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]

-1/24*(b*x^2+a)^9/a/x^24+1/88*b*(b*x^2+a)^9/a^2/x^22-1/440*b^2*(b*x^2+a)^9/a^3/x^20+1/3960*b^3*(b*x^2+a)^9/a^4
/x^18

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Rubi [A]  time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 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]

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 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]]

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*}

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Mathematica [A]  time = 0.00, size = 106, normalized size = 1.26 \[ -\frac {a^8}{24 x^{24}}-\frac {4 a^7 b}{11 x^{22}}-\frac {7 a^6 b^2}{5 x^{20}}-\frac {28 a^5 b^3}{9 x^{18}}-\frac {35 a^4 b^4}{8 x^{16}}-\frac {4 a^3 b^5}{x^{14}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {4 a b^7}{5 x^{10}}-\frac {b^8}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^25,x]

[Out]

-1/24*a^8/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)

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fricas [A]  time = 0.88, size = 92, normalized size = 1.10 \[ -\frac {495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \]

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

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giac [A]  time = 1.20, size = 92, normalized size = 1.10 \[ -\frac {495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \]

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

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maple [A]  time = 0.01, size = 91, normalized size = 1.08 \[ -\frac {b^{8}}{8 x^{8}}-\frac {4 a \,b^{7}}{5 x^{10}}-\frac {7 a^{2} b^{6}}{3 x^{12}}-\frac {4 a^{3} b^{5}}{x^{14}}-\frac {35 a^{4} b^{4}}{8 x^{16}}-\frac {28 a^{5} b^{3}}{9 x^{18}}-\frac {7 a^{6} b^{2}}{5 x^{20}}-\frac {4 a^{7} b}{11 x^{22}}-\frac {a^{8}}{24 x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 92, normalized size = 1.10 \[ -\frac {495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \]

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

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mupad [B]  time = 4.97, size = 92, normalized size = 1.10 \[ -\frac {\frac {a^8}{24}+\frac {4\,a^7\,b\,x^2}{11}+\frac {7\,a^6\,b^2\,x^4}{5}+\frac {28\,a^5\,b^3\,x^6}{9}+\frac {35\,a^4\,b^4\,x^8}{8}+4\,a^3\,b^5\,x^{10}+\frac {7\,a^2\,b^6\,x^{12}}{3}+\frac {4\,a\,b^7\,x^{14}}{5}+\frac {b^8\,x^{16}}{8}}{x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^25,x)

[Out]

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

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sympy [A]  time = 1.08, size = 99, normalized size = 1.18 \[ \frac {- 165 a^{8} - 1440 a^{7} b x^{2} - 5544 a^{6} b^{2} x^{4} - 12320 a^{5} b^{3} x^{6} - 17325 a^{4} b^{4} x^{8} - 15840 a^{3} b^{5} x^{10} - 9240 a^{2} b^{6} x^{12} - 3168 a b^{7} x^{14} - 495 b^{8} x^{16}}{3960 x^{24}} \]

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)

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