\(\int \frac {x^{10}}{(a+b x^2)^{9/2}} \, dx\) [527]

Optimal result

Integrand size = 15, antiderivative size = 131 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}-\frac {9 x^7}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac {3 x^5}{5 b^3 \left (a+b x^2\right )^{3/2}}-\frac {3 x^3}{b^4 \sqrt {a+b x^2}}+\frac {9 x \sqrt {a+b x^2}}{2 b^5}-\frac {9 a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \] Output:

-1/7*x^9/b/(b*x^2+a)^(7/2)-9/35*x^7/b^2/(b*x^2+a)^(5/2)-3/5*x^5/b^3/(b*x^2 
+a)^(3/2)-3*x^3/b^4/(b*x^2+a)^(1/2)+9/2*x*(b*x^2+a)^(1/2)/b^5-9/2*a*arctan 
h(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {315 a^4 x+1050 a^3 b x^3+1218 a^2 b^2 x^5+528 a b^3 x^7+35 b^4 x^9}{70 b^5 \left (a+b x^2\right )^{7/2}}-\frac {9 a \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{11/2}} \] Input:

Integrate[x^10/(a + b*x^2)^(9/2),x]
 

Output:

(315*a^4*x + 1050*a^3*b*x^3 + 1218*a^2*b^2*x^5 + 528*a*b^3*x^7 + 35*b^4*x^ 
9)/(70*b^5*(a + b*x^2)^(7/2)) - (9*a*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[ 
a + b*x^2])])/b^(11/2)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {252, 252, 252, 252, 262, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 \int \frac {x^8}{\left (b x^2+a\right )^{7/2}}dx}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 \left (\frac {7 \int \frac {x^6}{\left (b x^2+a\right )^{5/2}}dx}{5 b}-\frac {x^7}{5 b \left (a+b x^2\right )^{5/2}}\right )}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {x^4}{\left (b x^2+a\right )^{3/2}}dx}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^2\right )^{5/2}}\right )}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {x^2}{\sqrt {b x^2+a}}dx}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^2\right )^{5/2}}\right )}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^2\right )^{5/2}}\right )}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^2\right )^{5/2}}\right )}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right )}{b}-\frac {x^3}{b \sqrt {a+b x^2}}\right )}{3 b}-\frac {x^5}{3 b \left (a+b x^2\right )^{3/2}}\right )}{5 b}-\frac {x^7}{5 b \left (a+b x^2\right )^{5/2}}\right )}{7 b}-\frac {x^9}{7 b \left (a+b x^2\right )^{7/2}}\)

Input:

Int[x^10/(a + b*x^2)^(9/2),x]
 

Output:

-1/7*x^9/(b*(a + b*x^2)^(7/2)) + (9*(-1/5*x^7/(b*(a + b*x^2)^(5/2)) + (7*( 
-1/3*x^5/(b*(a + b*x^2)^(3/2)) + (5*(-(x^3/(b*Sqrt[a + b*x^2])) + (3*((x*S 
qrt[a + b*x^2])/(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2 
))))/b))/(3*b)))/(5*b)))/(7*b)
 

Defintions of rubi rules used

rule 219

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

rule 224

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72

method	result	size
pseudoelliptic	\(-\frac {9 \left (\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}} a -\left (\frac {b^{\frac {9}{2}} x^{8}}{9}+\frac {176 a \,b^{\frac {7}{2}} x^{6}}{105}+\frac {58 a^{2} b^{\frac {5}{2}} x^{4}}{15}+\frac {10 a^{3} b^{\frac {3}{2}} x^{2}}{3}+a^{4} \sqrt {b}\right ) x \right )}{2 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{\frac {11}{2}}}\)	\(94\)
default	\(\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\)	\(127\)
risch	\(\frac {x \sqrt {b \,x^{2}+a}}{2 b^{5}}-\frac {53 a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{560 b^{7} \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}-\frac {571 a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 b^{6} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {97 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{35 b^{6} \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {9 a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {11}{2}}}+\frac {571 a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 b^{6} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {97 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{35 b^{6} \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {a^{3} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{112 b^{7} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {a^{3} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{112 b^{7} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {53 a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{560 b^{7} \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}\)	\(559\)

Input:

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

Output:

-9/2/(b*x^2+a)^(7/2)/b^(11/2)*(arctanh((b*x^2+a)^(1/2)/x/b^(1/2))*(b*x^2+a 
)^(7/2)*a-(1/9*b^(9/2)*x^8+176/105*a*b^(7/2)*x^6+58/15*a^2*b^(5/2)*x^4+10/ 
3*a^3*b^(3/2)*x^2+a^4*b^(1/2))*x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.74 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {315 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (35 \, b^{5} x^{9} + 528 \, a b^{4} x^{7} + 1218 \, a^{2} b^{3} x^{5} + 1050 \, a^{3} b^{2} x^{3} + 315 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{140 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}, \frac {315 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (35 \, b^{5} x^{9} + 528 \, a b^{4} x^{7} + 1218 \, a^{2} b^{3} x^{5} + 1050 \, a^{3} b^{2} x^{3} + 315 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{70 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}\right ] \] Input:

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

Output:

[1/140*(315*(a*b^4*x^8 + 4*a^2*b^3*x^6 + 6*a^3*b^2*x^4 + 4*a^4*b*x^2 + a^5 
)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(35*b^5*x^9 
+ 528*a*b^4*x^7 + 1218*a^2*b^3*x^5 + 1050*a^3*b^2*x^3 + 315*a^4*b*x)*sqrt( 
b*x^2 + a))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4* 
b^6), 1/70*(315*(a*b^4*x^8 + 4*a^2*b^3*x^6 + 6*a^3*b^2*x^4 + 4*a^4*b*x^2 + 
 a^5)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (35*b^5*x^9 + 528*a*b^ 
4*x^7 + 1218*a^2*b^3*x^5 + 1050*a^3*b^2*x^3 + 315*a^4*b*x)*sqrt(b*x^2 + a) 
)/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3181 vs. \(2 (122) = 244\).

