3.424 \(\int \frac{(a+b x^2)^{9/2}}{x^{15}} \, dx\)

Optimal. Leaf size=179 \[ \frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}} \]

[Out]

(-3*b^4*Sqrt[a + b*x^2])/(256*x^6) - (3*b^5*Sqrt[a + b*x^2])/(1024*a*x^4) + (9*b^6*Sqrt[a + b*x^2])/(2048*a^2*
x^2) - (3*b^3*(a + b*x^2)^(3/2))/(128*x^8) - (3*b^2*(a + b*x^2)^(5/2))/(80*x^10) - (3*b*(a + b*x^2)^(7/2))/(56
*x^12) - (a + b*x^2)^(9/2)/(14*x^14) - (9*b^7*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2048*a^(5/2))

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Rubi [A]  time = 0.121738, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^15,x]

[Out]

(-3*b^4*Sqrt[a + b*x^2])/(256*x^6) - (3*b^5*Sqrt[a + b*x^2])/(1024*a*x^4) + (9*b^6*Sqrt[a + b*x^2])/(2048*a^2*
x^2) - (3*b^3*(a + b*x^2)^(3/2))/(128*x^8) - (3*b^2*(a + b*x^2)^(5/2))/(80*x^10) - (3*b*(a + b*x^2)^(7/2))/(56
*x^12) - (a + b*x^2)^(9/2)/(14*x^14) - (9*b^7*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2048*a^(5/2))

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^{15}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{9/2}}{x^8} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{28} (9 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{7/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{32} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{256} \left (9 b^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{512} \left (3 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}-\frac{\left (9 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{2048 a}\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{\left (9 b^7\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{4096 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{\left (9 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2048 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0123766, size = 39, normalized size = 0.22 \[ \frac{b^7 \left (a+b x^2\right )^{11/2} \, _2F_1\left (\frac{11}{2},8;\frac{13}{2};\frac{b x^2}{a}+1\right )}{11 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^15,x]

[Out]

(b^7*(a + b*x^2)^(11/2)*Hypergeometric2F1[11/2, 8, 13/2, 1 + (b*x^2)/a])/(11*a^8)

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Maple [A]  time = 0.29, size = 253, normalized size = 1.4 \begin{align*} -{\frac{1}{14\,a{x}^{14}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{b}{56\,{a}^{2}{x}^{12}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{2}}{560\,{a}^{3}{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{3}}{4480\,{a}^{4}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{4}}{8960\,{a}^{5}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{5}}{7168\,{a}^{6}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{6}}{2048\,{a}^{7}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{7}}{2048\,{a}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{b}^{7}}{14336\,{a}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{9\,{b}^{7}}{10240\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{7}}{2048\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{b}^{7}}{2048}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{9\,{b}^{7}}{2048\,{a}^{3}}\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^15,x)

[Out]

-1/14/a/x^14*(b*x^2+a)^(11/2)+1/56*b/a^2/x^12*(b*x^2+a)^(11/2)-1/560*b^2/a^3/x^10*(b*x^2+a)^(11/2)-1/4480*b^3/
a^4/x^8*(b*x^2+a)^(11/2)-1/8960*b^4/a^5/x^6*(b*x^2+a)^(11/2)-1/7168*b^5/a^6/x^4*(b*x^2+a)^(11/2)-1/2048*b^6/a^
7/x^2*(b*x^2+a)^(11/2)+1/2048*b^7/a^7*(b*x^2+a)^(9/2)+9/14336*b^7/a^6*(b*x^2+a)^(7/2)+9/10240*b^7/a^5*(b*x^2+a
)^(5/2)+3/2048*b^7/a^4*(b*x^2+a)^(3/2)-9/2048*b^7/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+9/2048*b^7/a^3
*(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^15,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.1011, size = 628, normalized size = 3.51 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{7} x^{14} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (315 \, a b^{6} x^{12} - 210 \, a^{2} b^{5} x^{10} - 14168 \, a^{3} b^{4} x^{8} - 39056 \, a^{4} b^{3} x^{6} - 44928 \, a^{5} b^{2} x^{4} - 24320 \, a^{6} b x^{2} - 5120 \, a^{7}\right )} \sqrt{b x^{2} + a}}{143360 \, a^{3} x^{14}}, \frac{315 \, \sqrt{-a} b^{7} x^{14} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (315 \, a b^{6} x^{12} - 210 \, a^{2} b^{5} x^{10} - 14168 \, a^{3} b^{4} x^{8} - 39056 \, a^{4} b^{3} x^{6} - 44928 \, a^{5} b^{2} x^{4} - 24320 \, a^{6} b x^{2} - 5120 \, a^{7}\right )} \sqrt{b x^{2} + a}}{71680 \, a^{3} x^{14}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^15,x, algorithm="fricas")

