} x^{19} + \frac {35}{8} \, a^{4} b^{4} x^{16} + \frac {56}{13} \, a^{5} b^{3} x^{13} + \frac {14}{5} \, a^{6} b^{2} x^{10} + \frac {8}{7} \, a^{7} b x^{7} + \frac {1}{4} \, a^{8} x^{4} \]

[In]

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

[Out]

1/28*b^8*x^28 + 8/25*a*b^7*x^25 + 14/11*a^2*b^6*x^22 + 56/19*a^3*b^5*x^19 + 35/8*a^4*b^4*x^16 + 56/13*a^5*b^3*
x^13 + 14/5*a^6*b^2*x^10 + 8/7*a^7*b*x^7 + 1/4*a^8*x^4

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \left (a+b x^3\right )^8 \, dx=\frac {a^8\,x^4}{4}+\frac {8\,a^7\,b\,x^7}{7}+\frac {14\,a^6\,b^2\,x^{10}}{5}+\frac {56\,a^5\,b^3\,x^{13}}{13}+\frac {35\,a^4\,b^4\,x^{16}}{8}+\frac {56\,a^3\,b^5\,x^{19}}{19}+\frac {14\,a^2\,b^6\,x^{22}}{11}+\frac {8\,a\,b^7\,x^{25}}{25}+\frac {b^8\,x^{28}}{28} \]

[In]

int(x^3*(a + b*x^3)^8,x)

[Out]

(a^8*x^4)/4 + (b^8*x^28)/28 + (8*a^7*b*x^7)/7 + (8*a*b^7*x^25)/25 + (14*a^6*b^2*x^10)/5 + (56*a^5*b^3*x^13)/13
+ (35*a^4*b^4*x^16)/8 + (56*a^3*b^5*x^19)/19 + (14*a^2*b^6*x^22)/11