3.67 \(\int \frac{(a+b x^2)^5}{x^{17}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b^2 \left (a+b x^2\right )^6}{336 a^3 x^{12}}+\frac{b \left (a+b x^2\right )^6}{56 a^2 x^{14}}-\frac{\left (a+b x^2\right )^6}{16 a x^{16}} \]

[Out]

-(a + b*x^2)^6/(16*a*x^16) + (b*(a + b*x^2)^6)/(56*a^2*x^14) - (b^2*(a + b*x^2)^6)/(336*a^3*x^12)

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Rubi [A]  time = 0.0280627, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ -\frac{b^2 \left (a+b x^2\right )^6}{336 a^3 x^{12}}+\frac{b \left (a+b x^2\right )^6}{56 a^2 x^{14}}-\frac{\left (a+b x^2\right )^6}{16 a x^{16}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^17,x]

[Out]

-(a + b*x^2)^6/(16*a*x^16) + (b*(a + b*x^2)^6)/(56*a^2*x^14) - (b^2*(a + b*x^2)^6)/(336*a^3*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 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^2\right )^5}{x^{17}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^9} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^6}{16 a x^{16}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^8} \, dx,x,x^2\right )}{8 a}\\ &=-\frac{\left (a+b x^2\right )^6}{16 a x^{16}}+\frac{b \left (a+b x^2\right )^6}{56 a^2 x^{14}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^7} \, dx,x,x^2\right )}{56 a^2}\\ &=-\frac{\left (a+b x^2\right )^6}{16 a x^{16}}+\frac{b \left (a+b x^2\right )^6}{56 a^2 x^{14}}-\frac{b^2 \left (a+b x^2\right )^6}{336 a^3 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0042791, size = 67, normalized size = 1.08 \[ -\frac{5 a^3 b^2}{6 x^{12}}-\frac{a^2 b^3}{x^{10}}-\frac{5 a^4 b}{14 x^{14}}-\frac{a^5}{16 x^{16}}-\frac{5 a b^4}{8 x^8}-\frac{b^5}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^17,x]

[Out]

-a^5/(16*x^16) - (5*a^4*b)/(14*x^14) - (5*a^3*b^2)/(6*x^12) - (a^2*b^3)/x^10 - (5*a*b^4)/(8*x^8) - b^5/(6*x^6)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.9 \begin{align*} -{\frac{5\,a{b}^{4}}{8\,{x}^{8}}}-{\frac{{a}^{2}{b}^{3}}{{x}^{10}}}-{\frac{5\,{a}^{3}{b}^{2}}{6\,{x}^{12}}}-{\frac{{a}^{5}}{16\,{x}^{16}}}-{\frac{{b}^{5}}{6\,{x}^{6}}}-{\frac{5\,{a}^{4}b}{14\,{x}^{14}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^17,x)

[Out]

-5/8*a*b^4/x^8-a^2*b^3/x^10-5/6*a^3*b^2/x^12-1/16*a^5/x^16-1/6*b^5/x^6-5/14*a^4*b/x^14

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Maxima [A]  time = 1.82797, size = 80, normalized size = 1.29 \begin{align*} -\frac{56 \, b^{5} x^{10} + 210 \, a b^{4} x^{8} + 336 \, a^{2} b^{3} x^{6} + 280 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} + 21 \, a^{5}}{336 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^17,x, algorithm="maxima")

[Out]

-1/336*(56*b^5*x^10 + 210*a*b^4*x^8 + 336*a^2*b^3*x^6 + 280*a^3*b^2*x^4 + 120*a^4*b*x^2 + 21*a^5)/x^16

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Fricas [A]  time = 1.28982, size = 140, normalized size = 2.26 \begin{align*} -\frac{56 \, b^{5} x^{10} + 210 \, a b^{4} x^{8} + 336 \, a^{2} b^{3} x^{6} + 280 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} + 21 \, a^{5}}{336 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^17,x, algorithm="fricas")

[Out]

-1/336*(56*b^5*x^10 + 210*a*b^4*x^8 + 336*a^2*b^3*x^6 + 280*a^3*b^2*x^4 + 120*a^4*b*x^2 + 21*a^5)/x^16

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Sympy [A]  time = 0.687703, size = 63, normalized size = 1.02 \begin{align*} - \frac{21 a^{5} + 120 a^{4} b x^{2} + 280 a^{3} b^{2} x^{4} + 336 a^{2} b^{3} x^{6} + 210 a b^{4} x^{8} + 56 b^{5} x^{10}}{336 x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**17,x)

[Out]

-(21*a**5 + 120*a**4*b*x**2 + 280*a**3*b**2*x**4 + 336*a**2*b**3*x**6 + 210*a*b**4*x**8 + 56*b**5*x**10)/(336*
x**16)

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Giac [A]  time = 2.6719, size = 80, normalized size = 1.29 \begin{align*} -\frac{56 \, b^{5} x^{10} + 210 \, a b^{4} x^{8} + 336 \, a^{2} b^{3} x^{6} + 280 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} + 21 \, a^{5}}{336 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^17,x, algorithm="giac")

[Out]

-1/336*(56*b^5*x^10 + 210*a*b^4*x^8 + 336*a^2*b^3*x^6 + 280*a^3*b^2*x^4 + 120*a^4*b*x^2 + 21*a^5)/x^16