[Out]

[1/143360*(315*sqrt(a)*b^7*x^14*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(315*a*b^6*x^12 - 210*
a^2*b^5*x^10 - 14168*a^3*b^4*x^8 - 39056*a^4*b^3*x^6 - 44928*a^5*b^2*x^4 - 24320*a^6*b*x^2 - 5120*a^7)*sqrt(b*
x^2 + a))/(a^3*x^14), 1/71680*(315*sqrt(-a)*b^7*x^14*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (315*a*b^6*x^12 - 210*
a^2*b^5*x^10 - 14168*a^3*b^4*x^8 - 39056*a^4*b^3*x^6 - 44928*a^5*b^2*x^4 - 24320*a^6*b*x^2 - 5120*a^7)*sqrt(b*
x^2 + a))/(a^3*x^14)]

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Sympy [A]  time = 22.8739, size = 231, normalized size = 1.29 \begin{align*} - \frac{a^{5}}{14 \sqrt{b} x^{15} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{23 a^{4} \sqrt{b}}{56 x^{13} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{541 a^{3} b^{\frac{3}{2}}}{560 x^{11} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5249 a^{2} b^{\frac{5}{2}}}{4480 x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{6653 a b^{\frac{7}{2}}}{8960 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{1027 b^{\frac{9}{2}}}{5120 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{11}{2}}}{2048 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{9 b^{\frac{13}{2}}}{2048 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{9 b^{7} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2048 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**15,x)

[Out]

-a**5/(14*sqrt(b)*x**15*sqrt(a/(b*x**2) + 1)) - 23*a**4*sqrt(b)/(56*x**13*sqrt(a/(b*x**2) + 1)) - 541*a**3*b**
(3/2)/(560*x**11*sqrt(a/(b*x**2) + 1)) - 5249*a**2*b**(5/2)/(4480*x**9*sqrt(a/(b*x**2) + 1)) - 6653*a*b**(7/2)
/(8960*x**7*sqrt(a/(b*x**2) + 1)) - 1027*b**(9/2)/(5120*x**5*sqrt(a/(b*x**2) + 1)) + 3*b**(11/2)/(2048*a*x**3*
sqrt(a/(b*x**2) + 1)) + 9*b**(13/2)/(2048*a**2*x*sqrt(a/(b*x**2) + 1)) - 9*b**7*asinh(sqrt(a)/(sqrt(b)*x))/(20
48*a**(5/2))

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Giac [A]  time = 2.35695, size = 184, normalized size = 1.03 \begin{align*} \frac{1}{71680} \, b^{7}{\left (\frac{315 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} - 2100 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a - 8393 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{2} + 9216 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{3} - 5943 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{4} + 2100 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{5} - 315 \, \sqrt{b x^{2} + a} a^{6}}{a^{2} b^{7} x^{14}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^15,x, algorithm="giac")

[Out]

1/71680*b^7*(315*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (315*(b*x^2 + a)^(13/2) - 2100*(b*x^2 + a)^
(11/2)*a - 8393*(b*x^2 + a)^(9/2)*a^2 + 9216*(b*x^2 + a)^(7/2)*a^3 - 5943*(b*x^2 + a)^(5/2)*a^4 + 2100*(b*x^2
+ a)^(3/2)*a^5 - 315*sqrt(b*x^2 + a)*a^6)/(a^2*b^7*x^14))