3.520 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{11}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}+\frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]

[Out]

((7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(80*a*x^8) + (b*(7*A*b - 10*a*B)*Sqrt[a + b*x
^2])/(480*a^2*x^6) - (b^2*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(384*a^3*x^4) + (b^3
*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(256*a^4*x^2) - (A*(a + b*x^2)^(3/2))/(10*a*x
^10) - (b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(9/2))

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Rubi [A]  time = 0.371806, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{9/2}}+\frac{b^3 \sqrt{a+b x^2} (7 A b-10 a B)}{256 a^4 x^2}-\frac{b^2 \sqrt{a+b x^2} (7 A b-10 a B)}{384 a^3 x^4}+\frac{b \sqrt{a+b x^2} (7 A b-10 a B)}{480 a^2 x^6}+\frac{\sqrt{a+b x^2} (7 A b-10 a B)}{80 a x^8}-\frac{A \left (a+b x^2\right )^{3/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]

[Out]

((7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(80*a*x^8) + (b*(7*A*b - 10*a*B)*Sqrt[a + b*x
^2])/(480*a^2*x^6) - (b^2*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(384*a^3*x^4) + (b^3
*(7*A*b - 10*a*B)*Sqrt[a + b*x^2])/(256*a^4*x^2) - (A*(a + b*x^2)^(3/2))/(10*a*x
^10) - (b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(9/2))

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Rubi in Sympy [A]  time = 31.8797, size = 177, normalized size = 0.94 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{10 a x^{10}} + \frac{\sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{80 a x^{8}} + \frac{b \sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{480 a^{2} x^{6}} - \frac{b^{2} \sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{384 a^{3} x^{4}} + \frac{b^{3} \sqrt{a + b x^{2}} \left (7 A b - 10 B a\right )}{256 a^{4} x^{2}} - \frac{b^{4} \left (7 A b - 10 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{256 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**11,x)

[Out]

-A*(a + b*x**2)**(3/2)/(10*a*x**10) + sqrt(a + b*x**2)*(7*A*b - 10*B*a)/(80*a*x*
*8) + b*sqrt(a + b*x**2)*(7*A*b - 10*B*a)/(480*a**2*x**6) - b**2*sqrt(a + b*x**2
)*(7*A*b - 10*B*a)/(384*a**3*x**4) + b**3*sqrt(a + b*x**2)*(7*A*b - 10*B*a)/(256
*a**4*x**2) - b**4*(7*A*b - 10*B*a)*atanh(sqrt(a + b*x**2)/sqrt(a))/(256*a**(9/2
))

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Mathematica [A]  time = 0.251981, size = 169, normalized size = 0.89 \[ -\frac{b^4 (7 A b-10 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{256 a^{9/2}}+\frac{b^4 \log (x) (7 A b-10 a B)}{256 a^{9/2}}+\sqrt{a+b x^2} \left (-\frac{b^3 (10 a B-7 A b)}{256 a^4 x^2}+\frac{b^2 (10 a B-7 A b)}{384 a^3 x^4}-\frac{b (10 a B-7 A b)}{480 a^2 x^6}+\frac{-10 a B-A b}{80 a x^8}-\frac{A}{10 x^{10}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^11,x]

[Out]

(-A/(10*x^10) + (-(A*b) - 10*a*B)/(80*a*x^8) - (b*(-7*A*b + 10*a*B))/(480*a^2*x^
6) + (b^2*(-7*A*b + 10*a*B))/(384*a^3*x^4) - (b^3*(-7*A*b + 10*a*B))/(256*a^4*x^
2))*Sqrt[a + b*x^2] + (b^4*(7*A*b - 10*a*B)*Log[x])/(256*a^(9/2)) - (b^4*(7*A*b
- 10*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(256*a^(9/2))

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Maple [A]  time = 0.038, size = 281, normalized size = 1.5 \[ -{\frac{A}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,Ab}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}A}{96\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,A{b}^{3}}{128\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{4}}{256\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,A{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{7\,A{b}^{5}}{256\,{a}^{5}}\sqrt{b{x}^{2}+a}}-{\frac{B}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Bb}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,B{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,B{b}^{4}}{128\,{a}^{4}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^11,x)

