3.274 \(\int \frac{(a+b x^3)^5}{x^{31}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{5 a^3 b^2}{12 x^{24}}-\frac{10 a^2 b^3}{21 x^{21}}-\frac{5 a^4 b}{27 x^{27}}-\frac{a^5}{30 x^{30}}-\frac{5 a b^4}{18 x^{18}}-\frac{b^5}{15 x^{15}} \]

[Out]

-a^5/(30*x^30) - (5*a^4*b)/(27*x^27) - (5*a^3*b^2)/(12*x^24) - (10*a^2*b^3)/(21*x^21) - (5*a*b^4)/(18*x^18) -
b^5/(15*x^15)

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Rubi [A]  time = 0.0299066, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{5 a^3 b^2}{12 x^{24}}-\frac{10 a^2 b^3}{21 x^{21}}-\frac{5 a^4 b}{27 x^{27}}-\frac{a^5}{30 x^{30}}-\frac{5 a b^4}{18 x^{18}}-\frac{b^5}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^31,x]

[Out]

-a^5/(30*x^30) - (5*a^4*b)/(27*x^27) - (5*a^3*b^2)/(12*x^24) - (10*a^2*b^3)/(21*x^21) - (5*a*b^4)/(18*x^18) -
b^5/(15*x^15)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5}{x^{31}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{11}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{11}}+\frac{5 a^4 b}{x^{10}}+\frac{10 a^3 b^2}{x^9}+\frac{10 a^2 b^3}{x^8}+\frac{5 a b^4}{x^7}+\frac{b^5}{x^6}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5}{30 x^{30}}-\frac{5 a^4 b}{27 x^{27}}-\frac{5 a^3 b^2}{12 x^{24}}-\frac{10 a^2 b^3}{21 x^{21}}-\frac{5 a b^4}{18 x^{18}}-\frac{b^5}{15 x^{15}}\\ \end{align*}

Mathematica [A]  time = 0.0041028, size = 69, normalized size = 1. \[ -\frac{5 a^3 b^2}{12 x^{24}}-\frac{10 a^2 b^3}{21 x^{21}}-\frac{5 a^4 b}{27 x^{27}}-\frac{a^5}{30 x^{30}}-\frac{5 a b^4}{18 x^{18}}-\frac{b^5}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^31,x]

[Out]

-a^5/(30*x^30) - (5*a^4*b)/(27*x^27) - (5*a^3*b^2)/(12*x^24) - (10*a^2*b^3)/(21*x^21) - (5*a*b^4)/(18*x^18) -
b^5/(15*x^15)

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Maple [A]  time = 0.007, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{30\,{x}^{30}}}-{\frac{5\,{a}^{4}b}{27\,{x}^{27}}}-{\frac{5\,{a}^{3}{b}^{2}}{12\,{x}^{24}}}-{\frac{10\,{a}^{2}{b}^{3}}{21\,{x}^{21}}}-{\frac{5\,a{b}^{4}}{18\,{x}^{18}}}-{\frac{{b}^{5}}{15\,{x}^{15}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^31,x)

[Out]

-1/30*a^5/x^30-5/27*a^4*b/x^27-5/12*a^3*b^2/x^24-10/21*a^2*b^3/x^21-5/18*a*b^4/x^18-1/15*b^5/x^15

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Maxima [A]  time = 0.993416, size = 80, normalized size = 1.16 \begin{align*} -\frac{252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^31,x, algorithm="maxima")

[Out]

-1/3780*(252*b^5*x^15 + 1050*a*b^4*x^12 + 1800*a^2*b^3*x^9 + 1575*a^3*b^2*x^6 + 700*a^4*b*x^3 + 126*a^5)/x^30

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Fricas [A]  time = 1.66605, size = 150, normalized size = 2.17 \begin{align*} -\frac{252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^31,x, algorithm="fricas")

[Out]

-1/3780*(252*b^5*x^15 + 1050*a*b^4*x^12 + 1800*a^2*b^3*x^9 + 1575*a^3*b^2*x^6 + 700*a^4*b*x^3 + 126*a^5)/x^30

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Sympy [A]  time = 1.29798, size = 63, normalized size = 0.91 \begin{align*} - \frac{126 a^{5} + 700 a^{4} b x^{3} + 1575 a^{3} b^{2} x^{6} + 1800 a^{2} b^{3} x^{9} + 1050 a b^{4} x^{12} + 252 b^{5} x^{15}}{3780 x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**31,x)

[Out]

-(126*a**5 + 700*a**4*b*x**3 + 1575*a**3*b**2*x**6 + 1800*a**2*b**3*x**9 + 1050*a*b**4*x**12 + 252*b**5*x**15)
/(3780*x**30)

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Giac [A]  time = 1.11883, size = 80, normalized size = 1.16 \begin{align*} -\frac{252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^31,x, algorithm="giac")

[Out]

-1/3780*(252*b^5*x^15 + 1050*a*b^4*x^12 + 1800*a^2*b^3*x^9 + 1575*a^3*b^2*x^6 + 700*a^4*b*x^3 + 126*a^5)/x^30