3.537 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{11}} \, dx\)

Optimal. Leaf size=184 \[ \frac{3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{7/2}}-\frac{3 b^3 \sqrt{a+b x^2} (A b-2 a B)}{256 a^3 x^2}+\frac{b^2 \sqrt{a+b x^2} (A b-2 a B)}{128 a^2 x^4}+\frac{\left (a+b x^2\right )^{3/2} (A b-2 a B)}{16 a x^8}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{32 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{10 a x^{10}} \]

[Out]

(b*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(32*a*x^6) + (b^2*(A*b - 2*a*B)*Sqrt[a + b*x^2
])/(128*a^2*x^4) - (3*b^3*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(256*a^3*x^2) + ((A*b -
2*a*B)*(a + b*x^2)^(3/2))/(16*a*x^8) - (A*(a + b*x^2)^(5/2))/(10*a*x^10) + (3*b
^4*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.367655, antiderivative size = 184, 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 \[ \frac{3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{7/2}}-\frac{3 b^3 \sqrt{a+b x^2} (A b-2 a B)}{256 a^3 x^2}+\frac{b^2 \sqrt{a+b x^2} (A b-2 a B)}{128 a^2 x^4}+\frac{\left (a+b x^2\right )^{3/2} (A b-2 a B)}{16 a x^8}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{32 a x^6}-\frac{A \left (a+b x^2\right )^{5/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^11,x]

[Out]

(b*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(32*a*x^6) + (b^2*(A*b - 2*a*B)*Sqrt[a + b*x^2
])/(128*a^2*x^4) - (3*b^3*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(256*a^3*x^2) + ((A*b -
2*a*B)*(a + b*x^2)^(3/2))/(16*a*x^8) - (A*(a + b*x^2)^(5/2))/(10*a*x^10) + (3*b
^4*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(256*a^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.9427, size = 170, normalized size = 0.92 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{10 a x^{10}} + \frac{b \sqrt{a + b x^{2}} \left (A b - 2 B a\right )}{32 a x^{6}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (\frac{A b}{2} - B a\right )}{8 a x^{8}} + \frac{b^{2} \sqrt{a + b x^{2}} \left (\frac{A b}{2} - B a\right )}{64 a^{2} x^{4}} - \frac{3 b^{3} \sqrt{a + b x^{2}} \left (\frac{A b}{2} - B a\right )}{128 a^{3} x^{2}} + \frac{3 b^{4} \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**11,x)

[Out]

-A*(a + b*x**2)**(5/2)/(10*a*x**10) + b*sqrt(a + b*x**2)*(A*b - 2*B*a)/(32*a*x**
6) + (a + b*x**2)**(3/2)*(A*b/2 - B*a)/(8*a*x**8) + b**2*sqrt(a + b*x**2)*(A*b/2
- B*a)/(64*a**2*x**4) - 3*b**3*sqrt(a + b*x**2)*(A*b/2 - B*a)/(128*a**3*x**2) +
3*b**4*(A*b/2 - B*a)*atanh(sqrt(a + b*x**2)/sqrt(a))/(128*a**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.290653, size = 164, normalized size = 0.89 \[ \frac{3 b^4 (A b-2 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{256 a^{7/2}}-\frac{3 b^4 \log (x) (A b-2 a B)}{256 a^{7/2}}+\sqrt{a+b x^2} \left (\frac{3 b^3 (2 a B-A b)}{256 a^3 x^2}-\frac{b^2 (2 a B-A b)}{128 a^2 x^4}+\frac{-10 a B-11 A b}{80 x^8}-\frac{b (30 a B+A b)}{160 a x^6}-\frac{a A}{10 x^{10}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^11,x]

[Out]

(-(a*A)/(10*x^10) + (-11*A*b - 10*a*B)/(80*x^8) - (b*(A*b + 30*a*B))/(160*a*x^6)
- (b^2*(-(A*b) + 2*a*B))/(128*a^2*x^4) + (3*b^3*(-(A*b) + 2*a*B))/(256*a^3*x^2)
)*Sqrt[a + b*x^2] - (3*b^4*(A*b - 2*a*B)*Log[x])/(256*a^(7/2)) + (3*b^4*(A*b - 2
*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(256*a^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.046, size = 317, normalized size = 1.7 \[ -{\frac{A}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{16\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}A}{32\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{3}}{128\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{4}}{256\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{b}^{5}}{256\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,A{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,A{b}^{5}}{256\,{a}^{4}}\sqrt{b{x}^{2}+a}}-{\frac{B}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{16\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{3\,B{b}^{4}}{128\,{a}^{3}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^11,x)

