\(\int x^3 (a+b x^3)^5 \, dx\) [276]

Optimal result

Integrand size = 13, antiderivative size = 66 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {a^5 x^4}{4}+\frac {5}{7} a^4 b x^7+a^3 b^2 x^{10}+\frac {10}{13} a^2 b^3 x^{13}+\frac {5}{16} a b^4 x^{16}+\frac {b^5 x^{19}}{19} \]

[Out]

1/4*a^5*x^4+5/7*a^4*b*x^7+a^3*b^2*x^10+10/13*a^2*b^3*x^13+5/16*a*b^4*x^16+1/19*b^5*x^19

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, 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} \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {a^5 x^4}{4}+\frac {5}{7} a^4 b x^7+a^3 b^2 x^{10}+\frac {10}{13} a^2 b^3 x^{13}+\frac {5}{16} a b^4 x^{16}+\frac {b^5 x^{19}}{19} \]

[In]

Int[x^3*(a + b*x^3)^5,x]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^7)/7 + a^3*b^2*x^10 + (10*a^2*b^3*x^13)/13 + (5*a*b^4*x^16)/16 + (b^5*x^19)/19

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^5 x^3+5 a^4 b x^6+10 a^3 b^2 x^9+10 a^2 b^3 x^{12}+5 a b^4 x^{15}+b^5 x^{18}\right ) \, dx \\ & = \frac {a^5 x^4}{4}+\frac {5}{7} a^4 b x^7+a^3 b^2 x^{10}+\frac {10}{13} a^2 b^3 x^{13}+\frac {5}{16} a b^4 x^{16}+\frac {b^5 x^{19}}{19} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {a^5 x^4}{4}+\frac {5}{7} a^4 b x^7+a^3 b^2 x^{10}+\frac {10}{13} a^2 b^3 x^{13}+\frac {5}{16} a b^4 x^{16}+\frac {b^5 x^{19}}{19} \]

[In]

Integrate[x^3*(a + b*x^3)^5,x]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^7)/7 + a^3*b^2*x^10 + (10*a^2*b^3*x^13)/13 + (5*a*b^4*x^16)/16 + (b^5*x^19)/19

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{7} a^{4} b \,x^{7}+a^{3} b^{2} x^{10}+\frac {10}{13} a^{2} b^{3} x^{13}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{19} b^{5} x^{19}\)	\(57\)
default	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{7} a^{4} b \,x^{7}+a^{3} b^{2} x^{10}+\frac {10}{13} a^{2} b^{3} x^{13}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{19} b^{5} x^{19}\)	\(57\)
norman	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{7} a^{4} b \,x^{7}+a^{3} b^{2} x^{10}+\frac {10}{13} a^{2} b^{3} x^{13}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{19} b^{5} x^{19}\)	\(57\)
risch	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{7} a^{4} b \,x^{7}+a^{3} b^{2} x^{10}+\frac {10}{13} a^{2} b^{3} x^{13}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{19} b^{5} x^{19}\)	\(57\)
parallelrisch	\(\frac {1}{4} a^{5} x^{4}+\frac {5}{7} a^{4} b \,x^{7}+a^{3} b^{2} x^{10}+\frac {10}{13} a^{2} b^{3} x^{13}+\frac {5}{16} a \,b^{4} x^{16}+\frac {1}{19} b^{5} x^{19}\)	\(57\)

[In]

int(x^3*(b*x^3+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/4*a^5*x^4+5/7*a^4*b*x^7+a^3*b^2*x^10+10/13*a^2*b^3*x^13+5/16*a*b^4*x^16+1/19*b^5*x^19

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {1}{19} \, b^{5} x^{19} + \frac {5}{16} \, a b^{4} x^{16} + \frac {10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{4} \, a^{5} x^{4} \]

[In]

integrate(x^3*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {a^{5} x^{4}}{4} + \frac {5 a^{4} b x^{7}}{7} + a^{3} b^{2} x^{10} + \frac {10 a^{2} b^{3} x^{13}}{13} + \frac {5 a b^{4} x^{16}}{16} + \frac {b^{5} x^{19}}{19} \]

[In]

integrate(x**3*(b*x**3+a)**5,x)

[Out]

a**5*x**4/4 + 5*a**4*b*x**7/7 + a**3*b**2*x**10 + 10*a**2*b**3*x**13/13 + 5*a*b**4*x**16/16 + b**5*x**19/19

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {1}{19} \, b^{5} x^{19} + \frac {5}{16} \, a b^{4} x^{16} + \frac {10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{4} \, a^{5} x^{4} \]

[In]

integrate(x^3*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {1}{19} \, b^{5} x^{19} + \frac {5}{16} \, a b^{4} x^{16} + \frac {10}{13} \, a^{2} b^{3} x^{13} + a^{3} b^{2} x^{10} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{4} \, a^{5} x^{4} \]

[In]

integrate(x^3*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/19*b^5*x^19 + 5/16*a*b^4*x^16 + 10/13*a^2*b^3*x^13 + a^3*b^2*x^10 + 5/7*a^4*b*x^7 + 1/4*a^5*x^4

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b x^3\right )^5 \, dx=\frac {a^5\,x^4}{4}+\frac {5\,a^4\,b\,x^7}{7}+a^3\,b^2\,x^{10}+\frac {10\,a^2\,b^3\,x^{13}}{13}+\frac {5\,a\,b^4\,x^{16}}{16}+\frac {b^5\,x^{19}}{19} \]

[In]

int(x^3*(a + b*x^3)^5,x)

[Out]

(a^5*x^4)/4 + (b^5*x^19)/19 + (5*a^4*b*x^7)/7 + (5*a*b^4*x^16)/16 + a^3*b^2*x^10 + (10*a^2*b^3*x^13)/13