3.519 \(\int \frac{x^8}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=106 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}-\frac{x}{b^4 \sqrt{a+b x^2}}-\frac{x^3}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^5}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^7}{7 b \left (a+b x^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.133731, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}-\frac{x}{b^4 \sqrt{a+b x^2}}-\frac{x^3}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^5}{5 b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^7}{7 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^2)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.6918, size = 92, normalized size = 0.87 \[ - \frac{x^{7}}{7 b \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{x^{5}}{5 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{x^{3}}{3 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x}{b^{4} \sqrt{a + b x^{2}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**2+a)**(9/2),x)

[Out]

-x**7/(7*b*(a + b*x**2)**(7/2)) - x**5/(5*b**2*(a + b*x**2)**(5/2)) - x**3/(3*b*
*3*(a + b*x**2)**(3/2)) - x/(b**4*sqrt(a + b*x**2)) + atanh(sqrt(b)*x/sqrt(a + b
*x**2))/b**(9/2)

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Mathematica [A]  time = 0.132754, size = 80, normalized size = 0.75 \[ \frac{\log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{9/2}}-\frac{x \left (105 a^3+350 a^2 b x^2+406 a b^2 x^4+176 b^3 x^6\right )}{105 b^4 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^2)^(9/2),x]

[Out]

-(x*(105*a^3 + 350*a^2*b*x^2 + 406*a*b^2*x^4 + 176*b^3*x^6))/(105*b^4*(a + b*x^2
)^(7/2)) + Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]/b^(9/2)

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Maple [A]  time = 0.024, size = 88, normalized size = 0.8 \[ -{\frac{{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{x}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{1\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^2+a)^(9/2),x)

[Out]

-1/7*x^7/b/(b*x^2+a)^(7/2)-1/5*x^5/b^2/(b*x^2+a)^(5/2)-1/3*x^3/b^3/(b*x^2+a)^(3/
2)-x/b^4/(b*x^2+a)^(1/2)+1/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271229, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (176 \, b^{3} x^{7} + 406 \, a b^{2} x^{5} + 350 \, a^{2} b x^{3} + 105 \, a^{3} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 105 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{210 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \sqrt{b}}, -\frac{{\left (176 \, b^{3} x^{7} + 406 \, a b^{2} x^{5} + 350 \, a^{2} b x^{3} + 105 \, a^{3} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 105 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{105 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[-1/210*(2*(176*b^3*x^7 + 406*a*b^2*x^5 + 350*a^2*b*x^3 + 105*a^3*x)*sqrt(b*x^2
+ a)*sqrt(b) - 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)*l
og(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((b^8*x^8 + 4*a*b^7*x^6 + 6*
a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)*sqrt(b)), -1/105*((176*b^3*x^7 + 406*a*b^
2*x^5 + 350*a^2*b*x^3 + 105*a^3*x)*sqrt(b*x^2 + a)*sqrt(-b) - 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)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a))
)/((b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x^4 + 4*a^3*b^5*x^2 + a^4*b^4)*sqrt(-b))]

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Sympy [A]  time = 31.4447, size = 2980, normalized size = 28.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**2+a)**(9/2),x)

[Out]

105*a**(205/2)*b**45*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)
*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a
) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105
/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a)
+ 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2
)*x**12*sqrt(1 + b*x**2/a)) + 630*a**(203/2)*b**46*x**2*sqrt(1 + b*x**2/a)*asinh
(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2
)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 +
b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)
*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b
*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 1575*a**(201/2)
*b**47*x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2
)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*
a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*
sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a*
*(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*s
qrt(1 + b*x**2/a)) + 2100*a**(199/2)*b**48*x**6*sqrt(1 + b*x**2/a)*asinh(sqrt(b)
*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(10
1/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a
) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107
/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a)
+ 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 1575*a**(197/2)*b**49*x
**8*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1
+ b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/
2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 +
b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)
*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 +
b*x**2/a)) + 630*a**(195/2)*b**50*x**10*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(
a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**
2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100
*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8
*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a
**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 105*a**(193/2)*b**51*x**12*sqrt
(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**
2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(1
03/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/
a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109
/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a
)) - 105*a**102*b**(91/2)*x/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a
**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*s
qrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a*
*(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sq
rt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 665*a**
101*b**(93/2)*x**3/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)
*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b
*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*
b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*
x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 1771*a**100*b**(
95/2)*x**5/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101
/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a)
+ 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/
2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a)
+ 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 2549*a**99*b**(97/2)*x**
7/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*
sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a
**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*s
qrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**
(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 2096*a**98*b**(99/2)*x**9/(105*a*
*(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 +
b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)
*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b
*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b
**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 934*a**97*b**(101/2)*x**11/(105*a**(205/2)
*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a
) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105
/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a)
+ 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2
)*x**12*sqrt(1 + b*x**2/a)) - 176*a**96*b**(103/2)*x**13/(105*a**(205/2)*b**(99/
2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575
*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6
*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a
**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*
sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.224491, size = 105, normalized size = 0.99 \[ -\frac{{\left (2 \,{\left (x^{2}{\left (\frac{88 \, x^{2}}{b} + \frac{203 \, a}{b^{2}}\right )} + \frac{175 \, a^{2}}{b^{3}}\right )} x^{2} + \frac{105 \, a^{3}}{b^{4}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

-1/105*(2*(x^2*(88*x^2/b + 203*a/b^2) + 175*a^2/b^3)*x^2 + 105*a^3/b^4)*x/(b*x^2
+ a)^(7/2) - ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)