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

Optimal. Leaf size=99 \[ \frac{28}{19} a^2 b^6 x^{19}+\frac{7}{2} a^3 b^5 x^{16}+\frac{70}{13} a^4 b^4 x^{13}+\frac{28}{5} a^5 b^3 x^{10}+4 a^6 b^2 x^7+2 a^7 b x^4+a^8 x+\frac{4}{11} a b^7 x^{22}+\frac{b^8 x^{25}}{25} \]

[Out]

a^8*x + 2*a^7*b*x^4 + 4*a^6*b^2*x^7 + (28*a^5*b^3*x^10)/5 + (70*a^4*b^4*x^13)/13 + (7*a^3*b^5*x^16)/2 + (28*a^
2*b^6*x^19)/19 + (4*a*b^7*x^22)/11 + (b^8*x^25)/25

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Rubi [A]  time = 0.037282, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {194} \[ \frac{28}{19} a^2 b^6 x^{19}+\frac{7}{2} a^3 b^5 x^{16}+\frac{70}{13} a^4 b^4 x^{13}+\frac{28}{5} a^5 b^3 x^{10}+4 a^6 b^2 x^7+2 a^7 b x^4+a^8 x+\frac{4}{11} a b^7 x^{22}+\frac{b^8 x^{25}}{25} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8,x]

[Out]

a^8*x + 2*a^7*b*x^4 + 4*a^6*b^2*x^7 + (28*a^5*b^3*x^10)/5 + (70*a^4*b^4*x^13)/13 + (7*a^3*b^5*x^16)/2 + (28*a^
2*b^6*x^19)/19 + (4*a*b^7*x^22)/11 + (b^8*x^25)/25

Rule 194

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

Rubi steps

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

Mathematica [A]  time = 0.0014218, size = 99, normalized size = 1. \[ \frac{28}{19} a^2 b^6 x^{19}+\frac{7}{2} a^3 b^5 x^{16}+\frac{70}{13} a^4 b^4 x^{13}+\frac{28}{5} a^5 b^3 x^{10}+4 a^6 b^2 x^7+2 a^7 b x^4+a^8 x+\frac{4}{11} a b^7 x^{22}+\frac{b^8 x^{25}}{25} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8,x]

[Out]

a^8*x + 2*a^7*b*x^4 + 4*a^6*b^2*x^7 + (28*a^5*b^3*x^10)/5 + (70*a^4*b^4*x^13)/13 + (7*a^3*b^5*x^16)/2 + (28*a^
2*b^6*x^19)/19 + (4*a*b^7*x^22)/11 + (b^8*x^25)/25

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Maple [A]  time = 0.002, size = 88, normalized size = 0.9 \begin{align*}{a}^{8}x+2\,{a}^{7}b{x}^{4}+4\,{a}^{6}{b}^{2}{x}^{7}+{\frac{28\,{a}^{5}{b}^{3}{x}^{10}}{5}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{13}}{13}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{16}}{2}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{19}}{19}}+{\frac{4\,a{b}^{7}{x}^{22}}{11}}+{\frac{{b}^{8}{x}^{25}}{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^8*x+2*a^7*b*x^4+4*a^6*b^2*x^7+28/5*a^5*b^3*x^10+70/13*a^4*b^4*x^13+7/2*a^3*b^5*x^16+28/19*a^2*b^6*x^19+4/11*
a*b^7*x^22+1/25*b^8*x^25

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Maxima [A]  time = 0.952367, size = 117, normalized size = 1.18 \begin{align*} \frac{1}{25} \, b^{8} x^{25} + \frac{4}{11} \, a b^{7} x^{22} + \frac{28}{19} \, a^{2} b^{6} x^{19} + \frac{7}{2} \, a^{3} b^{5} x^{16} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 4/11*a*b^7*x^22 + 28/19*a^2*b^6*x^19 + 7/2*a^3*b^5*x^16 + 70/13*a^4*b^4*x^13 + 28/5*a^5*b^3*x^
10 + 4*a^6*b^2*x^7 + 2*a^7*b*x^4 + a^8*x

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Fricas [A]  time = 1.64422, size = 205, normalized size = 2.07 \begin{align*} \frac{1}{25} x^{25} b^{8} + \frac{4}{11} x^{22} b^{7} a + \frac{28}{19} x^{19} b^{6} a^{2} + \frac{7}{2} x^{16} b^{5} a^{3} + \frac{70}{13} x^{13} b^{4} a^{4} + \frac{28}{5} x^{10} b^{3} a^{5} + 4 x^{7} b^{2} a^{6} + 2 x^{4} b a^{7} + x a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*x^25*b^8 + 4/11*x^22*b^7*a + 28/19*x^19*b^6*a^2 + 7/2*x^16*b^5*a^3 + 70/13*x^13*b^4*a^4 + 28/5*x^10*b^3*a
^5 + 4*x^7*b^2*a^6 + 2*x^4*b*a^7 + x*a^8

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Sympy [A]  time = 0.097392, size = 100, normalized size = 1.01 \begin{align*} a^{8} x + 2 a^{7} b x^{4} + 4 a^{6} b^{2} x^{7} + \frac{28 a^{5} b^{3} x^{10}}{5} + \frac{70 a^{4} b^{4} x^{13}}{13} + \frac{7 a^{3} b^{5} x^{16}}{2} + \frac{28 a^{2} b^{6} x^{19}}{19} + \frac{4 a b^{7} x^{22}}{11} + \frac{b^{8} x^{25}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x + 2*a**7*b*x**4 + 4*a**6*b**2*x**7 + 28*a**5*b**3*x**10/5 + 70*a**4*b**4*x**13/13 + 7*a**3*b**5*x**16/2
+ 28*a**2*b**6*x**19/19 + 4*a*b**7*x**22/11 + b**8*x**25/25

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Giac [A]  time = 1.09711, size = 117, normalized size = 1.18 \begin{align*} \frac{1}{25} \, b^{8} x^{25} + \frac{4}{11} \, a b^{7} x^{22} + \frac{28}{19} \, a^{2} b^{6} x^{19} + \frac{7}{2} \, a^{3} b^{5} x^{16} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{28}{5} \, a^{5} b^{3} x^{10} + 4 \, a^{6} b^{2} x^{7} + 2 \, a^{7} b x^{4} + a^{8} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*b^8*x^25 + 4/11*a*b^7*x^22 + 28/19*a^2*b^6*x^19 + 7/2*a^3*b^5*x^16 + 70/13*a^4*b^4*x^13 + 28/5*a^5*b^3*x^
10 + 4*a^6*b^2*x^7 + 2*a^7*b*x^4 + a^8*x