3.104 \(\int \frac{(a+b x^2)^8}{x^{25}} \, dx\)

Optimal. Leaf size=84 \[ \frac{b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{\left (a+b x^2\right )^9}{24 a x^{24}} \]

[Out]

-(a + b*x^2)^9/(24*a*x^24) + (b*(a + b*x^2)^9)/(88*a^2*x^22) - (b^2*(a + b*x^2)^9)/(440*a^3*x^20) + (b^3*(a +
b*x^2)^9)/(3960*a^4*x^18)

________________________________________________________________________________________

Rubi [A]  time = 0.0414562, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{\left (a+b x^2\right )^9}{24 a x^{24}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^25,x]

[Out]

-(a + b*x^2)^9/(24*a*x^24) + (b*(a + b*x^2)^9)/(88*a^2*x^22) - (b^2*(a + b*x^2)^9)/(440*a^3*x^20) + (b^3*(a +
b*x^2)^9)/(3960*a^4*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^{25}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^2\right )}{8 a}\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^2\right )}{44 a^2}\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^2\right )}{440 a^3}\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac{b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}}\\ \end{align*}

Mathematica [A]  time = 0.0040744, size = 106, normalized size = 1.26 \[ -\frac{7 a^6 b^2}{5 x^{20}}-\frac{28 a^5 b^3}{9 x^{18}}-\frac{35 a^4 b^4}{8 x^{16}}-\frac{4 a^3 b^5}{x^{14}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{4 a^7 b}{11 x^{22}}-\frac{a^8}{24 x^{24}}-\frac{4 a b^7}{5 x^{10}}-\frac{b^8}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^25,x]

[Out]

-a^8/(24*x^24) - (4*a^7*b)/(11*x^22) - (7*a^6*b^2)/(5*x^20) - (28*a^5*b^3)/(9*x^18) - (35*a^4*b^4)/(8*x^16) -
(4*a^3*b^5)/x^14 - (7*a^2*b^6)/(3*x^12) - (4*a*b^7)/(5*x^10) - b^8/(8*x^8)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 91, normalized size = 1.1 \begin{align*} -{\frac{{b}^{8}}{8\,{x}^{8}}}-{\frac{{a}^{8}}{24\,{x}^{24}}}-{\frac{35\,{a}^{4}{b}^{4}}{8\,{x}^{16}}}-{\frac{4\,a{b}^{7}}{5\,{x}^{10}}}-4\,{\frac{{a}^{3}{b}^{5}}{{x}^{14}}}-{\frac{7\,{a}^{6}{b}^{2}}{5\,{x}^{20}}}-{\frac{4\,{a}^{7}b}{11\,{x}^{22}}}-{\frac{7\,{a}^{2}{b}^{6}}{3\,{x}^{12}}}-{\frac{28\,{a}^{5}{b}^{3}}{9\,{x}^{18}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^25,x)

[Out]

-1/8*b^8/x^8-1/24*a^8/x^24-35/8*a^4*b^4/x^16-4/5*a*b^7/x^10-4*a^3*b^5/x^14-7/5*a^6*b^2/x^20-4/11*a^7*b/x^22-7/
3*a^2*b^6/x^12-28/9*a^5*b^3/x^18

________________________________________________________________________________________

Maxima [A]  time = 2.7703, size = 124, normalized size = 1.48 \begin{align*} -\frac{495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^25,x, algorithm="maxima")

[Out]

-1/3960*(495*b^8*x^16 + 3168*a*b^7*x^14 + 9240*a^2*b^6*x^12 + 15840*a^3*b^5*x^10 + 17325*a^4*b^4*x^8 + 12320*a
^5*b^3*x^6 + 5544*a^6*b^2*x^4 + 1440*a^7*b*x^2 + 165*a^8)/x^24

________________________________________________________________________________________

Fricas [A]  time = 1.23595, size = 235, normalized size = 2.8 \begin{align*} -\frac{495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^25,x, algorithm="fricas")

[Out]

-1/3960*(495*b^8*x^16 + 3168*a*b^7*x^14 + 9240*a^2*b^6*x^12 + 15840*a^3*b^5*x^10 + 17325*a^4*b^4*x^8 + 12320*a
^5*b^3*x^6 + 5544*a^6*b^2*x^4 + 1440*a^7*b*x^2 + 165*a^8)/x^24

________________________________________________________________________________________

Sympy [A]  time = 1.32326, size = 99, normalized size = 1.18 \begin{align*} - \frac{165 a^{8} + 1440 a^{7} b x^{2} + 5544 a^{6} b^{2} x^{4} + 12320 a^{5} b^{3} x^{6} + 17325 a^{4} b^{4} x^{8} + 15840 a^{3} b^{5} x^{10} + 9240 a^{2} b^{6} x^{12} + 3168 a b^{7} x^{14} + 495 b^{8} x^{16}}{3960 x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**25,x)

[Out]

-(165*a**8 + 1440*a**7*b*x**2 + 5544*a**6*b**2*x**4 + 12320*a**5*b**3*x**6 + 17325*a**4*b**4*x**8 + 15840*a**3
*b**5*x**10 + 9240*a**2*b**6*x**12 + 3168*a*b**7*x**14 + 495*b**8*x**16)/(3960*x**24)

________________________________________________________________________________________

Giac [A]  time = 2.86287, size = 124, normalized size = 1.48 \begin{align*} -\frac{495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^25,x, algorithm="giac")

[Out]

-1/3960*(495*b^8*x^16 + 3168*a*b^7*x^14 + 9240*a^2*b^6*x^12 + 15840*a^3*b^5*x^10 + 17325*a^4*b^4*x^8 + 12320*a
^5*b^3*x^6 + 5544*a^6*b^2*x^4 + 1440*a^7*b*x^2 + 165*a^8)/x^24