3.423 \(\int \frac {(a+b x^2)^{9/2}}{x^{13}} \, dx\)

Optimal. Leaf size=155 \[ \frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{1024 a^{3/2}}-\frac {21 b^5 \sqrt {a+b x^2}}{1024 a x^2}-\frac {21 b^4 \sqrt {a+b x^2}}{512 x^4}-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}} \]

[Out]

-7/128*b^3*(b*x^2+a)^(3/2)/x^6-21/320*b^2*(b*x^2+a)^(5/2)/x^8-3/40*b*(b*x^2+a)^(7/2)/x^10-1/12*(b*x^2+a)^(9/2)
/x^12+21/1024*b^6*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)-21/512*b^4*(b*x^2+a)^(1/2)/x^4-21/1024*b^5*(b*x^2+a
)^(1/2)/a/x^2

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Rubi [A]  time = 0.10, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ \frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{1024 a^{3/2}}-\frac {21 b^5 \sqrt {a+b x^2}}{1024 a x^2}-\frac {21 b^4 \sqrt {a+b x^2}}{512 x^4}-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^13,x]

[Out]

(-21*b^4*Sqrt[a + b*x^2])/(512*x^4) - (21*b^5*Sqrt[a + b*x^2])/(1024*a*x^2) - (7*b^3*(a + b*x^2)^(3/2))/(128*x
^6) - (21*b^2*(a + b*x^2)^(5/2))/(320*x^8) - (3*b*(a + b*x^2)^(7/2))/(40*x^10) - (a + b*x^2)^(9/2)/(12*x^12) +
(21*b^6*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(1024*a^(3/2))

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 51

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*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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^{13}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{9/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{7/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}+\frac {1}{80} \left (21 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}+\frac {1}{128} \left (21 b^3\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}+\frac {1}{256} \left (21 b^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {21 b^4 \sqrt {a+b x^2}}{512 x^4}-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}+\frac {\left (21 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{1024}\\ &=-\frac {21 b^4 \sqrt {a+b x^2}}{512 x^4}-\frac {21 b^5 \sqrt {a+b x^2}}{1024 a x^2}-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}-\frac {\left (21 b^6\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2048 a}\\ &=-\frac {21 b^4 \sqrt {a+b x^2}}{512 x^4}-\frac {21 b^5 \sqrt {a+b x^2}}{1024 a x^2}-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}-\frac {\left (21 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{1024 a}\\ &=-\frac {21 b^4 \sqrt {a+b x^2}}{512 x^4}-\frac {21 b^5 \sqrt {a+b x^2}}{1024 a x^2}-\frac {7 b^3 \left (a+b x^2\right )^{3/2}}{128 x^6}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{320 x^8}-\frac {3 b \left (a+b x^2\right )^{7/2}}{40 x^{10}}-\frac {\left (a+b x^2\right )^{9/2}}{12 x^{12}}+\frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{1024 a^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 39, normalized size = 0.25 \[ -\frac {b^6 \left (a+b x^2\right )^{11/2} \, _2F_1\left (\frac {11}{2},7;\frac {13}{2};\frac {b x^2}{a}+1\right )}{11 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^13,x]

[Out]

-1/11*(b^6*(a + b*x^2)^(11/2)*Hypergeometric2F1[11/2, 7, 13/2, 1 + (b*x^2)/a])/a^7

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fricas [A]  time = 1.08, size = 223, normalized size = 1.44 \[ \left [\frac {315 \, \sqrt {a} b^{6} x^{12} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (315 \, a b^{5} x^{10} + 4910 \, a^{2} b^{4} x^{8} + 11432 \, a^{3} b^{3} x^{6} + 12144 \, a^{4} b^{2} x^{4} + 6272 \, a^{5} b x^{2} + 1280 \, a^{6}\right )} \sqrt {b x^{2} + a}}{30720 \, a^{2} x^{12}}, -\frac {315 \, \sqrt {-a} b^{6} x^{12} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (315 \, a b^{5} x^{10} + 4910 \, a^{2} b^{4} x^{8} + 11432 \, a^{3} b^{3} x^{6} + 12144 \, a^{4} b^{2} x^{4} + 6272 \, a^{5} b x^{2} + 1280 \, a^{6}\right )} \sqrt {b x^{2} + a}}{15360 \, a^{2} x^{12}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^13,x, algorithm="fricas")

