∫ 1____ x(a+bx2)9∕2 dx [528]

Optimal. Leaf size=95 \[ \frac {1}{7 a \left (a+b x^2\right )^{7/2}}+\frac {1}{5 a^2 \left (a+b x^2\right )^{5/2}}+\frac {1}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {1}{a^4 \sqrt {a+b x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \]

1/7/a/(b*x^2+a)^(7/2)+1/5/a^2/(b*x^2+a)^(5/2)+1/3/a^3/(b*x^2+a)^(3/2)-arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(9/2)

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Rubi [A]
time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {1}{a^4 \sqrt {a+b x^2}}+\frac {1}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac {1}{5 a^2 \left (a+b x^2\right )^{5/2}}+\frac {1}{7 a \left (a+b x^2\right )^{7/2}} \end {gather*}

1/(7*a*(a + b*x^2)^(7/2)) + 1/(5*a^2*(a + b*x^2)^(5/2)) + 1/(3*a^3*(a + b*x^2)^(3/2)) + 1/(a^4*Sqrt[a + b*x^2]

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]

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,

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

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]]

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

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Mathematica [A]
time = 0.07, size = 76, normalized size = 0.80 \begin {gather*} \frac {176 a^3+406 a^2 b x^2+350 a b^2 x^4+105 b^3 x^6}{105 a^4 \left (a+b x^2\right )^{7/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \end {gather*}

(176*a^3 + 406*a^2*b*x^2 + 350*a*b^2*x^4 + 105*b^3*x^6)/(105*a^4*(a + b*x^2)^(7/2)) - ArcTanh[Sqrt[a + b*x^2]/

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method	result	size

default	\(\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\)	\(100\)

1/7/a/(b*x^2+a)^(7/2)+1/a*(1/5/a/(b*x^2+a)^(5/2)+1/a*(1/3/a/(b*x^2+a)^(3/2)+1/a*(1/a/(b*x^2+a)^(1/2)-1/a^(3/2)

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Maxima [A]
time = 0.31, size = 73, normalized size = 0.77 \begin {gather*} -\frac {\operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {1}{\sqrt {b x^{2} + a} a^{4}} + \frac {1}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {1}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {1}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} \end {gather*}

-arcsinh(a/(sqrt(a*b)*abs(x)))/a^(9/2) + 1/(sqrt(b*x^2 + a)*a^4) + 1/3/((b*x^2 + a)^(3/2)*a^3) + 1/5/((b*x^2 +

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (75) = 150\).
time = 1.85, size = 329, normalized size = 3.46 \begin {gather*} \left [\frac {105 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (105 \, a b^{3} x^{6} + 350 \, a^{2} b^{2} x^{4} + 406 \, a^{3} b x^{2} + 176 \, a^{4}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{5} b^{4} x^{8} + 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} + 4 \, a^{8} b x^{2} + a^{9}\right )}}, \frac {105 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, a b^{3} x^{6} + 350 \, a^{2} b^{2} x^{4} + 406 \, a^{3} b x^{2} + 176 \, a^{4}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{4} x^{8} + 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} + 4 \, a^{8} b x^{2} + a^{9}\right )}}\right ] \end {gather*}

[1/210*(105*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a

)*sqrt(a) + 2*a)/x^2) + 2*(105*a*b^3*x^6 + 350*a^2*b^2*x^4 + 406*a^3*b*x^2 + 176*a^4)*sqrt(b*x^2 + a))/(a^5*b^

4*x^8 + 4*a^6*b^3*x^6 + 6*a^7*b^2*x^4 + 4*a^8*b*x^2 + a^9), 1/105*(105*(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4

+ 4*a^3*b*x^2 + a^4)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (105*a*b^3*x^6 + 350*a^2*b^2*x^4 + 406*a^3*b*

x^2 + 176*a^4)*sqrt(b*x^2 + a))/(a^5*b^4*x^8 + 4*a^6*b^3*x^6 + 6*a^7*b^2*x^4 + 4*a^8*b*x^2 + a^9)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5250 vs. \(2 (85) = 170\).
time = 3.90, size = 5250, normalized size = 55.26 \begin {gather*} \text {Too large to display} \end {gather*}

352*a**32*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/

2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**

(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 105*a

**32*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x*

*6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**

7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 210*a**32*log(s

qrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**

3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)

*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 2924*a**31*

b*x**2*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*

b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59

/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 1050*a**

31*b*x**2*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b*

*3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2

)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 2100*a**31

*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*

a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25

200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20)

+ 10852*a**30*b**2*x**4*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 +

25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**

12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*

x**20) + 4725*a**30*b**2*x**4*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4

+ 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x*

*12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10

*x**20) - 9450*a**30*b**2*x**4*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(6

9/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a*

*(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*

a**(53/2)*b**10*x**20) + 23630*a**29*b**3*x**6*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 945

0*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 4

4100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18

+ 210*a**(53/2)*b**10*x**20) + 12600*a**29*b**3*x**6*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9

450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 +

44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**

18 + 210*a**(53/2)*b**10*x**20) - 25200*a**29*b**3*x**6*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(

71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63

/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a*

*(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 33280*a**28*b**4*x**8*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 21

00*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920

*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 +

2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 22050*a**28*b**4*x**8*log(b*x**2/a)/(210*a**(73/2) +

2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 529

20*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16

+ 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 44100*a**28*b**4*x**8*log(sqrt(1 + b*x**2/a) + 1)/(

210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)

*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(5

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Giac [A]
time = 0.54, size = 81, normalized size = 0.85 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {105 \, {\left (b x^{2} + a\right )}^{3} + 35 \, {\left (b x^{2} + a\right )}^{2} a + 21 \, {\left (b x^{2} + a\right )} a^{2} + 15 \, a^{3}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} \end {gather*}

arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) + 1/105*(105*(b*x^2 + a)^3 + 35*(b*x^2 + a)^2*a + 21*(b*x^2 +

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Mupad [B]
time = 4.85, size = 75, normalized size = 0.79 \begin {gather*} \frac {\frac {b\,x^2+a}{5\,a^2}+\frac {1}{7\,a}+\frac {{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {{\left (b\,x^2+a\right )}^3}{a^4}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}} \end {gather*}

((a + b*x^2)/(5*a^2) + 1/(7*a) + (a + b*x^2)^2/(3*a^3) + (a + b*x^2)^3/a^4)/(a + b*x^2)^(7/2) - atanh((a + b*x

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3.6.28 \(\int \frac {1}{x (a+b x^2)^{9/2}} \, dx\) [528]