3.280 \(\int \frac{(a+b x^3)^5}{x^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{10}{7} a^2 b^3 x^7+\frac{5}{2} a^3 b^2 x^4+5 a^4 b x-\frac{a^5}{2 x^2}+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13} \]

[Out]

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

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Rubi [A]  time = 0.0221762, antiderivative size = 65, 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} \[ \frac{10}{7} a^2 b^3 x^7+\frac{5}{2} a^3 b^2 x^4+5 a^4 b x-\frac{a^5}{2 x^2}+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^3,x]

[Out]

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

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 \frac{\left (a+b x^3\right )^5}{x^3} \, dx &=\int \left (5 a^4 b+\frac{a^5}{x^3}+10 a^3 b^2 x^3+10 a^2 b^3 x^6+5 a b^4 x^9+b^5 x^{12}\right ) \, dx\\ &=-\frac{a^5}{2 x^2}+5 a^4 b x+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^2 b^3 x^7+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0038263, size = 65, normalized size = 1. \[ \frac{10}{7} a^2 b^3 x^7+\frac{5}{2} a^3 b^2 x^4+5 a^4 b x-\frac{a^5}{2 x^2}+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{2\,{x}^{2}}}+5\,{a}^{4}bx+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{7}}{7}}+{\frac{a{b}^{4}{x}^{10}}{2}}+{\frac{{b}^{5}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.957067, size = 74, normalized size = 1.14 \begin{align*} \frac{1}{13} \, b^{5} x^{13} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{2} \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x - \frac{a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66314, size = 138, normalized size = 2.12 \begin{align*} \frac{14 \, b^{5} x^{15} + 91 \, a b^{4} x^{12} + 260 \, a^{2} b^{3} x^{9} + 455 \, a^{3} b^{2} x^{6} + 910 \, a^{4} b x^{3} - 91 \, a^{5}}{182 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/182*(14*b^5*x^15 + 91*a*b^4*x^12 + 260*a^2*b^3*x^9 + 455*a^3*b^2*x^6 + 910*a^4*b*x^3 - 91*a^5)/x^2

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Sympy [A]  time = 0.381013, size = 61, normalized size = 0.94 \begin{align*} - \frac{a^{5}}{2 x^{2}} + 5 a^{4} b x + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{10 a^{2} b^{3} x^{7}}{7} + \frac{a b^{4} x^{10}}{2} + \frac{b^{5} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10958, size = 74, normalized size = 1.14 \begin{align*} \frac{1}{13} \, b^{5} x^{13} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{2} \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x - \frac{a^{5}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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