Time = 8.55 (sec) , antiderivative size = 3181, normalized size of antiderivative = 24.28 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-315*a**(311/2)*b**66*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**( 
309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt 
(1 + b*x**2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400 
*a**(303/2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2 
)*x**8*sqrt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x** 
2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 1890*a**(309/2 
)*b**67*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b* 
*(143/2)*sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x* 
*2/a) + 1050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/ 
2)*b**(149/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sq 
rt(1 + b*x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70 
*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 4725*a**(307/2)*b**68*x 
**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)* 
sqrt(1 + b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1 
050*a**(305/2)*b**(147/2)*x**4*sqrt(1 + b*x**2/a) + 1400*a**(303/2)*b**(14 
9/2)*x**6*sqrt(1 + b*x**2/a) + 1050*a**(301/2)*b**(151/2)*x**8*sqrt(1 + b* 
x**2/a) + 420*a**(299/2)*b**(153/2)*x**10*sqrt(1 + b*x**2/a) + 70*a**(297/ 
2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 6300*a**(305/2)*b**69*x**6*sqrt( 
1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(70*a**(309/2)*b**(143/2)*sqrt(1 + 
b*x**2/a) + 420*a**(307/2)*b**(145/2)*x**2*sqrt(1 + b*x**2/a) + 1050*a*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (105) = 210\).

Time = 0.06 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.18 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x^{9}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {9 \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} a x}{70 \, b} + \frac {3 \, a x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{10 \, b^{2}} + \frac {3 \, a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{2 \, b^{3}} + \frac {9 \, a^{2} x^{3}}{2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} - \frac {417 \, a x}{70 \, \sqrt {b x^{2} + a} b^{5}} - \frac {51 \, a^{2} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} + \frac {261 \, a^{3} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}} - \frac {9 \, a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {11}{2}}} \] Input:

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

Output:

1/2*x^9/((b*x^2 + a)^(7/2)*b) + 9/70*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a* 
x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/ 
((b*x^2 + a)^(7/2)*b^4))*a*x/b + 3/10*a*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 
20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b^2 + 3/ 
2*a*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^3 + 9/ 
2*a^2*x^3/((b*x^2 + a)^(5/2)*b^4) - 417/70*a*x/(sqrt(b*x^2 + a)*b^5) - 51/ 
70*a^2*x/((b*x^2 + a)^(3/2)*b^5) + 261/70*a^3*x/((b*x^2 + a)^(5/2)*b^5) - 
9/2*a*arcsinh(b*x/sqrt(a*b))/b^(11/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.69 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left (x^{2} {\left (\frac {35 \, x^{2}}{b} + \frac {528 \, a}{b^{2}}\right )} + \frac {1218 \, a^{2}}{b^{3}}\right )} x^{2} + \frac {1050 \, a^{3}}{b^{4}}\right )} x^{2} + \frac {315 \, a^{4}}{b^{5}}\right )} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {9 \, a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} \] Input:

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

Output:

1/70*(((x^2*(35*x^2/b + 528*a/b^2) + 1218*a^2/b^3)*x^2 + 1050*a^3/b^4)*x^2 
 + 315*a^4/b^5)*x/(b*x^2 + a)^(7/2) + 9/2*a*log(abs(-sqrt(b)*x + sqrt(b*x^ 
2 + a)))/b^(11/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^{10}}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.59 \[ \int \frac {x^{10}}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {315 \sqrt {b \,x^{2}+a}\, a^{4} b x +1050 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{3}+1218 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{5}+528 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{7}+35 \sqrt {b \,x^{2}+a}\, b^{5} x^{9}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5}-1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,x^{2}-1890 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{4}-1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{6}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}-213 \sqrt {b}\, a^{5}-852 \sqrt {b}\, a^{4} b \,x^{2}-1278 \sqrt {b}\, a^{3} b^{2} x^{4}-852 \sqrt {b}\, a^{2} b^{3} x^{6}-213 \sqrt {b}\, a \,b^{4} x^{8}}{70 b^{6} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

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

Output:

(315*sqrt(a + b*x**2)*a**4*b*x + 1050*sqrt(a + b*x**2)*a**3*b**2*x**3 + 12 
18*sqrt(a + b*x**2)*a**2*b**3*x**5 + 528*sqrt(a + b*x**2)*a*b**4*x**7 + 35 
*sqrt(a + b*x**2)*b**5*x**9 - 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)* 
x)/sqrt(a))*a**5 - 1260*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a) 
)*a**4*b*x**2 - 1890*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a 
**3*b**2*x**4 - 1260*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a 
**2*b**3*x**6 - 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a* 
b**4*x**8 - 213*sqrt(b)*a**5 - 852*sqrt(b)*a**4*b*x**2 - 1278*sqrt(b)*a**3 
*b**2*x**4 - 852*sqrt(b)*a**2*b**3*x**6 - 213*sqrt(b)*a*b**4*x**8)/(70*b** 
6*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))