3.273 \(\int \frac{(a+b x^3)^5}{x^{28}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{10 a^3 b^2}{21 x^{21}}-\frac{5 a^2 b^3}{9 x^{18}}-\frac{5 a^4 b}{24 x^{24}}-\frac{a^5}{27 x^{27}}-\frac{a b^4}{3 x^{15}}-\frac{b^5}{12 x^{12}} \]

[Out]

-a^5/(27*x^27) - (5*a^4*b)/(24*x^24) - (10*a^3*b^2)/(21*x^21) - (5*a^2*b^3)/(9*x^18) - (a*b^4)/(3*x^15) - b^5/
(12*x^12)

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Rubi [A]  time = 0.0310307, 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} \[ -\frac{10 a^3 b^2}{21 x^{21}}-\frac{5 a^2 b^3}{9 x^{18}}-\frac{5 a^4 b}{24 x^{24}}-\frac{a^5}{27 x^{27}}-\frac{a b^4}{3 x^{15}}-\frac{b^5}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^28,x]

[Out]

-a^5/(27*x^27) - (5*a^4*b)/(24*x^24) - (10*a^3*b^2)/(21*x^21) - (5*a^2*b^3)/(9*x^18) - (a*b^4)/(3*x^15) - b^5/
(12*x^12)

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^{28}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{10}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{10}}+\frac{5 a^4 b}{x^9}+\frac{10 a^3 b^2}{x^8}+\frac{10 a^2 b^3}{x^7}+\frac{5 a b^4}{x^6}+\frac{b^5}{x^5}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5}{27 x^{27}}-\frac{5 a^4 b}{24 x^{24}}-\frac{10 a^3 b^2}{21 x^{21}}-\frac{5 a^2 b^3}{9 x^{18}}-\frac{a b^4}{3 x^{15}}-\frac{b^5}{12 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0041865, size = 69, normalized size = 1. \[ -\frac{10 a^3 b^2}{21 x^{21}}-\frac{5 a^2 b^3}{9 x^{18}}-\frac{5 a^4 b}{24 x^{24}}-\frac{a^5}{27 x^{27}}-\frac{a b^4}{3 x^{15}}-\frac{b^5}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^28,x]

[Out]

-a^5/(27*x^27) - (5*a^4*b)/(24*x^24) - (10*a^3*b^2)/(21*x^21) - (5*a^2*b^3)/(9*x^18) - (a*b^4)/(3*x^15) - b^5/
(12*x^12)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{27\,{x}^{27}}}-{\frac{5\,{a}^{4}b}{24\,{x}^{24}}}-{\frac{10\,{a}^{3}{b}^{2}}{21\,{x}^{21}}}-{\frac{5\,{a}^{2}{b}^{3}}{9\,{x}^{18}}}-{\frac{a{b}^{4}}{3\,{x}^{15}}}-{\frac{{b}^{5}}{12\,{x}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^28,x)

[Out]

-1/27*a^5/x^27-5/24*a^4*b/x^24-10/21*a^3*b^2/x^21-5/9*a^2*b^3/x^18-1/3*a*b^4/x^15-1/12*b^5/x^12

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Maxima [A]  time = 0.989007, size = 80, normalized size = 1.16 \begin{align*} -\frac{126 \, b^{5} x^{15} + 504 \, a b^{4} x^{12} + 840 \, a^{2} b^{3} x^{9} + 720 \, a^{3} b^{2} x^{6} + 315 \, a^{4} b x^{3} + 56 \, a^{5}}{1512 \, x^{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^28,x, algorithm="maxima")

[Out]

-1/1512*(126*b^5*x^15 + 504*a*b^4*x^12 + 840*a^2*b^3*x^9 + 720*a^3*b^2*x^6 + 315*a^4*b*x^3 + 56*a^5)/x^27

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Fricas [A]  time = 1.57762, size = 144, normalized size = 2.09 \begin{align*} -\frac{126 \, b^{5} x^{15} + 504 \, a b^{4} x^{12} + 840 \, a^{2} b^{3} x^{9} + 720 \, a^{3} b^{2} x^{6} + 315 \, a^{4} b x^{3} + 56 \, a^{5}}{1512 \, x^{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^28,x, algorithm="fricas")

[Out]

-1/1512*(126*b^5*x^15 + 504*a*b^4*x^12 + 840*a^2*b^3*x^9 + 720*a^3*b^2*x^6 + 315*a^4*b*x^3 + 56*a^5)/x^27

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Sympy [A]  time = 1.28932, size = 63, normalized size = 0.91 \begin{align*} - \frac{56 a^{5} + 315 a^{4} b x^{3} + 720 a^{3} b^{2} x^{6} + 840 a^{2} b^{3} x^{9} + 504 a b^{4} x^{12} + 126 b^{5} x^{15}}{1512 x^{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**28,x)

[Out]

-(56*a**5 + 315*a**4*b*x**3 + 720*a**3*b**2*x**6 + 840*a**2*b**3*x**9 + 504*a*b**4*x**12 + 126*b**5*x**15)/(15
12*x**27)

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Giac [A]  time = 1.11498, size = 80, normalized size = 1.16 \begin{align*} -\frac{126 \, b^{5} x^{15} + 504 \, a b^{4} x^{12} + 840 \, a^{2} b^{3} x^{9} + 720 \, a^{3} b^{2} x^{6} + 315 \, a^{4} b x^{3} + 56 \, a^{5}}{1512 \, x^{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^28,x, algorithm="giac")

[Out]

-1/1512*(126*b^5*x^15 + 504*a*b^4*x^12 + 840*a^2*b^3*x^9 + 720*a^3*b^2*x^6 + 315*a^4*b*x^3 + 56*a^5)/x^27