[Out]

-1/10*A*(b*x^2+a)^(5/2)/a/x^10+1/16*A*b/a^2/x^8*(b*x^2+a)^(5/2)-1/32*A*b^2/a^3/x
^6*(b*x^2+a)^(5/2)+1/128*A*b^3/a^4/x^4*(b*x^2+a)^(5/2)+1/256*A*b^4/a^5/x^2*(b*x^
2+a)^(5/2)-1/256*A*b^5/a^5*(b*x^2+a)^(3/2)+3/256*A*b^5/a^(7/2)*ln((2*a+2*a^(1/2)
*(b*x^2+a)^(1/2))/x)-3/256*A*b^5/a^4*(b*x^2+a)^(1/2)-1/8*B/a/x^8*(b*x^2+a)^(5/2)
+1/16*B*b/a^2/x^6*(b*x^2+a)^(5/2)-1/64*B*b^2/a^3/x^4*(b*x^2+a)^(5/2)-1/128*B*b^3
/a^4/x^2*(b*x^2+a)^(5/2)+1/128*B*b^4/a^4*(b*x^2+a)^(3/2)-3/128*B*b^4/a^(5/2)*ln(
(2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+3/128*B*b^4/a^3*(b*x^2+a)^(1/2)

_______________________________________________________________________________________

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)^(3/2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.420074, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (2 \, B a b^{4} - A b^{5}\right )} x^{10} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (15 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{8} - 10 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{6} - 128 \, A a^{4} - 8 \,{\left (30 \, B a^{3} b + A a^{2} b^{2}\right )} x^{4} - 16 \,{\left (10 \, B a^{4} + 11 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{2560 \, a^{\frac{7}{2}} x^{10}}, -\frac{15 \,{\left (2 \, B a b^{4} - A b^{5}\right )} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (15 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{8} - 10 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{6} - 128 \, A a^{4} - 8 \,{\left (30 \, B a^{3} b + A a^{2} b^{2}\right )} x^{4} - 16 \,{\left (10 \, B a^{4} + 11 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{1280 \, \sqrt{-a} a^{3} x^{10}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^11,x, algorithm="fricas")

[Out]

[-1/2560*(15*(2*B*a*b^4 - A*b^5)*x^10*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2
+ a)*a)/x^2) - 2*(15*(2*B*a*b^3 - A*b^4)*x^8 - 10*(2*B*a^2*b^2 - A*a*b^3)*x^6 -
128*A*a^4 - 8*(30*B*a^3*b + A*a^2*b^2)*x^4 - 16*(10*B*a^4 + 11*A*a^3*b)*x^2)*sq
rt(b*x^2 + a)*sqrt(a))/(a^(7/2)*x^10), -1/1280*(15*(2*B*a*b^4 - A*b^5)*x^10*arct
an(sqrt(-a)/sqrt(b*x^2 + a)) - (15*(2*B*a*b^3 - A*b^4)*x^8 - 10*(2*B*a^2*b^2 - A
*a*b^3)*x^6 - 128*A*a^4 - 8*(30*B*a^3*b + A*a^2*b^2)*x^4 - 16*(10*B*a^4 + 11*A*a
^3*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^10)]

_______________________________________________________________________________________

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)**(3/2)*(B*x**2+A)/x**11,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.252197, size = 286, normalized size = 1.55 \[ \frac{\frac{15 \,{\left (2 \, B a b^{5} - A b^{6}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{30 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} B a b^{5} - 140 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 140 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 30 \, \sqrt{b x^{2} + a} B a^{5} b^{5} - 15 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} A b^{6} + 70 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A a b^{6} - 128 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a^{2} b^{6} - 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 15 \, \sqrt{b x^{2} + a} A a^{4} b^{6}}{a^{3} b^{5} x^{10}}}{1280 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^11,x, algorithm="giac")

[Out]

1/1280*(15*(2*B*a*b^5 - A*b^6)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) +
(30*(b*x^2 + a)^(9/2)*B*a*b^5 - 140*(b*x^2 + a)^(7/2)*B*a^2*b^5 + 140*(b*x^2 +
a)^(3/2)*B*a^4*b^5 - 30*sqrt(b*x^2 + a)*B*a^5*b^5 - 15*(b*x^2 + a)^(9/2)*A*b^6 +
70*(b*x^2 + a)^(7/2)*A*a*b^6 - 128*(b*x^2 + a)^(5/2)*A*a^2*b^6 - 70*(b*x^2 + a)
^(3/2)*A*a^3*b^6 + 15*sqrt(b*x^2 + a)*A*a^4*b^6)/(a^3*b^5*x^10))/b