[Out]

-1/10*A*(b*x^2+a)^(3/2)/a/x^10+7/80*A*b/a^2/x^8*(b*x^2+a)^(3/2)-7/96*A*b^2/a^3/x
^6*(b*x^2+a)^(3/2)+7/128*A*b^3/a^4/x^4*(b*x^2+a)^(3/2)-7/256*A*b^4/a^5/x^2*(b*x^
2+a)^(3/2)-7/256*A*b^5/a^(9/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+7/256*A*b^5
/a^5*(b*x^2+a)^(1/2)-1/8*B/a/x^8*(b*x^2+a)^(3/2)+5/48*B*b/a^2/x^6*(b*x^2+a)^(3/2
)-5/64*B*b^2/a^3/x^4*(b*x^2+a)^(3/2)+5/128*B*b^3/a^4/x^2*(b*x^2+a)^(3/2)+5/128*B
*b^4/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-5/128*B*b^4/a^4*(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((B*x^2 + A)*sqrt(b*x^2 + a)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.410732, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{10} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (15 \,{\left (10 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} - 10 \,{\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 384 \, A a^{4} + 8 \,{\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{7680 \, a^{\frac{9}{2}} x^{10}}, \frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (15 \,{\left (10 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} - 10 \,{\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 384 \, A a^{4} + 8 \,{\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{3840 \, \sqrt{-a} a^{4} x^{10}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^11,x, algorithm="fricas")

[Out]

[-1/7680*(15*(10*B*a*b^4 - 7*A*b^5)*x^10*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*
x^2 + a)*a)/x^2) + 2*(15*(10*B*a*b^3 - 7*A*b^4)*x^8 - 10*(10*B*a^2*b^2 - 7*A*a*b
^3)*x^6 + 384*A*a^4 + 8*(10*B*a^3*b - 7*A*a^2*b^2)*x^4 + 48*(10*B*a^4 + A*a^3*b)
*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(9/2)*x^10), 1/3840*(15*(10*B*a*b^4 - 7*A*b^5)
*x^10*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (15*(10*B*a*b^3 - 7*A*b^4)*x^8 - 10*(10
*B*a^2*b^2 - 7*A*a*b^3)*x^6 + 384*A*a^4 + 8*(10*B*a^3*b - 7*A*a^2*b^2)*x^4 + 48*
(10*B*a^4 + A*a^3*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^4*x^10)]

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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((B*x**2+A)*(b*x**2+a)**(1/2)/x**11,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239782, size = 311, normalized size = 1.65 \[ -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{150 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} B a b^{5} - 700 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 1280 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{3} b^{5} - 580 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x^{2} + a} B a^{5} b^{5} - 105 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} A b^{6} + 490 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A a b^{6} - 896 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a^{2} b^{6} + 790 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 105 \, \sqrt{b x^{2} + a} A a^{4} b^{6}}{a^{4} b^{5} x^{10}}}{3840 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^11,x, algorithm="giac")

[Out]

-1/3840*(15*(10*B*a*b^5 - 7*A*b^6)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^
4) + (150*(b*x^2 + a)^(9/2)*B*a*b^5 - 700*(b*x^2 + a)^(7/2)*B*a^2*b^5 + 1280*(b*
x^2 + a)^(5/2)*B*a^3*b^5 - 580*(b*x^2 + a)^(3/2)*B*a^4*b^5 - 150*sqrt(b*x^2 + a)
*B*a^5*b^5 - 105*(b*x^2 + a)^(9/2)*A*b^6 + 490*(b*x^2 + a)^(7/2)*A*a*b^6 - 896*(
b*x^2 + a)^(5/2)*A*a^2*b^6 + 790*(b*x^2 + a)^(3/2)*A*a^3*b^6 + 105*sqrt(b*x^2 +
a)*A*a^4*b^6)/(a^4*b^5*x^10))/b