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

Optimal. Leaf size=131 \[ -\frac{9 x^7}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{3 x^5}{5 b^3 \left (a+b x^2\right )^{3/2}}-\frac{3 x^3}{b^4 \sqrt{a+b x^2}}+\frac{9 x \sqrt{a+b x^2}}{2 b^5}-\frac{9 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}-\frac{x^9}{7 b \left (a+b x^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.0535724, antiderivative size = 131, normalized size of antiderivative = 1., 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 = {288, 321, 217, 206} \[ -\frac{9 x^7}{35 b^2 \left (a+b x^2\right )^{5/2}}-\frac{3 x^5}{5 b^3 \left (a+b x^2\right )^{3/2}}-\frac{3 x^3}{b^4 \sqrt{a+b x^2}}+\frac{9 x \sqrt{a+b x^2}}{2 b^5}-\frac{9 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}-\frac{x^9}{7 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(a + b*x^2)^(9/2),x]

[Out]

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

Rule 288

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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{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 \int \frac{x^8}{\left (a+b x^2\right )^{7/2}} \, dx}{7 b}\\ &=-\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{9 \int \frac{x^6}{\left (a+b x^2\right )^{5/2}} \, dx}{5 b^2}\\ &=-\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 \int \frac{x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{b^3}\\ &=-\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 \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{b^4}\\ &=-\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) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{2 b^5}\\ &=-\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) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2 b^5}\\ &=-\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 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.188903, size = 114, normalized size = 0.87 \[ \frac{\sqrt{b} x \left (1218 a^2 b^2 x^4+1050 a^3 b x^2+315 a^4+528 a b^3 x^6+35 b^4 x^8\right )-315 a^{3/2} \left (a+b x^2\right )^3 \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{70 b^{11/2} \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[b]*x*(315*a^4 + 1050*a^3*b*x^2 + 1218*a^2*b^2*x^4 + 528*a*b^3*x^6 + 35*b^4*x^8) - 315*a^(3/2)*(a + b*x^2
)^3*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(70*b^(11/2)*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.029, size = 111, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,a{x}^{7}}{14\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,a{x}^{5}}{10\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{3\,a{x}^{3}}{2\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{9\,ax}{2\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{9\,a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^9/b/(b*x^2+a)^(7/2)+9/14/b^2*a*x^7/(b*x^2+a)^(7/2)+9/10/b^3*a*x^5/(b*x^2+a)^(5/2)+3/2/b^4*a*x^3/(b*x^2+a
)^(3/2)+9/2/b^5*a*x/(b*x^2+a)^(1/2)-9/2/b^(11/2)*a*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(x^10/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43223, size = 798, normalized size = 6.09 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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)*sqr
t(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)]

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Sympy [B]  time = 11.9994, size = 3181, normalized size = 24.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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) + 14
00*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**(3
09/2)*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) + 42
0*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)) - 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) + 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)) - 6300*a**(305/2)*b**6
9*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**(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**(15
3/2)*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**(303/2)*b**70*x**
8*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**(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)) - 1890*a**(301/2)*b**71*x**10*sq
rt(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**(14
5/2)*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**1
0*sqrt(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) - 315*a**(299/2)*b**72*x**12*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*sqr
t(1 + b*x**2/a) + 70*a**(297/2)*b**(155/2)*x**12*sqrt(1 + b*x**2/a)) + 315*a**155*b**(133/2)*x/(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*sqr
t(1 + b*x**2/a)) + 1995*a**154*b**(135/2)*x**3/(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**(1
49/2)*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)) + 5313*a**153*b**(137/2)*x**5/(
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**(15
5/2)*x**12*sqrt(1 + b*x**2/a)) + 7647*a**152*b**(139/2)*x**7/(70*a**(309/2)*b**(143/2)*sqrt(1 + b*x**2/a) + 42
0*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)) + 6323*a**151*b**
(141/2)*x**9/(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) + 10
50*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)) + 2907*a**150*b**(143/2)*x**11/(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)) +
633*a**149*b**(145/2)*x**13/(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)) + 35*a**148*b**(147/2)*x**15/(70*a**(309/2)*b**(14
3/2)*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))

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Giac [A]  time = 1.88799, size = 123, normalized size = 0.94 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)