3.271 \(\int \frac{(a+b x^3)^5}{x^{22}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^3\right )^6}{126 a^2 x^{18}}-\frac{\left (a+b x^3\right )^6}{21 a x^{21}} \]

[Out]

-(a + b*x^3)^6/(21*a*x^21) + (b*(a + b*x^3)^6)/(126*a^2*x^18)

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Rubi [A]  time = 0.0168073, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b \left (a+b x^3\right )^6}{126 a^2 x^{18}}-\frac{\left (a+b x^3\right )^6}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^22,x]

[Out]

-(a + b*x^3)^6/(21*a*x^21) + (b*(a + b*x^3)^6)/(126*a^2*x^18)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5}{x^{22}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^8} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^6}{21 a x^{21}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^7} \, dx,x,x^3\right )}{21 a}\\ &=-\frac{\left (a+b x^3\right )^6}{21 a x^{21}}+\frac{b \left (a+b x^3\right )^6}{126 a^2 x^{18}}\\ \end{align*}

Mathematica [A]  time = 0.0040881, size = 69, normalized size = 1.72 \[ -\frac{2 a^3 b^2}{3 x^{15}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{5 a^4 b}{18 x^{18}}-\frac{a^5}{21 x^{21}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^22,x]

[Out]

-a^5/(21*x^21) - (5*a^4*b)/(18*x^18) - (2*a^3*b^2)/(3*x^15) - (5*a^2*b^3)/(6*x^12) - (5*a*b^4)/(9*x^9) - b^5/(
6*x^6)

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Maple [A]  time = 0.005, size = 58, normalized size = 1.5 \begin{align*} -{\frac{2\,{a}^{3}{b}^{2}}{3\,{x}^{15}}}-{\frac{5\,{a}^{4}b}{18\,{x}^{18}}}-{\frac{{b}^{5}}{6\,{x}^{6}}}-{\frac{5\,a{b}^{4}}{9\,{x}^{9}}}-{\frac{5\,{a}^{2}{b}^{3}}{6\,{x}^{12}}}-{\frac{{a}^{5}}{21\,{x}^{21}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^22,x)

[Out]

-2/3*a^3*b^2/x^15-5/18*a^4*b/x^18-1/6*b^5/x^6-5/9*a*b^4/x^9-5/6*a^2*b^3/x^12-1/21*a^5/x^21

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Maxima [A]  time = 0.987401, size = 80, normalized size = 2. \begin{align*} -\frac{21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^22,x, algorithm="maxima")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*b*x^3 + 6*a^5)/x^21

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Fricas [A]  time = 1.57436, size = 136, normalized size = 3.4 \begin{align*} -\frac{21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^22,x, algorithm="fricas")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*b*x^3 + 6*a^5)/x^21

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Sympy [A]  time = 0.769148, size = 63, normalized size = 1.58 \begin{align*} - \frac{6 a^{5} + 35 a^{4} b x^{3} + 84 a^{3} b^{2} x^{6} + 105 a^{2} b^{3} x^{9} + 70 a b^{4} x^{12} + 21 b^{5} x^{15}}{126 x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**22,x)

[Out]

-(6*a**5 + 35*a**4*b*x**3 + 84*a**3*b**2*x**6 + 105*a**2*b**3*x**9 + 70*a*b**4*x**12 + 21*b**5*x**15)/(126*x**
21)

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Giac [A]  time = 1.1118, size = 80, normalized size = 2. \begin{align*} -\frac{21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^22,x, algorithm="giac")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*b*x^3 + 6*a^5)/x^21