3.422 \(\int \frac {(a+b x^2)^{9/2}}{x^{11}} \, dx\)

Optimal. Leaf size=131 \[ -\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}}-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8} \]

[Out]

-21/128*b^3*(b*x^2+a)^(3/2)/x^4-21/160*b^2*(b*x^2+a)^(5/2)/x^6-9/80*b*(b*x^2+a)^(7/2)/x^8-1/10*(b*x^2+a)^(9/2)
/x^10-63/256*b^5*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)-63/256*b^4*(b*x^2+a)^(1/2)/x^2

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Rubi [A]  time = 0.08, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^11,x]

[Out]

(-63*b^4*Sqrt[a + b*x^2])/(256*x^2) - (21*b^3*(a + b*x^2)^(3/2))/(128*x^4) - (21*b^2*(a + b*x^2)^(5/2))/(160*x
^6) - (9*b*(a + b*x^2)^(7/2))/(80*x^8) - (a + b*x^2)^(9/2)/(10*x^10) - (63*b^5*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]
])/(256*Sqrt[a])

Rule 47

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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 )^{9/2}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{9/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{20} (9 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{7/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{160} \left (63 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{64} \left (21 b^3\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{256} \left (63 b^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{512} \left (63 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{256} \left (63 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 109, normalized size = 0.83 \[ -\frac {128 a^5+784 a^4 b x^2+2024 a^3 b^2 x^4+2858 a^2 b^3 x^6+315 b^5 x^{10} \sqrt {\frac {b x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )+2455 a b^4 x^8+965 b^5 x^{10}}{1280 x^{10} \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^11,x]

[Out]

-1/1280*(128*a^5 + 784*a^4*b*x^2 + 2024*a^3*b^2*x^4 + 2858*a^2*b^3*x^6 + 2455*a*b^4*x^8 + 965*b^5*x^10 + 315*b
^5*x^10*Sqrt[1 + (b*x^2)/a]*ArcTanh[Sqrt[1 + (b*x^2)/a]])/(x^10*Sqrt[a + b*x^2])

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fricas [A]  time = 1.05, size = 202, normalized size = 1.54 \[ \left [\frac {315 \, \sqrt {a} b^{5} x^{10} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{2560 \, a x^{10}}, \frac {315 \, \sqrt {-a} b^{5} x^{10} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{1280 \, a x^{10}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^11,x, algorithm="fricas")

[Out]

[1/2560*(315*sqrt(a)*b^5*x^10*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(965*a*b^4*x^8 + 1490*a^
2*b^3*x^6 + 1368*a^3*b^2*x^4 + 656*a^4*b*x^2 + 128*a^5)*sqrt(b*x^2 + a))/(a*x^10), 1/1280*(315*sqrt(-a)*b^5*x^
10*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (965*a*b^4*x^8 + 1490*a^2*b^3*x^6 + 1368*a^3*b^2*x^4 + 656*a^4*b*x^2 + 1
28*a^5)*sqrt(b*x^2 + a))/(a*x^10)]

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giac [A]  time = 1.09, size = 121, normalized size = 0.92 \[ \frac {\frac {315 \, b^{6} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {965 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{6} - 2370 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b^{6} + 2688 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b^{6} - 1470 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{6} + 315 \, \sqrt {b x^{2} + a} a^{4} b^{6}}{b^{5} x^{10}}}{1280 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^11,x, algorithm="giac")

[Out]

1/1280*(315*b^6*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) - (965*(b*x^2 + a)^(9/2)*b^6 - 2370*(b*x^2 + a)^(7/2
)*a*b^6 + 2688*(b*x^2 + a)^(5/2)*a^2*b^6 - 1470*(b*x^2 + a)^(3/2)*a^3*b^6 + 315*sqrt(b*x^2 + a)*a^4*b^6)/(b^5*
x^10))/b

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maple [B]  time = 0.05, size = 213, normalized size = 1.63 \[ -\frac {63 b^{5} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{256 \sqrt {a}}+\frac {63 \sqrt {b \,x^{2}+a}\, b^{5}}{256 a}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}{256 a^{2}}+\frac {63 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{5}}{1280 a^{3}}+\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}{256 a^{4}}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{5}}{256 a^{5}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{4}}{256 a^{5} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{3}}{128 a^{4} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{2}}{160 a^{3} x^{6}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{80 a^{2} x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{10 a \,x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^11,x)