[Out]

[1/30720*(315*sqrt(a)*b^6*x^12*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(315*a*b^5*x^10 + 4910*
a^2*b^4*x^8 + 11432*a^3*b^3*x^6 + 12144*a^4*b^2*x^4 + 6272*a^5*b*x^2 + 1280*a^6)*sqrt(b*x^2 + a))/(a^2*x^12),
-1/15360*(315*sqrt(-a)*b^6*x^12*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (315*a*b^5*x^10 + 4910*a^2*b^4*x^8 + 11432*
a^3*b^3*x^6 + 12144*a^4*b^2*x^4 + 6272*a^5*b*x^2 + 1280*a^6)*sqrt(b*x^2 + a))/(a^2*x^12)]

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giac [A]  time = 1.10, size = 143, normalized size = 0.92 \[ -\frac {\frac {315 \, b^{7} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{7} + 3335 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a b^{7} - 5058 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} b^{7} + 4158 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} b^{7} - 1785 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} b^{7} + 315 \, \sqrt {b x^{2} + a} a^{5} b^{7}}{a b^{6} x^{12}}}{15360 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^13,x, algorithm="giac")

[Out]

-1/15360*(315*b^7*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + (315*(b*x^2 + a)^(11/2)*b^7 + 3335*(b*x^2 +
a)^(9/2)*a*b^7 - 5058*(b*x^2 + a)^(7/2)*a^2*b^7 + 4158*(b*x^2 + a)^(5/2)*a^3*b^7 - 1785*(b*x^2 + a)^(3/2)*a^4*
b^7 + 315*sqrt(b*x^2 + a)*a^5*b^7)/(a*b^6*x^12))/b

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maple [A]  time = 0.13, size = 233, normalized size = 1.50 \[ \frac {21 b^{6} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{1024 a^{\frac {3}{2}}}-\frac {21 \sqrt {b \,x^{2}+a}\, b^{6}}{1024 a^{2}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{6}}{1024 a^{3}}-\frac {21 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{6}}{5120 a^{4}}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{6}}{1024 a^{5}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{6}}{3072 a^{6}}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{5}}{3072 a^{6} x^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{4}}{1536 a^{5} x^{4}}+\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{3}}{1920 a^{4} x^{6}}+\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{2}}{960 a^{3} x^{8}}+\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{120 a^{2} x^{10}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{12 a \,x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^13,x)

[Out]

-1/12/a/x^12*(b*x^2+a)^(11/2)+1/120/a^2*b/x^10*(b*x^2+a)^(11/2)+1/960/a^3*b^2/x^8*(b*x^2+a)^(11/2)+1/1920/a^4*
b^3/x^6*(b*x^2+a)^(11/2)+1/1536/a^5*b^4/x^4*(b*x^2+a)^(11/2)+7/3072/a^6*b^5/x^2*(b*x^2+a)^(11/2)-7/3072/a^6*b^
6*(b*x^2+a)^(9/2)-3/1024/a^5*b^6*(b*x^2+a)^(7/2)-21/5120/a^4*b^6*(b*x^2+a)^(5/2)-7/1024/a^3*b^6*(b*x^2+a)^(3/2
)+21/1024/a^(3/2)*b^6*ln((2*a+2*(b*x^2+a)^(1/2)*a^(1/2))/x)-21/1024/a^2*b^6*(b*x^2+a)^(1/2)

