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

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

[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))

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Rubi [A]  time = 0.101011, antiderivative size = 155, normalized size of antiderivative = 1., 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 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 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]

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*}

Mathematica [C]  time = 0.0122679, 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]

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

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Maple [A]  time = 0.113, size = 233, normalized size = 1.5 \begin{align*} -{\frac{1}{12\,a{x}^{12}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{b}{120\,{a}^{2}{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{2}}{960\,{a}^{3}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{3}}{1920\,{a}^{4}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{4}}{1536\,{a}^{5}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{7\,{b}^{5}}{3072\,{a}^{6}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{7\,{b}^{6}}{3072\,{a}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}-{\frac{3\,{b}^{6}}{1024\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{21\,{b}^{6}}{5120\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{6}}{1024\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{21\,{b}^{6}}{1024}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{21\,{b}^{6}}{1024\,{a}^{2}}\sqrt{b{x}^{2}+a}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.84903, size = 571, normalized size = 3.68 \begin{align*} \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 ] \end{align*}

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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Sympy [A]  time = 16.2194, size = 204, normalized size = 1.32 \begin{align*} - \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}}} \end{align*}

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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Giac [A]  time = 2.37557, size = 165, normalized size = 1.06 \begin{align*} -\frac{1}{15360} \, b^{6}{\left (\frac{315 \, \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}} + 3335 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a - 5058 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} + 4158 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} - 1785 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4} + 315 \, \sqrt{b x^{2} + a} a^{5}}{a b^{6} x^{12}}\right )} \end{align*}

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