3.173 \(\int \frac{A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac{x \left (A b^4-a^4 F\right )}{a b^4 \left (a+b x^2\right )^{7/2}}+\frac{x^5 \left (a \left (-58 a^3 F+3 a b^2 C+4 b^3 B\right )+24 A b^4\right )}{15 a^3 b^2 \left (a+b x^2\right )^{7/2}}+\frac{x^3 \left (-10 a^4 F+a b^3 B+6 A b^4\right )}{3 a^2 b^3 \left (a+b x^2\right )^{7/2}}+\frac{x^7 \left (a \left (-176 a^3 F+15 a^2 b D+6 a b^2 C+8 b^3 B\right )+48 A b^4\right )}{105 a^4 b \left (a+b x^2\right )^{7/2}}+\frac{F \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

[Out]

((A*b^4 - a^4*F)*x)/(a*b^4*(a + b*x^2)^(7/2)) + ((6*A*b^4 + a*b^3*B - 10*a^4*F)*
x^3)/(3*a^2*b^3*(a + b*x^2)^(7/2)) + ((24*A*b^4 + a*(4*b^3*B + 3*a*b^2*C - 58*a^
3*F))*x^5)/(15*a^3*b^2*(a + b*x^2)^(7/2)) + ((48*A*b^4 + a*(8*b^3*B + 6*a*b^2*C
+ 15*a^2*b*D - 176*a^3*F))*x^7)/(105*a^4*b*(a + b*x^2)^(7/2)) + (F*ArcTanh[(Sqrt
[b]*x)/Sqrt[a + b*x^2]])/b^(9/2)

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Rubi [A]  time = 0.947392, antiderivative size = 252, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x \left (a \left (122 a^3 F-45 a^2 b D+3 a b^2 C+4 b^3 B\right )+24 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{x \left (a \left (-22 a^3 F+15 a^2 b D-8 a b^2 C+b^3 B\right )+6 A b^4\right )}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac{x \left (\frac{A}{a}-\frac{a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}+\frac{x \left (a \left (-176 a^3 F+15 a^2 b D+6 a b^2 C+8 b^3 B\right )+48 A b^4\right )}{105 a^4 b^4 \sqrt{a+b x^2}}+\frac{F \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2 + C*x^4 + D*x^6 + F*x^8)/(a + b*x^2)^(9/2),x]

[Out]

((A/a - (b^3*B - a*b^2*C + a^2*b*D - a^3*F)/b^4)*x)/(7*(a + b*x^2)^(7/2)) + ((6*
A*b^4 + a*(b^3*B - 8*a*b^2*C + 15*a^2*b*D - 22*a^3*F))*x)/(35*a^2*b^4*(a + b*x^2
)^(5/2)) + ((24*A*b^4 + a*(4*b^3*B + 3*a*b^2*C - 45*a^2*b*D + 122*a^3*F))*x)/(10
5*a^3*b^4*(a + b*x^2)^(3/2)) + ((48*A*b^4 + a*(8*b^3*B + 6*a*b^2*C + 15*a^2*b*D
- 176*a^3*F))*x)/(105*a^4*b^4*Sqrt[a + b*x^2]) + (F*ArcTanh[(Sqrt[b]*x)/Sqrt[a +
b*x^2]])/b^(9/2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((F*x**8+D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.424071, size = 176, normalized size = 0.82 \[ \frac{x \left (-105 a^7 F-350 a^6 b F x^2-406 a^5 b^2 F x^4-176 a^4 b^3 F x^6+a^3 b^4 \left (105 A+35 B x^2+21 C x^4+15 D x^6\right )+2 a^2 b^5 x^2 \left (105 A+14 B x^2+3 C x^4\right )+8 a b^6 x^4 \left (21 A+B x^2\right )+48 A b^7 x^6\right )}{105 a^4 b^4 \left (a+b x^2\right )^{7/2}}+\frac{F \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2 + C*x^4 + D*x^6 + F*x^8)/(a + b*x^2)^(9/2),x]

[Out]

(x*(-105*a^7*F - 350*a^6*b*F*x^2 - 406*a^5*b^2*F*x^4 + 48*A*b^7*x^6 - 176*a^4*b^
3*F*x^6 + 8*a*b^6*x^4*(21*A + B*x^2) + 2*a^2*b^5*x^2*(105*A + 14*B*x^2 + 3*C*x^4
) + a^3*b^4*(105*A + 35*B*x^2 + 21*C*x^4 + 15*D*x^6)))/(105*a^4*b^4*(a + b*x^2)^
(7/2)) + (F*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/b^(9/2)

