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

Optimal. Leaf size=261 \[ \frac{x^3 \left (a \left (162 a^3 F-71 a^2 b D+15 a b^2 C+6 b^3 B\right )+8 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )+4 A b^4\right )}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac{x^3 \left (A b^4-a \left (a^3 (-F)+a^2 b D-a b^2 C+b^3 B\right )\right )}{7 a b^4 \left (a+b x^2\right )^{7/2}}+\frac{(2 b D-9 a F) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}-\frac{x (b D-4 a F)}{b^5 \sqrt{a+b x^2}}+\frac{F x \sqrt{a+b x^2}}{2 b^5} \]

[Out]

((A*b^4 - a*(b^3*B - a*b^2*C + a^2*b*D - a^3*F))*x^3)/(7*a*b^4*(a + b*x^2)^(7/2)
) + ((4*A*b^4 + a*(3*b^3*B - 10*a*b^2*C + 17*a^2*b*D - 24*a^3*F))*x^3)/(35*a^2*b
^4*(a + b*x^2)^(5/2)) + ((8*A*b^4 + a*(6*b^3*B + 15*a*b^2*C - 71*a^2*b*D + 162*a
^3*F))*x^3)/(105*a^3*b^4*(a + b*x^2)^(3/2)) - ((b*D - 4*a*F)*x)/(b^5*Sqrt[a + b*
x^2]) + (F*x*Sqrt[a + b*x^2])/(2*b^5) + ((2*b*D - 9*a*F)*ArcTanh[(Sqrt[b]*x)/Sqr
t[a + b*x^2]])/(2*b^(11/2))

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Rubi [A]  time = 1.54255, antiderivative size = 257, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216 \[ \frac{x^3 \left (a \left (162 a^3 F-71 a^2 b D+15 a b^2 C+6 b^3 B\right )+8 A b^4\right )}{105 a^3 b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^3 \left (a \left (-24 a^3 F+17 a^2 b D-10 a b^2 C+3 b^3 B\right )+4 A b^4\right )}{35 a^2 b^4 \left (a+b x^2\right )^{5/2}}+\frac{x^3 \left (\frac{A}{a}-\frac{a^3 (-F)+a^2 b D-a b^2 C+b^3 B}{b^4}\right )}{7 \left (a+b x^2\right )^{7/2}}+\frac{(2 b D-9 a F) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{11/2}}-\frac{x (b D-4 a F)}{b^5 \sqrt{a+b x^2}}+\frac{F x \sqrt{a+b x^2}}{2 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(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^3)/(7*(a + b*x^2)^(7/2)) + ((
4*A*b^4 + a*(3*b^3*B - 10*a*b^2*C + 17*a^2*b*D - 24*a^3*F))*x^3)/(35*a^2*b^4*(a
+ b*x^2)^(5/2)) + ((8*A*b^4 + a*(6*b^3*B + 15*a*b^2*C - 71*a^2*b*D + 162*a^3*F))
*x^3)/(105*a^3*b^4*(a + b*x^2)^(3/2)) - ((b*D - 4*a*F)*x)/(b^5*Sqrt[a + b*x^2])
+ (F*x*Sqrt[a + b*x^2])/(2*b^5) + ((2*b*D - 9*a*F)*ArcTanh[(Sqrt[b]*x)/Sqrt[a +
b*x^2]])/(2*b^(11/2))

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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(x**2*(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 = 0.437863, size = 201, normalized size = 0.77 \[ \frac{x \left (945 a^7 F-210 a^6 b \left (D-15 F x^2\right )+14 a^5 b^2 x^2 \left (261 F x^2-50 D\right )+4 a^4 b^3 x^4 \left (396 F x^2-203 D\right )+a^3 b^4 x^6 \left (105 F x^2-352 D\right )+2 a^2 b^5 x^2 \left (35 A+21 B x^2+15 C x^4\right )+4 a b^6 x^4 \left (14 A+3 B x^2\right )+16 A b^7 x^6\right )}{210 a^3 b^5 \left (a+b x^2\right )^{7/2}}+\frac{(2 b D-9 a F) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(945*a^7*F + 16*A*b^7*x^6 + 4*a*b^6*x^4*(14*A + 3*B*x^2) - 210*a^6*b*(D - 15*
F*x^2) + a^3*b^4*x^6*(-352*D + 105*F*x^2) + 14*a^5*b^2*x^2*(-50*D + 261*F*x^2) +
4*a^4*b^3*x^4*(-203*D + 396*F*x^2) + 2*a^2*b^5*x^2*(35*A + 21*B*x^2 + 15*C*x^4)
))/(210*a^3*b^5*(a + b*x^2)^(7/2)) + ((2*b*D - 9*a*F)*Log[b*x + Sqrt[b]*Sqrt[a +
b*x^2]])/(2*b^(11/2))

