3.278 \(\int (a+b x^3)^5 \, dx\)

Optimal. Leaf size=61 \[ a^2 b^3 x^{10}+\frac{10}{7} a^3 b^2 x^7+\frac{5}{4} a^4 b x^4+a^5 x+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{16}}{16} \]

[Out]

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

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Rubi [A]  time = 0.0193903, antiderivative size = 61, 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} \[ a^2 b^3 x^{10}+\frac{10}{7} a^3 b^2 x^7+\frac{5}{4} a^4 b x^4+a^5 x+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.00106, size = 61, normalized size = 1. \[ a^2 b^3 x^{10}+\frac{10}{7} a^3 b^2 x^7+\frac{5}{4} a^4 b x^4+a^5 x+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5,x]

[Out]

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

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Maple [A]  time = 0., size = 54, normalized size = 0.9 \begin{align*} x{a}^{5}+{\frac{5\,{a}^{4}b{x}^{4}}{4}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{7}}{7}}+{a}^{2}{b}^{3}{x}^{10}+{\frac{5\,a{b}^{4}{x}^{13}}{13}}+{\frac{{b}^{5}{x}^{16}}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964643, size = 72, normalized size = 1.18 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{4} \, a^{4} b x^{4} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51685, size = 123, normalized size = 2.02 \begin{align*} \frac{1}{16} x^{16} b^{5} + \frac{5}{13} x^{13} b^{4} a + x^{10} b^{3} a^{2} + \frac{10}{7} x^{7} b^{2} a^{3} + \frac{5}{4} x^{4} b a^{4} + x a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.088176, size = 60, normalized size = 0.98 \begin{align*} a^{5} x + \frac{5 a^{4} b x^{4}}{4} + \frac{10 a^{3} b^{2} x^{7}}{7} + a^{2} b^{3} x^{10} + \frac{5 a b^{4} x^{13}}{13} + \frac{b^{5} x^{16}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11108, size = 72, normalized size = 1.18 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{13} \, a b^{4} x^{13} + a^{2} b^{3} x^{10} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{4} \, a^{4} b x^{4} + a^{5} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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