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Maple [B]  time = 0.01, size = 427, normalized size = 2. \[{\frac{Ax}{7\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{6\,Ax}{35\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,Ax}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,Ax}{35\,{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{Bx}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Bx}{35\,ab} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,Bx}{105\,{a}^{2}b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,Bx}{105\,{a}^{3}b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{C{x}^{3}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{3\,Cxa}{28\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Cx}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Cx}{35\,a{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Cx}{35\,{a}^{2}{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{D{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,D{x}^{3}a}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,Dx{a}^{2}}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Dxa}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Dx}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Dx}{7\,a{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{F{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{F{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{F{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Fx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{F\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((F*x^8+D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)

[Out]

1/7*A*x/a/(b*x^2+a)^(7/2)+6/35*A/a^2*x/(b*x^2+a)^(5/2)+8/35*A/a^3*x/(b*x^2+a)^(3
/2)+16/35*A/a^4*x/(b*x^2+a)^(1/2)-1/7*B/b*x/(b*x^2+a)^(7/2)+1/35*B/a/b*x/(b*x^2+
a)^(5/2)+4/105*B*x/a^2/b/(b*x^2+a)^(3/2)+8/105*B*x/a^3/b/(b*x^2+a)^(1/2)-1/4*C*x
^3/b/(b*x^2+a)^(7/2)-3/28*C*a/b^2*x/(b*x^2+a)^(7/2)+3/140*C/b^2*x/(b*x^2+a)^(5/2
)+1/35*C/a/b^2*x/(b*x^2+a)^(3/2)+2/35*C/a^2/b^2*x/(b*x^2+a)^(1/2)-1/2*D*x^5/b/(b
*x^2+a)^(7/2)-5/8*D*a/b^2*x^3/(b*x^2+a)^(7/2)-15/56*D*a^2/b^3*x/(b*x^2+a)^(7/2)+
3/56*D*a/b^3*x/(b*x^2+a)^(5/2)+1/14*D/b^3*x/(b*x^2+a)^(3/2)+1/7*D/a/b^3*x/(b*x^2
+a)^(1/2)-1/7*F*x^7/b/(b*x^2+a)^(7/2)-1/5*F/b^2*x^5/(b*x^2+a)^(5/2)-1/3*F/b^3*x^
3/(b*x^2+a)^(3/2)-F/b^4*x/(b*x^2+a)^(1/2)+F/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((F*x^8 + D*x^6 + C*x^4 + B*x^2 + A)/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.604811, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left ({\left (176 \, F a^{4} b^{3} - 15 \, D a^{3} b^{4} - 6 \, C a^{2} b^{5} - 8 \, B a b^{6} - 48 \, A b^{7}\right )} x^{7} + 7 \,{\left (58 \, F a^{5} b^{2} - 3 \, C a^{3} b^{4} - 4 \, B a^{2} b^{5} - 24 \, A a b^{6}\right )} x^{5} + 35 \,{\left (10 \, F a^{6} b - B a^{3} b^{4} - 6 \, A a^{2} b^{5}\right )} x^{3} + 105 \,{\left (F a^{7} - A a^{3} b^{4}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 105 \,{\left (F a^{4} b^{4} x^{8} + 4 \, F a^{5} b^{3} x^{6} + 6 \, F a^{6} b^{2} x^{4} + 4 \, F a^{7} b x^{2} + F a^{8}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{210 \,{\left (a^{4} b^{8} x^{8} + 4 \, a^{5} b^{7} x^{6} + 6 \, a^{6} b^{6} x^{4} + 4 \, a^{7} b^{5} x^{2} + a^{8} b^{4}\right )} \sqrt{b}}, -\frac{{\left ({\left (176 \, F a^{4} b^{3} - 15 \, D a^{3} b^{4} - 6 \, C a^{2} b^{5} - 8 \, B a b^{6} - 48 \, A b^{7}\right )} x^{7} + 7 \,{\left (58 \, F a^{5} b^{2} - 3 \, C a^{3} b^{4} - 4 \, B a^{2} b^{5} - 24 \, A a b^{6}\right )} x^{5} + 35 \,{\left (10 \, F a^{6} b - B a^{3} b^{4} - 6 \, A a^{2} b^{5}\right )} x^{3} + 105 \,{\left (F a^{7} - A a^{3} b^{4}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 105 \,{\left (F a^{4} b^{4} x^{8} + 4 \, F a^{5} b^{3} x^{6} + 6 \, F a^{6} b^{2} x^{4} + 4 \, F a^{7} b x^{2} + F a^{8}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{105 \,{\left (a^{4} b^{8} x^{8} + 4 \, a^{5} b^{7} x^{6} + 6 \, a^{6} b^{6} x^{4} + 4 \, a^{7} b^{5} x^{2} + a^{8} b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F*x^8 + D*x^6 + C*x^4 + B*x^2 + A)/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[-1/210*(2*((176*F*a^4*b^3 - 15*D*a^3*b^4 - 6*C*a^2*b^5 - 8*B*a*b^6 - 48*A*b^7)*
x^7 + 7*(58*F*a^5*b^2 - 3*C*a^3*b^4 - 4*B*a^2*b^5 - 24*A*a*b^6)*x^5 + 35*(10*F*a
^6*b - B*a^3*b^4 - 6*A*a^2*b^5)*x^3 + 105*(F*a^7 - A*a^3*b^4)*x)*sqrt(b*x^2 + a)
*sqrt(b) - 105*(F*a^4*b^4*x^8 + 4*F*a^5*b^3*x^6 + 6*F*a^6*b^2*x^4 + 4*F*a^7*b*x^
2 + F*a^8)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((a^4*b^8*x^8 +
4*a^5*b^7*x^6 + 6*a^6*b^6*x^4 + 4*a^7*b^5*x^2 + a^8*b^4)*sqrt(b)), -1/105*(((176
*F*a^4*b^3 - 15*D*a^3*b^4 - 6*C*a^2*b^5 - 8*B*a*b^6 - 48*A*b^7)*x^7 + 7*(58*F*a^
5*b^2 - 3*C*a^3*b^4 - 4*B*a^2*b^5 - 24*A*a*b^6)*x^5 + 35*(10*F*a^6*b - B*a^3*b^4
- 6*A*a^2*b^5)*x^3 + 105*(F*a^7 - A*a^3*b^4)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 105*
(F*a^4*b^4*x^8 + 4*F*a^5*b^3*x^6 + 6*F*a^6*b^2*x^4 + 4*F*a^7*b*x^2 + F*a^8)*arct
an(sqrt(-b)*x/sqrt(b*x^2 + a)))/((a^4*b^8*x^8 + 4*a^5*b^7*x^6 + 6*a^6*b^6*x^4 +
4*a^7*b^5*x^2 + a^8*b^4)*sqrt(-b))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F*x**8+D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228094, size = 275, normalized size = 1.29 \[ -\frac{{\left ({\left (x^{2}{\left (\frac{{\left (176 \, F a^{4} b^{6} - 15 \, D a^{3} b^{7} - 6 \, C a^{2} b^{8} - 8 \, B a b^{9} - 48 \, A b^{10}\right )} x^{2}}{a^{4} b^{7}} + \frac{7 \,{\left (58 \, F a^{5} b^{5} - 3 \, C a^{3} b^{7} - 4 \, B a^{2} b^{8} - 24 \, A a b^{9}\right )}}{a^{4} b^{7}}\right )} + \frac{35 \,{\left (10 \, F a^{6} b^{4} - B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )}}{a^{4} b^{7}}\right )} x^{2} + \frac{105 \,{\left (F a^{7} b^{3} - A a^{3} b^{7}\right )}}{a^{4} b^{7}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{F{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F*x^8 + D*x^6 + C*x^4 + B*x^2 + A)/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

-1/105*((x^2*((176*F*a^4*b^6 - 15*D*a^3*b^7 - 6*C*a^2*b^8 - 8*B*a*b^9 - 48*A*b^1
0)*x^2/(a^4*b^7) + 7*(58*F*a^5*b^5 - 3*C*a^3*b^7 - 4*B*a^2*b^8 - 24*A*a*b^9)/(a^
4*b^7)) + 35*(10*F*a^6*b^4 - B*a^3*b^7 - 6*A*a^2*b^8)/(a^4*b^7))*x^2 + 105*(F*a^
7*b^3 - A*a^3*b^7)/(a^4*b^7))*x/(b*x^2 + a)^(7/2) - F*ln(abs(-sqrt(b)*x + sqrt(b
*x^2 + a)))/b^(9/2)