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Maple [B]  time = 0.016, size = 478, normalized size = 1.8 \[ -{\frac{Dx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Cxa}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Cx}{7\,a{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{9\,Fax}{2\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,aC{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Ax}{35\,ab} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{Ax}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{3}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,Bx}{140\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{C{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Cx}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{D{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{D{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{D{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,Ax}{105\,{a}^{3}b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Bxa}{28\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{Bx}{35\,a{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{3\,Fa{x}^{3}}{2\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{F{x}^{9}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{9\,Fa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}}+{\frac{2\,Bx}{35\,{a}^{2}{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,Cx{a}^{2}}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{4\,Ax}{105\,{a}^{2}b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{D\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{9\,Fa{x}^{5}}{10\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{9\,Fa{x}^{7}}{14\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-D*x/b^4/(b*x^2+a)^(1/2)+3/56*C*a/b^3*x/(b*x^2+a)^(5/2)+1/7*C/a/b^3*x/(b*x^2+a)^
(1/2)+9/2*F*a/b^5*x/(b*x^2+a)^(1/2)-5/8*C*a/b^2*x^3/(b*x^2+a)^(7/2)+1/35*A/a/b*x
/(b*x^2+a)^(5/2)-1/7*A/b*x/(b*x^2+a)^(7/2)-1/4*B*x^3/b/(b*x^2+a)^(7/2)+3/140*B/b
^2*x/(b*x^2+a)^(5/2)-1/2*C*x^5/b/(b*x^2+a)^(7/2)+1/14*C/b^3*x/(b*x^2+a)^(3/2)-1/
7*D*x^7/b/(b*x^2+a)^(7/2)-1/5*D/b^2*x^5/(b*x^2+a)^(5/2)-1/3*D/b^3*x^3/(b*x^2+a)^
(3/2)+8/105*A/a^3/b*x/(b*x^2+a)^(1/2)-3/28*B*a/b^2*x/(b*x^2+a)^(7/2)+1/35*B/a/b^
2*x/(b*x^2+a)^(3/2)+3/2*F*a/b^4*x^3/(b*x^2+a)^(3/2)+1/2*F*x^9/b/(b*x^2+a)^(7/2)-
9/2*F*a/b^(11/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+2/35*B*x/a^2/b^2/(b*x^2+a)^(1/2)-
15/56*C*a^2/b^3*x/(b*x^2+a)^(7/2)+4/105*A/a^2/b*x/(b*x^2+a)^(3/2)+D/b^(9/2)*ln(x
*b^(1/2)+(b*x^2+a)^(1/2))+9/10*F*a/b^3*x^5/(b*x^2+a)^(5/2)+9/14*F*a/b^2*x^7/(b*x
^2+a)^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.604369, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (105 \, F a^{3} b^{4} x^{9} + 2 \,{\left (792 \, F a^{4} b^{3} - 176 \, D a^{3} b^{4} + 15 \, C a^{2} b^{5} + 6 \, B a b^{6} + 8 \, A b^{7}\right )} x^{7} + 14 \,{\left (261 \, F a^{5} b^{2} - 58 \, D a^{4} b^{3} + 3 \, B a^{2} b^{5} + 4 \, A a b^{6}\right )} x^{5} + 70 \,{\left (45 \, F a^{6} b - 10 \, D a^{5} b^{2} + A a^{2} b^{5}\right )} x^{3} + 105 \,{\left (9 \, F a^{7} - 2 \, D a^{6} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 105 \,{\left (9 \, F a^{8} - 2 \, D a^{7} b +{\left (9 \, F a^{4} b^{4} - 2 \, D a^{3} b^{5}\right )} x^{8} + 4 \,{\left (9 \, F a^{5} b^{3} - 2 \, D a^{4} b^{4}\right )} x^{6} + 6 \,{\left (9 \, F a^{6} b^{2} - 2 \, D a^{5} b^{3}\right )} x^{4} + 4 \,{\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{420 \,{\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )} \sqrt{b}}, \frac{{\left (105 \, F a^{3} b^{4} x^{9} + 2 \,{\left (792 \, F a^{4} b^{3} - 176 \, D a^{3} b^{4} + 15 \, C a^{2} b^{5} + 6 \, B a b^{6} + 8 \, A b^{7}\right )} x^{7} + 14 \,{\left (261 \, F a^{5} b^{2} - 58 \, D a^{4} b^{3} + 3 \, B a^{2} b^{5} + 4 \, A a b^{6}\right )} x^{5} + 70 \,{\left (45 \, F a^{6} b - 10 \, D a^{5} b^{2} + A a^{2} b^{5}\right )} x^{3} + 105 \,{\left (9 \, F a^{7} - 2 \, D a^{6} b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 105 \,{\left (9 \, F a^{8} - 2 \, D a^{7} b +{\left (9 \, F a^{4} b^{4} - 2 \, D a^{3} b^{5}\right )} x^{8} + 4 \,{\left (9 \, F a^{5} b^{3} - 2 \, D a^{4} b^{4}\right )} x^{6} + 6 \,{\left (9 \, F a^{6} b^{2} - 2 \, D a^{5} b^{3}\right )} x^{4} + 4 \,{\left (9 \, F a^{7} b - 2 \, D a^{6} b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{210 \,{\left (a^{3} b^{9} x^{8} + 4 \, a^{4} b^{8} x^{6} + 6 \, a^{5} b^{7} x^{4} + 4 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\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)*x^2/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[1/420*(2*(105*F*a^3*b^4*x^9 + 2*(792*F*a^4*b^3 - 176*D*a^3*b^4 + 15*C*a^2*b^5 +
6*B*a*b^6 + 8*A*b^7)*x^7 + 14*(261*F*a^5*b^2 - 58*D*a^4*b^3 + 3*B*a^2*b^5 + 4*A
*a*b^6)*x^5 + 70*(45*F*a^6*b - 10*D*a^5*b^2 + A*a^2*b^5)*x^3 + 105*(9*F*a^7 - 2*
D*a^6*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - 105*(9*F*a^8 - 2*D*a^7*b + (9*F*a^4*b^4 -
2*D*a^3*b^5)*x^8 + 4*(9*F*a^5*b^3 - 2*D*a^4*b^4)*x^6 + 6*(9*F*a^6*b^2 - 2*D*a^5*
b^3)*x^4 + 4*(9*F*a^7*b - 2*D*a^6*b^2)*x^2)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^
2 + a)*sqrt(b)))/((a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2 +
a^7*b^5)*sqrt(b)), 1/210*((105*F*a^3*b^4*x^9 + 2*(792*F*a^4*b^3 - 176*D*a^3*b^4
+ 15*C*a^2*b^5 + 6*B*a*b^6 + 8*A*b^7)*x^7 + 14*(261*F*a^5*b^2 - 58*D*a^4*b^3 +
3*B*a^2*b^5 + 4*A*a*b^6)*x^5 + 70*(45*F*a^6*b - 10*D*a^5*b^2 + A*a^2*b^5)*x^3 +
105*(9*F*a^7 - 2*D*a^6*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 105*(9*F*a^8 - 2*D*a^7*b
+ (9*F*a^4*b^4 - 2*D*a^3*b^5)*x^8 + 4*(9*F*a^5*b^3 - 2*D*a^4*b^4)*x^6 + 6*(9*F*
a^6*b^2 - 2*D*a^5*b^3)*x^4 + 4*(9*F*a^7*b - 2*D*a^6*b^2)*x^2)*arctan(sqrt(-b)*x/
sqrt(b*x^2 + a)))/((a^3*b^9*x^8 + 4*a^4*b^8*x^6 + 6*a^5*b^7*x^4 + 4*a^6*b^6*x^2
+ a^7*b^5)*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(x**2*(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 = 0.229006, size = 302, normalized size = 1.16 \[ \frac{{\left ({\left ({\left ({\left (\frac{105 \, F x^{2}}{b} + \frac{2 \,{\left (792 \, F a^{4} b^{7} - 176 \, D a^{3} b^{8} + 15 \, C a^{2} b^{9} + 6 \, B a b^{10} + 8 \, A b^{11}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac{14 \,{\left (261 \, F a^{5} b^{6} - 58 \, D a^{4} b^{7} + 3 \, B a^{2} b^{9} + 4 \, A a b^{10}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac{70 \,{\left (45 \, F a^{6} b^{5} - 10 \, D a^{5} b^{6} + A a^{2} b^{9}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac{105 \,{\left (9 \, F a^{7} b^{4} - 2 \, D a^{6} b^{5}\right )}}{a^{3} b^{9}}\right )} x}{210 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{{\left (9 \, F a - 2 \, D b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{11}{2}}} \]

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

[Out]

1/210*((((105*F*x^2/b + 2*(792*F*a^4*b^7 - 176*D*a^3*b^8 + 15*C*a^2*b^9 + 6*B*a*
b^10 + 8*A*b^11)/(a^3*b^9))*x^2 + 14*(261*F*a^5*b^6 - 58*D*a^4*b^7 + 3*B*a^2*b^9
+ 4*A*a*b^10)/(a^3*b^9))*x^2 + 70*(45*F*a^6*b^5 - 10*D*a^5*b^6 + A*a^2*b^9)/(a^
3*b^9))*x^2 + 105*(9*F*a^7*b^4 - 2*D*a^6*b^5)/(a^3*b^9))*x/(b*x^2 + a)^(7/2) + 1
/2*(9*F*a - 2*D*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(11/2)