3.554 \(\int \frac{\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^{11}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}+\frac{b^3 \sqrt{a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}+\frac{b^2 \sqrt{a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac{\left (a+b x^2\right )^{5/2} (3 A b-10 a B)}{80 a x^8}+\frac{b \left (a+b x^2\right )^{3/2} (3 A b-10 a B)}{96 a x^6}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}} \]

[Out]

(b^2*(3*A*b - 10*a*B)*Sqrt[a + b*x^2])/(128*a*x^4) + (b^3*(3*A*b - 10*a*B)*Sqrt[
a + b*x^2])/(256*a^2*x^2) + (b*(3*A*b - 10*a*B)*(a + b*x^2)^(3/2))/(96*a*x^6) +
((3*A*b - 10*a*B)*(a + b*x^2)^(5/2))/(80*a*x^8) - (A*(a + b*x^2)^(7/2))/(10*a*x^
10) - (b^4*(3*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(5/2))

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Rubi [A]  time = 0.367694, 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 \[ -\frac{b^4 (3 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}+\frac{b^3 \sqrt{a+b x^2} (3 A b-10 a B)}{256 a^2 x^2}+\frac{b^2 \sqrt{a+b x^2} (3 A b-10 a B)}{128 a x^4}+\frac{\left (a+b x^2\right )^{5/2} (3 A b-10 a B)}{80 a x^8}+\frac{b \left (a+b x^2\right )^{3/2} (3 A b-10 a B)}{96 a x^6}-\frac{A \left (a+b x^2\right )^{7/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^11,x]

[Out]

(b^2*(3*A*b - 10*a*B)*Sqrt[a + b*x^2])/(128*a*x^4) + (b^3*(3*A*b - 10*a*B)*Sqrt[
a + b*x^2])/(256*a^2*x^2) + (b*(3*A*b - 10*a*B)*(a + b*x^2)^(3/2))/(96*a*x^6) +
((3*A*b - 10*a*B)*(a + b*x^2)^(5/2))/(80*a*x^8) - (A*(a + b*x^2)^(7/2))/(10*a*x^
10) - (b^4*(3*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(5/2))

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Rubi in Sympy [A]  time = 30.8619, size = 173, normalized size = 0.92 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{10 a x^{10}} + \frac{b^{2} \sqrt{a + b x^{2}} \left (3 A b - 10 B a\right )}{128 a x^{4}} + \frac{b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 A b - 10 B a\right )}{96 a x^{6}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (3 A b - 10 B a\right )}{80 a x^{8}} + \frac{b^{3} \sqrt{a + b x^{2}} \left (3 A b - 10 B a\right )}{256 a^{2} x^{2}} - \frac{b^{4} \left (3 A b - 10 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{256 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**11,x)

[Out]

-A*(a + b*x**2)**(7/2)/(10*a*x**10) + b**2*sqrt(a + b*x**2)*(3*A*b - 10*B*a)/(12
8*a*x**4) + b*(a + b*x**2)**(3/2)*(3*A*b - 10*B*a)/(96*a*x**6) + (a + b*x**2)**(
5/2)*(3*A*b - 10*B*a)/(80*a*x**8) + b**3*sqrt(a + b*x**2)*(3*A*b - 10*B*a)/(256*
a**2*x**2) - b**4*(3*A*b - 10*B*a)*atanh(sqrt(a + b*x**2)/sqrt(a))/(256*a**(5/2)
)

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Mathematica [A]  time = 0.32626, size = 167, normalized size = 0.88 \[ -\frac{b^4 (3 A b-10 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{256 a^{5/2}}+\frac{b^4 \log (x) (3 A b-10 a B)}{256 a^{5/2}}+\sqrt{a+b x^2} \left (-\frac{b^3 (10 a B-3 A b)}{256 a^2 x^2}-\frac{a^2 A}{10 x^{10}}-\frac{b^2 (118 a B+3 A b)}{384 a x^4}-\frac{a (10 a B+21 A b)}{80 x^8}-\frac{b (170 a B+93 A b)}{480 x^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^11,x]

[Out]

(-(a^2*A)/(10*x^10) - (a*(21*A*b + 10*a*B))/(80*x^8) - (b*(93*A*b + 170*a*B))/(4
80*x^6) - (b^2*(3*A*b + 118*a*B))/(384*a*x^4) - (b^3*(-3*A*b + 10*a*B))/(256*a^2
*x^2))*Sqrt[a + b*x^2] + (b^4*(3*A*b - 10*a*B)*Log[x])/(256*a^(5/2)) - (b^4*(3*A
*b - 10*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(256*a^(5/2))

