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

Optimal. Leaf size=150 \[ -\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{5 \sqrt{a+b x^2} (7 A b-4 a B)}{8 a^4 x^2}-\frac{5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt{a+b x^2}}-\frac{7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{4 a x^4 \left (a+b x^2\right )^{3/2}} \]

[Out]

-A/(4*a*x^4*(a + b*x^2)^(3/2)) - (7*A*b - 4*a*B)/(12*a^2*x^2*(a + b*x^2)^(3/2))
- (5*(7*A*b - 4*a*B))/(12*a^3*x^2*Sqrt[a + b*x^2]) + (5*(7*A*b - 4*a*B)*Sqrt[a +
b*x^2])/(8*a^4*x^2) - (5*b*(7*A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8
*a^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.295204, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{5 \sqrt{a+b x^2} (7 A b-4 a B)}{8 a^4 x^2}-\frac{5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt{a+b x^2}}-\frac{7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{4 a x^4 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(4*a*x^4*(a + b*x^2)^(3/2)) - (7*A*b - 4*a*B)/(12*a^2*x^2*(a + b*x^2)^(3/2))
- (5*(7*A*b - 4*a*B))/(12*a^3*x^2*Sqrt[a + b*x^2]) + (5*(7*A*b - 4*a*B)*Sqrt[a +
b*x^2])/(8*a^4*x^2) - (5*b*(7*A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8
*a^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.3753, size = 143, normalized size = 0.95 \[ - \frac{A}{4 a x^{4} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{7 A b - 4 B a}{12 a^{2} x^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{5 \left (7 A b - 4 B a\right )}{12 a^{3} x^{2} \sqrt{a + b x^{2}}} + \frac{5 \sqrt{a + b x^{2}} \left (7 A b - 4 B a\right )}{8 a^{4} x^{2}} - \frac{5 b \left (7 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(4*a*x**4*(a + b*x**2)**(3/2)) - (7*A*b - 4*B*a)/(12*a**2*x**2*(a + b*x**2)**
(3/2)) - 5*(7*A*b - 4*B*a)/(12*a**3*x**2*sqrt(a + b*x**2)) + 5*sqrt(a + b*x**2)*
(7*A*b - 4*B*a)/(8*a**4*x**2) - 5*b*(7*A*b - 4*B*a)*atanh(sqrt(a + b*x**2)/sqrt(
a))/(8*a**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.345666, size = 136, normalized size = 0.91 \[ \frac{\frac{\sqrt{a} \left (-6 a^3 \left (A+2 B x^2\right )+a^2 b x^2 \left (21 A-80 B x^2\right )+20 a b^2 x^4 \left (7 A-3 B x^2\right )+105 A b^3 x^6\right )}{x^4 \left (a+b x^2\right )^{3/2}}+15 b (4 a B-7 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+15 b \log (x) (7 A b-4 a B)}{24 a^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(105*A*b^3*x^6 + a^2*b*x^2*(21*A - 80*B*x^2) + 20*a*b^2*x^4*(7*A - 3*B
*x^2) - 6*a^3*(A + 2*B*x^2)))/(x^4*(a + b*x^2)^(3/2)) + 15*b*(7*A*b - 4*a*B)*Log
[x] + 15*b*(-7*A*b + 4*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(24*a^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 187, normalized size = 1.3 \[ -{\frac{A}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,Ab}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{b}^{2}A}{24\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{b}^{2}A}{8\,{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}-{\frac{B}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Bb}{6\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Bb}{2\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*A/a/x^4/(b*x^2+a)^(3/2)+7/8*A*b/a^2/x^2/(b*x^2+a)^(3/2)+35/24*A*b^2/a^3/(b*
x^2+a)^(3/2)+35/8*A*b^2/a^4/(b*x^2+a)^(1/2)-35/8*A*b^2/a^(9/2)*ln((2*a+2*a^(1/2)
*(b*x^2+a)^(1/2))/x)-1/2*B/a/x^2/(b*x^2+a)^(3/2)-5/6*B*b/a^2/(b*x^2+a)^(3/2)-5/2
*B*b/a^3/(b*x^2+a)^(1/2)+5/2*B*b/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)

_______________________________________________________________________________________

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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.259379, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (15 \,{\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 6 \, A a^{3} + 3 \,{\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} + 15 \,{\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + 2 \,{\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} +{\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{48 \,{\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )} \sqrt{a}}, -\frac{{\left (15 \,{\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 6 \, A a^{3} + 3 \,{\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 15 \,{\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + 2 \,{\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} +{\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{24 \,{\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(2*(15*(4*B*a*b^2 - 7*A*b^3)*x^6 + 20*(4*B*a^2*b - 7*A*a*b^2)*x^4 + 6*A*a
^3 + 3*(4*B*a^3 - 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a) + 15*((4*B*a*b^3 - 7*A
*b^4)*x^8 + 2*(4*B*a^2*b^2 - 7*A*a*b^3)*x^6 + (4*B*a^3*b - 7*A*a^2*b^2)*x^4)*log
(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2))/((a^4*b^2*x^8 + 2*a^5*b*x^
6 + a^6*x^4)*sqrt(a)), -1/24*((15*(4*B*a*b^2 - 7*A*b^3)*x^6 + 20*(4*B*a^2*b - 7*
A*a*b^2)*x^4 + 6*A*a^3 + 3*(4*B*a^3 - 7*A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a) -
15*((4*B*a*b^3 - 7*A*b^4)*x^8 + 2*(4*B*a^2*b^2 - 7*A*a*b^3)*x^6 + (4*B*a^3*b -
7*A*a^2*b^2)*x^4)*arctan(sqrt(-a)/sqrt(b*x^2 + a)))/((a^4*b^2*x^8 + 2*a^5*b*x^6
+ a^6*x^4)*sqrt(-a))]

_______________________________________________________________________________________

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)/x**5/(b*x**2+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.253074, size = 223, normalized size = 1.49 \[ -\frac{5 \,{\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{4}} - \frac{6 \,{\left (b x^{2} + a\right )} B a b + B a^{2} b - 9 \,{\left (b x^{2} + a\right )} A b^{2} - A a b^{2}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x^{2} + a} B a^{2} b - 11 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{2} + 13 \, \sqrt{b x^{2} + a} A a b^{2}}{8 \, a^{4} b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/8*(4*B*a*b - 7*A*b^2)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) - 1/3*(
6*(b*x^2 + a)*B*a*b + B*a^2*b - 9*(b*x^2 + a)*A*b^2 - A*a*b^2)/((b*x^2 + a)^(3/2
)*a^4) - 1/8*(4*(b*x^2 + a)^(3/2)*B*a*b - 4*sqrt(b*x^2 + a)*B*a^2*b - 11*(b*x^2
+ a)^(3/2)*A*b^2 + 13*sqrt(b*x^2 + a)*A*a*b^2)/(a^4*b^2*x^4)