3.433 \(\int \frac{(a+b x^2)^{9/2}}{x^{10}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}+b^{9/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9} \]

[Out]

-((b^4*Sqrt[a + b*x^2])/x) - (b^3*(a + b*x^2)^(3/2))/(3*x^3) - (b^2*(a + b*x^2)^(5/2))/(5*x^5) - (b*(a + b*x^2
)^(7/2))/(7*x^7) - (a + b*x^2)^(9/2)/(9*x^9) + b^(9/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]

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Rubi [A]  time = 0.0524736, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 217, 206} \[ -\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}+b^{9/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^10,x]

[Out]

-((b^4*Sqrt[a + b*x^2])/x) - (b^3*(a + b*x^2)^(3/2))/(3*x^3) - (b^2*(a + b*x^2)^(5/2))/(5*x^5) - (b*(a + b*x^2
)^(7/2))/(7*x^7) - (a + b*x^2)^(9/2)/(9*x^9) + b^(9/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{9/2}}{x^{10}} \, dx &=-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b \int \frac{\left (a+b x^2\right )^{7/2}}{x^8} \, dx\\ &=-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b^2 \int \frac{\left (a+b x^2\right )^{5/2}}{x^6} \, dx\\ &=-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b^3 \int \frac{\left (a+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b^4 \int \frac{\sqrt{a+b x^2}}{x^2} \, dx\\ &=-\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b^5 \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=-\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b^5 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=-\frac{b^4 \sqrt{a+b x^2}}{x}-\frac{b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac{\left (a+b x^2\right )^{9/2}}{9 x^9}+b^{9/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.009773, size = 54, normalized size = 0.44 \[ -\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left (-\frac{9}{2},-\frac{9}{2};-\frac{7}{2};-\frac{b x^2}{a}\right )}{9 x^9 \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^10,x]

[Out]

-(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -9/2, -7/2, -((b*x^2)/a)])/(9*x^9*Sqrt[1 + (b*x^2)/a])

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Maple [B]  time = 0.03, size = 206, normalized size = 1.7 \begin{align*} -{\frac{1}{9\,a{x}^{9}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{2\,b}{63\,{a}^{2}{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{8\,{b}^{2}}{315\,{a}^{3}{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{16\,{b}^{3}}{315\,{a}^{4}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{128\,{b}^{4}}{315\,{a}^{5}x} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{128\,{b}^{5}x}{315\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{16\,{b}^{5}x}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{b}^{5}x}{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{5}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{5}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{9}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^10,x)

[Out]

-1/9/a/x^9*(b*x^2+a)^(11/2)-2/63*b/a^2/x^7*(b*x^2+a)^(11/2)-8/315*b^2/a^3/x^5*(b*x^2+a)^(11/2)-16/315*b^3/a^4/
x^3*(b*x^2+a)^(11/2)-128/315*b^4/a^5/x*(b*x^2+a)^(11/2)+128/315*b^5/a^5*x*(b*x^2+a)^(9/2)+16/35*b^5/a^4*x*(b*x
^2+a)^(7/2)+8/15*b^5/a^3*x*(b*x^2+a)^(5/2)+2/3*b^5/a^2*x*(b*x^2+a)^(3/2)+b^5/a*x*(b*x^2+a)^(1/2)+b^(9/2)*ln(x*
b^(1/2)+(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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.11358, size = 452, normalized size = 3.65 \begin{align*} \left [\frac{315 \, b^{\frac{9}{2}} x^{9} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{630 \, x^{9}}, -\frac{315 \, \sqrt{-b} b^{4} x^{9} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, x^{9}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^10,x, algorithm="fricas")

[Out]

[1/630*(315*b^(9/2)*x^9*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(563*b^4*x^8 + 506*a*b^3*x^6 + 408
*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4)*sqrt(b*x^2 + a))/x^9, -1/315*(315*sqrt(-b)*b^4*x^9*arctan(sqrt(-b)*x/sq
rt(b*x^2 + a)) + (563*b^4*x^8 + 506*a*b^3*x^6 + 408*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4)*sqrt(b*x^2 + a))/x^9
]

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Sympy [A]  time = 10.3483, size = 160, normalized size = 1.29 \begin{align*} - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{9 x^{8}} - \frac{37 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 x^{6}} - \frac{136 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 x^{4}} - \frac{506 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{315 x^{2}} - \frac{563 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{315} - \frac{b^{\frac{9}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{9}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**10,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(9*x**8) - 37*a**3*b**(3/2)*sqrt(a/(b*x**2) + 1)/(63*x**6) - 136*a**2*b**(5
/2)*sqrt(a/(b*x**2) + 1)/(105*x**4) - 506*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/(315*x**2) - 563*b**(9/2)*sqrt(a/(b*
x**2) + 1)/315 - b**(9/2)*log(a/(b*x**2))/2 + b**(9/2)*log(sqrt(a/(b*x**2) + 1) + 1)

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Giac [B]  time = 2.87017, size = 373, normalized size = 3.01 \begin{align*} -\frac{1}{2} \, b^{\frac{9}{2}} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a b^{\frac{9}{2}} - 6300 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{2} b^{\frac{9}{2}} + 21000 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{3} b^{\frac{9}{2}} - 31500 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{4} b^{\frac{9}{2}} + 39438 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{5} b^{\frac{9}{2}} - 26292 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{6} b^{\frac{9}{2}} + 13968 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{7} b^{\frac{9}{2}} - 3492 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{8} b^{\frac{9}{2}} + 563 \, a^{9} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^10,x, algorithm="giac")

[Out]

-1/2*b^(9/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/315*(1575*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a*b^(9/2) - 6
300*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(9/2) + 21000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(9/2) - 31500*
(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(9/2) + 39438*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(9/2) - 26292*(sqrt
(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(9/2) + 13968*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(9/2) - 3492*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^8*b^(9/2) + 563*a^9*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9