3.309 \(\int x^3 (a+b x^3)^8 \, dx\)

Optimal. Leaf size=108 \[ \frac{14}{11} a^2 b^6 x^{22}+\frac{56}{19} a^3 b^5 x^{19}+\frac{35}{8} a^4 b^4 x^{16}+\frac{56}{13} a^5 b^3 x^{13}+\frac{14}{5} a^6 b^2 x^{10}+\frac{8}{7} a^7 b x^7+\frac{a^8 x^4}{4}+\frac{8}{25} a b^7 x^{25}+\frac{b^8 x^{28}}{28} \]

[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

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Rubi [A]  time = 0.0388762, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{14}{11} a^2 b^6 x^{22}+\frac{56}{19} a^3 b^5 x^{19}+\frac{35}{8} a^4 b^4 x^{16}+\frac{56}{13} a^5 b^3 x^{13}+\frac{14}{5} a^6 b^2 x^{10}+\frac{8}{7} a^7 b x^7+\frac{a^8 x^4}{4}+\frac{8}{25} a b^7 x^{25}+\frac{b^8 x^{28}}{28} \]

Antiderivative was successfully verified.

[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 270

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*} \int x^3 \left (a+b x^3\right )^8 \, dx &=\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]  time = 0.0027506, size = 108, normalized size = 1. \[ \frac{14}{11} a^2 b^6 x^{22}+\frac{56}{19} a^3 b^5 x^{19}+\frac{35}{8} a^4 b^4 x^{16}+\frac{56}{13} a^5 b^3 x^{13}+\frac{14}{5} a^6 b^2 x^{10}+\frac{8}{7} a^7 b x^7+\frac{a^8 x^4}{4}+\frac{8}{25} a b^7 x^{25}+\frac{b^8 x^{28}}{28} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.001, size = 91, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 0.954884, size = 122, normalized size = 1.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.66038, size = 224, normalized size = 2.07 \begin{align*} \frac{1}{28} x^{28} b^{8} + \frac{8}{25} x^{25} b^{7} a + \frac{14}{11} x^{22} b^{6} a^{2} + \frac{56}{19} x^{19} b^{5} a^{3} + \frac{35}{8} x^{16} b^{4} a^{4} + \frac{56}{13} x^{13} b^{3} a^{5} + \frac{14}{5} x^{10} b^{2} a^{6} + \frac{8}{7} x^{7} b a^{7} + \frac{1}{4} x^{4} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.11256, size = 107, normalized size = 0.99 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.11035, size = 122, normalized size = 1.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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