3.103 \(\int \frac{(a+b x^2)^8}{x^{23}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b^2 \left (a+b x^2\right )^9}{990 a^3 x^{18}}+\frac{b \left (a+b x^2\right )^9}{110 a^2 x^{20}}-\frac{\left (a+b x^2\right )^9}{22 a x^{22}} \]

[Out]

-(a + b*x^2)^9/(22*a*x^22) + (b*(a + b*x^2)^9)/(110*a^2*x^20) - (b^2*(a + b*x^2)^9)/(990*a^3*x^18)

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Rubi [A]  time = 0.0290133, 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} \[ -\frac{b^2 \left (a+b x^2\right )^9}{990 a^3 x^{18}}+\frac{b \left (a+b x^2\right )^9}{110 a^2 x^{20}}-\frac{\left (a+b x^2\right )^9}{22 a x^{22}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^23,x]

[Out]

-(a + b*x^2)^9/(22*a*x^22) + (b*(a + b*x^2)^9)/(110*a^2*x^20) - (b^2*(a + b*x^2)^9)/(990*a^3*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^2\right )^8}{x^{23}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^9}{22 a x^{22}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^2\right )}{11 a}\\ &=-\frac{\left (a+b x^2\right )^9}{22 a x^{22}}+\frac{b \left (a+b x^2\right )^9}{110 a^2 x^{20}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^2\right )}{110 a^2}\\ &=-\frac{\left (a+b x^2\right )^9}{22 a x^{22}}+\frac{b \left (a+b x^2\right )^9}{110 a^2 x^{20}}-\frac{b^2 \left (a+b x^2\right )^9}{990 a^3 x^{18}}\\ \end{align*}

Mathematica [A]  time = 0.0040393, size = 104, normalized size = 1.68 \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{7 a^5 b^3}{2 x^{16}}-\frac{5 a^4 b^4}{x^{14}}-\frac{14 a^3 b^5}{3 x^{12}}-\frac{14 a^2 b^6}{5 x^{10}}-\frac{2 a^7 b}{5 x^{20}}-\frac{a^8}{22 x^{22}}-\frac{a b^7}{x^8}-\frac{b^8}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^23,x]

[Out]

-a^8/(22*x^22) - (2*a^7*b)/(5*x^20) - (14*a^6*b^2)/(9*x^18) - (7*a^5*b^3)/(2*x^16) - (5*a^4*b^4)/x^14 - (14*a^
3*b^5)/(3*x^12) - (14*a^2*b^6)/(5*x^10) - (a*b^7)/x^8 - b^8/(6*x^6)

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Maple [A]  time = 0.005, size = 91, normalized size = 1.5 \begin{align*} -{\frac{a{b}^{7}}{{x}^{8}}}-{\frac{14\,{a}^{6}{b}^{2}}{9\,{x}^{18}}}-{\frac{14\,{a}^{2}{b}^{6}}{5\,{x}^{10}}}-{\frac{7\,{a}^{5}{b}^{3}}{2\,{x}^{16}}}-{\frac{14\,{a}^{3}{b}^{5}}{3\,{x}^{12}}}-{\frac{2\,{a}^{7}b}{5\,{x}^{20}}}-{\frac{{a}^{8}}{22\,{x}^{22}}}-{\frac{{b}^{8}}{6\,{x}^{6}}}-5\,{\frac{{a}^{4}{b}^{4}}{{x}^{14}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^23,x)

[Out]

-a*b^7/x^8-14/9*a^6*b^2/x^18-14/5*a^2*b^6/x^10-7/2*a^5*b^3/x^16-14/3*a^3*b^5/x^12-2/5*a^7*b/x^20-1/22*a^8/x^22
-1/6*b^8/x^6-5*a^4*b^4/x^14

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Maxima [A]  time = 2.03609, size = 124, normalized size = 2. \begin{align*} -\frac{165 \, b^{8} x^{16} + 990 \, a b^{7} x^{14} + 2772 \, a^{2} b^{6} x^{12} + 4620 \, a^{3} b^{5} x^{10} + 4950 \, a^{4} b^{4} x^{8} + 3465 \, a^{5} b^{3} x^{6} + 1540 \, a^{6} b^{2} x^{4} + 396 \, a^{7} b x^{2} + 45 \, a^{8}}{990 \, x^{22}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^23,x, algorithm="maxima")

[Out]

-1/990*(165*b^8*x^16 + 990*a*b^7*x^14 + 2772*a^2*b^6*x^12 + 4620*a^3*b^5*x^10 + 4950*a^4*b^4*x^8 + 3465*a^5*b^
3*x^6 + 1540*a^6*b^2*x^4 + 396*a^7*b*x^2 + 45*a^8)/x^22

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Fricas [A]  time = 1.25925, size = 225, normalized size = 3.63 \begin{align*} -\frac{165 \, b^{8} x^{16} + 990 \, a b^{7} x^{14} + 2772 \, a^{2} b^{6} x^{12} + 4620 \, a^{3} b^{5} x^{10} + 4950 \, a^{4} b^{4} x^{8} + 3465 \, a^{5} b^{3} x^{6} + 1540 \, a^{6} b^{2} x^{4} + 396 \, a^{7} b x^{2} + 45 \, a^{8}}{990 \, x^{22}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^23,x, algorithm="fricas")

[Out]

-1/990*(165*b^8*x^16 + 990*a*b^7*x^14 + 2772*a^2*b^6*x^12 + 4620*a^3*b^5*x^10 + 4950*a^4*b^4*x^8 + 3465*a^5*b^
3*x^6 + 1540*a^6*b^2*x^4 + 396*a^7*b*x^2 + 45*a^8)/x^22

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Sympy [A]  time = 1.28588, size = 99, normalized size = 1.6 \begin{align*} - \frac{45 a^{8} + 396 a^{7} b x^{2} + 1540 a^{6} b^{2} x^{4} + 3465 a^{5} b^{3} x^{6} + 4950 a^{4} b^{4} x^{8} + 4620 a^{3} b^{5} x^{10} + 2772 a^{2} b^{6} x^{12} + 990 a b^{7} x^{14} + 165 b^{8} x^{16}}{990 x^{22}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**23,x)

[Out]

-(45*a**8 + 396*a**7*b*x**2 + 1540*a**6*b**2*x**4 + 3465*a**5*b**3*x**6 + 4950*a**4*b**4*x**8 + 4620*a**3*b**5
*x**10 + 2772*a**2*b**6*x**12 + 990*a*b**7*x**14 + 165*b**8*x**16)/(990*x**22)

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Giac [A]  time = 1.86166, size = 124, normalized size = 2. \begin{align*} -\frac{165 \, b^{8} x^{16} + 990 \, a b^{7} x^{14} + 2772 \, a^{2} b^{6} x^{12} + 4620 \, a^{3} b^{5} x^{10} + 4950 \, a^{4} b^{4} x^{8} + 3465 \, a^{5} b^{3} x^{6} + 1540 \, a^{6} b^{2} x^{4} + 396 \, a^{7} b x^{2} + 45 \, a^{8}}{990 \, x^{22}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^23,x, algorithm="giac")

[Out]

-1/990*(165*b^8*x^16 + 990*a*b^7*x^14 + 2772*a^2*b^6*x^12 + 4620*a^3*b^5*x^10 + 4950*a^4*b^4*x^8 + 3465*a^5*b^
3*x^6 + 1540*a^6*b^2*x^4 + 396*a^7*b*x^2 + 45*a^8)/x^22