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maxima [A]  time = 1.51, size = 221, normalized size = 1.43 \[ \frac {21 \, b^{6} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{1024 \, a^{\frac {3}{2}}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{6}}{3072 \, a^{6}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{6}}{1024 \, a^{5}} - \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{6}}{5120 \, a^{4}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{6}}{1024 \, a^{3}} - \frac {21 \, \sqrt {b x^{2} + a} b^{6}}{1024 \, a^{2}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{5}}{3072 \, a^{6} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{1536 \, a^{5} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{1920 \, a^{4} x^{6}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{960 \, a^{3} x^{8}} + \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{120 \, a^{2} x^{10}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{12 \, a x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^13,x, algorithm="maxima")

[Out]

21/1024*b^6*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 7/3072*(b*x^2 + a)^(9/2)*b^6/a^6 - 3/1024*(b*x^2 + a)^(7/2
)*b^6/a^5 - 21/5120*(b*x^2 + a)^(5/2)*b^6/a^4 - 7/1024*(b*x^2 + a)^(3/2)*b^6/a^3 - 21/1024*sqrt(b*x^2 + a)*b^6
/a^2 + 7/3072*(b*x^2 + a)^(11/2)*b^5/(a^6*x^2) + 1/1536*(b*x^2 + a)^(11/2)*b^4/(a^5*x^4) + 1/1920*(b*x^2 + a)^
(11/2)*b^3/(a^4*x^6) + 1/960*(b*x^2 + a)^(11/2)*b^2/(a^3*x^8) + 1/120*(b*x^2 + a)^(11/2)*b/(a^2*x^10) - 1/12*(
b*x^2 + a)^(11/2)/(a*x^12)

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mupad [B]  time = 6.68, size = 123, normalized size = 0.79 \[ \frac {843\,a\,{\left (b\,x^2+a\right )}^{7/2}}{2560\,x^{12}}-\frac {667\,{\left (b\,x^2+a\right )}^{9/2}}{3072\,x^{12}}-\frac {21\,a^4\,\sqrt {b\,x^2+a}}{1024\,x^{12}}+\frac {119\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}{1024\,x^{12}}-\frac {693\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}{2560\,x^{12}}-\frac {21\,{\left (b\,x^2+a\right )}^{11/2}}{1024\,a\,x^{12}}-\frac {b^6\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,21{}\mathrm {i}}{1024\,a^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^13,x)

[Out]

(843*a*(a + b*x^2)^(7/2))/(2560*x^12) - (b^6*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*21i)/(1024*a^(3/2)) - (667*(
a + b*x^2)^(9/2))/(3072*x^12) - (21*a^4*(a + b*x^2)^(1/2))/(1024*x^12) + (119*a^3*(a + b*x^2)^(3/2))/(1024*x^1
2) - (693*a^2*(a + b*x^2)^(5/2))/(2560*x^12) - (21*(a + b*x^2)^(11/2))/(1024*a*x^12)

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sympy [A]  time = 14.98, size = 204, normalized size = 1.32 \[ - \frac {a^{5}}{12 \sqrt {b} x^{13} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {59 a^{4} \sqrt {b}}{120 x^{11} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {1151 a^{3} b^{\frac {3}{2}}}{960 x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2947 a^{2} b^{\frac {5}{2}}}{1920 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {8171 a b^{\frac {7}{2}}}{7680 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {1045 b^{\frac {9}{2}}}{3072 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {21 b^{\frac {11}{2}}}{1024 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {21 b^{6} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{1024 a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**13,x)

[Out]

-a**5/(12*sqrt(b)*x**13*sqrt(a/(b*x**2) + 1)) - 59*a**4*sqrt(b)/(120*x**11*sqrt(a/(b*x**2) + 1)) - 1151*a**3*b
**(3/2)/(960*x**9*sqrt(a/(b*x**2) + 1)) - 2947*a**2*b**(5/2)/(1920*x**7*sqrt(a/(b*x**2) + 1)) - 8171*a*b**(7/2
)/(7680*x**5*sqrt(a/(b*x**2) + 1)) - 1045*b**(9/2)/(3072*x**3*sqrt(a/(b*x**2) + 1)) - 21*b**(11/2)/(1024*a*x*s
qrt(a/(b*x**2) + 1)) + 21*b**6*asinh(sqrt(a)/(sqrt(b)*x))/(1024*a**(3/2))

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