[Out]

-1/10/a/x^10*(b*x^2+a)^(11/2)-1/80/a^2*b/x^8*(b*x^2+a)^(11/2)-1/160/a^3*b^2/x^6*(b*x^2+a)^(11/2)-1/128/a^4*b^3
/x^4*(b*x^2+a)^(11/2)-7/256/a^5*b^4/x^2*(b*x^2+a)^(11/2)+7/256/a^5*b^5*(b*x^2+a)^(9/2)+9/256/a^4*b^5*(b*x^2+a)
^(7/2)+63/1280/a^3*b^5*(b*x^2+a)^(5/2)+21/256/a^2*b^5*(b*x^2+a)^(3/2)-63/256/a^(1/2)*b^5*ln((2*a+2*(b*x^2+a)^(
1/2)*a^(1/2))/x)+63/256/a*b^5*(b*x^2+a)^(1/2)

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maxima [A]  time = 1.48, size = 201, normalized size = 1.53 \[ -\frac {63 \, b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, \sqrt {a}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{5}}{256 \, a^{5}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}}{256 \, a^{4}} + \frac {63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}}{1280 \, a^{3}} + \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}}{256 \, a^{2}} + \frac {63 \, \sqrt {b x^{2} + a} b^{5}}{256 \, a} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{256 \, a^{5} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{128 \, a^{4} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{160 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{80 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{10 \, a x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^11,x, algorithm="maxima")

[Out]

-63/256*b^5*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 7/256*(b*x^2 + a)^(9/2)*b^5/a^5 + 9/256*(b*x^2 + a)^(7/2)*
b^5/a^4 + 63/1280*(b*x^2 + a)^(5/2)*b^5/a^3 + 21/256*(b*x^2 + a)^(3/2)*b^5/a^2 + 63/256*sqrt(b*x^2 + a)*b^5/a
- 7/256*(b*x^2 + a)^(11/2)*b^4/(a^5*x^2) - 1/128*(b*x^2 + a)^(11/2)*b^3/(a^4*x^4) - 1/160*(b*x^2 + a)^(11/2)*b
^2/(a^3*x^6) - 1/80*(b*x^2 + a)^(11/2)*b/(a^2*x^8) - 1/10*(b*x^2 + a)^(11/2)/(a*x^10)

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mupad [B]  time = 6.62, size = 106, normalized size = 0.81 \[ \frac {237\,a\,{\left (b\,x^2+a\right )}^{7/2}}{128\,x^{10}}-\frac {193\,{\left (b\,x^2+a\right )}^{9/2}}{256\,x^{10}}-\frac {63\,a^4\,\sqrt {b\,x^2+a}}{256\,x^{10}}+\frac {147\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^{10}}-\frac {21\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}{10\,x^{10}}+\frac {b^5\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{256\,\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^11,x)

[Out]

(b^5*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*63i)/(256*a^(1/2)) - (193*(a + b*x^2)^(9/2))/(256*x^10) + (237*a*(a
+ b*x^2)^(7/2))/(128*x^10) - (63*a^4*(a + b*x^2)^(1/2))/(256*x^10) + (147*a^3*(a + b*x^2)^(3/2))/(128*x^10) -
(21*a^2*(a + b*x^2)^(5/2))/(10*x^10)

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sympy [A]  time = 9.31, size = 153, normalized size = 1.17 \[ - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{10 x^{9}} - \frac {41 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{80 x^{7}} - \frac {171 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{160 x^{5}} - \frac {149 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{128 x^{3}} - \frac {193 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{256 x} - \frac {63 b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{256 \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**11,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(10*x**9) - 41*a**3*b**(3/2)*sqrt(a/(b*x**2) + 1)/(80*x**7) - 171*a**2*b**(
5/2)*sqrt(a/(b*x**2) + 1)/(160*x**5) - 149*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/(128*x**3) - 193*b**(9/2)*sqrt(a/(b
*x**2) + 1)/(256*x) - 63*b**5*asinh(sqrt(a)/(sqrt(b)*x))/(256*sqrt(a))

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