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Maple [B]  time = 0.013, size = 353, normalized size = 1.9 \[ -{\frac{A}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,Ab}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}A}{160\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{b}^{3}}{640\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,A{b}^{4}}{1280\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,A{b}^{5}}{1280\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{5}}{256\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{3\,A{b}^{5}}{256\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{B}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bb}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{b}^{2}}{192\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,B{b}^{4}}{384\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,B{b}^{4}}{128\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)*(B*x^2+A)/x^11,x)

[Out]

-1/10*A*(b*x^2+a)^(7/2)/a/x^10+3/80*A*b/a^2/x^8*(b*x^2+a)^(7/2)-1/160*A*b^2/a^3/
x^6*(b*x^2+a)^(7/2)-1/640*A*b^3/a^4/x^4*(b*x^2+a)^(7/2)-3/1280*A*b^4/a^5/x^2*(b*
x^2+a)^(7/2)+3/1280*A*b^5/a^5*(b*x^2+a)^(5/2)+1/256*A*b^5/a^4*(b*x^2+a)^(3/2)-3/
256*A*b^5/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+3/256*A*b^5/a^3*(b*x^2+a
)^(1/2)-1/8*B/a/x^8*(b*x^2+a)^(7/2)+1/48*B*b/a^2/x^6*(b*x^2+a)^(7/2)+1/192*B*b^2
/a^3/x^4*(b*x^2+a)^(7/2)+1/128*B*b^3/a^4/x^2*(b*x^2+a)^(7/2)-1/128*B*b^4/a^4*(b*
x^2+a)^(5/2)-5/384*B*b^4/a^3*(b*x^2+a)^(3/2)+5/128*B*b^4/a^(3/2)*ln((2*a+2*a^(1/
2)*(b*x^2+a)^(1/2))/x)-5/128*B*b^4/a^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((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.419708, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} x^{10} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (15 \,{\left (10 \, B a b^{3} - 3 \, A b^{4}\right )} x^{8} + 10 \,{\left (118 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{6} + 384 \, A a^{4} + 8 \,{\left (170 \, B a^{3} b + 93 \, A a^{2} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{4} + 21 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{7680 \, a^{\frac{5}{2}} x^{10}}, \frac{15 \,{\left (10 \, B a b^{4} - 3 \, A b^{5}\right )} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (15 \,{\left (10 \, B a b^{3} - 3 \, A b^{4}\right )} x^{8} + 10 \,{\left (118 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} x^{6} + 384 \, A a^{4} + 8 \,{\left (170 \, B a^{3} b + 93 \, A a^{2} b^{2}\right )} x^{4} + 48 \,{\left (10 \, B a^{4} + 21 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{3840 \, \sqrt{-a} a^{2} x^{10}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^11,x, algorithm="fricas")

[Out]

[-1/7680*(15*(10*B*a*b^4 - 3*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 - 3*A*b^4)*x^8 + 10*(118*B*a^2*b^2 + 3*A*a*
b^3)*x^6 + 384*A*a^4 + 8*(170*B*a^3*b + 93*A*a^2*b^2)*x^4 + 48*(10*B*a^4 + 21*A*
a^3*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(5/2)*x^10), 1/3840*(15*(10*B*a*b^4 - 3*
A*b^5)*x^10*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (15*(10*B*a*b^3 - 3*A*b^4)*x^8 +
10*(118*B*a^2*b^2 + 3*A*a*b^3)*x^6 + 384*A*a^4 + 8*(170*B*a^3*b + 93*A*a^2*b^2)*
x^4 + 48*(10*B*a^4 + 21*A*a^3*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*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)**(5/2)*(B*x**2+A)/x**11,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246964, size = 311, normalized size = 1.65 \[ -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 3 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{150 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} B a b^{5} + 580 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a^{2} b^{5} - 1280 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{3} b^{5} + 700 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x^{2} + a} B a^{5} b^{5} - 45 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} A b^{6} + 210 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A a b^{6} + 384 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a^{2} b^{6} - 210 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 45 \, \sqrt{b x^{2} + a} A a^{4} b^{6}}{a^{2} b^{5} x^{10}}}{3840 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^11,x, algorithm="giac")

[Out]

-1/3840*(15*(10*B*a*b^5 - 3*A*b^6)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^
2) + (150*(b*x^2 + a)^(9/2)*B*a*b^5 + 580*(b*x^2 + a)^(7/2)*B*a^2*b^5 - 1280*(b*
x^2 + a)^(5/2)*B*a^3*b^5 + 700*(b*x^2 + a)^(3/2)*B*a^4*b^5 - 150*sqrt(b*x^2 + a)
*B*a^5*b^5 - 45*(b*x^2 + a)^(9/2)*A*b^6 + 210*(b*x^2 + a)^(7/2)*A*a*b^6 + 384*(b
*x^2 + a)^(5/2)*A*a^2*b^6 - 210*(b*x^2 + a)^(3/2)*A*a^3*b^6 + 45*sqrt(b*x^2 + a)
*A*a^4*b^6)/(a^2*b^5*x^10))/b