3.547 \(\int \frac{\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=146 \[ -\frac{\left (a+b x^2\right )^{5/2} (3 a B+4 A b)}{3 a x}+\frac{5 b x \left (a+b x^2\right )^{3/2} (3 a B+4 A b)}{12 a}+\frac{5}{8} b x \sqrt{a+b x^2} (3 a B+4 A b)+\frac{5}{8} a \sqrt{b} (3 a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{A \left (a+b x^2\right )^{7/2}}{3 a x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.168715, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (a+b x^2\right )^{5/2} (3 a B+4 A b)}{3 a x}+\frac{5 b x \left (a+b x^2\right )^{3/2} (3 a B+4 A b)}{12 a}+\frac{5}{8} b x \sqrt{a+b x^2} (3 a B+4 A b)+\frac{5}{8} a \sqrt{b} (3 a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{A \left (a+b x^2\right )^{7/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^4,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.1999, size = 138, normalized size = 0.95 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{3 a x^{3}} + \frac{5 a \sqrt{b} \left (4 A b + 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8} + \frac{5 b x \sqrt{a + b x^{2}} \left (4 A b + 3 B a\right )}{8} + \frac{5 b x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (4 A b + 3 B a\right )}{12 a} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (4 A b + 3 B a\right )}{3 a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.176564, size = 106, normalized size = 0.73 \[ \frac{\sqrt{a+b x^2} \left (-8 a^2 A+3 b x^4 (9 a B+4 A b)-8 a x^2 (3 a B+7 A b)+6 b^2 B x^6\right )}{24 x^3}+\frac{5}{8} a \sqrt{b} (3 a B+4 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^4,x]

[Out]

(Sqrt[a + b*x^2]*(-8*a^2*A - 8*a*(7*A*b + 3*a*B)*x^2 + 3*b*(4*A*b + 9*a*B)*x^4 +
6*b^2*B*x^6))/(24*x^3) + (5*a*Sqrt[b]*(4*A*b + 3*a*B)*Log[b*x + Sqrt[b]*Sqrt[a
+ b*x^2]])/8

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 204, normalized size = 1.4 \[ -{\frac{A}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Ab}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{4\,Ax{b}^{2}}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,Ax{b}^{2}}{3\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ax{b}^{2}}{2}\sqrt{b{x}^{2}+a}}+{\frac{5\,Aa}{2}{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }-{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{bBx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,bBx}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,Bxab}{8}\sqrt{b{x}^{2}+a}}+{\frac{15\,{a}^{2}B}{8}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)*(B*x^2+A)/x^4,x)

[Out]

-1/3*A*(b*x^2+a)^(7/2)/a/x^3-4/3*A*b/a^2/x*(b*x^2+a)^(7/2)+4/3*A*b^2/a^2*x*(b*x^
2+a)^(5/2)+5/3*A*b^2/a*x*(b*x^2+a)^(3/2)+5/2*A*b^2*x*(b*x^2+a)^(1/2)+5/2*A*b^(3/
2)*a*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-B/a/x*(b*x^2+a)^(7/2)+B*b/a*x*(b*x^2+a)^(5/2)
+5/4*B*b*x*(b*x^2+a)^(3/2)+15/8*B*b*a*x*(b*x^2+a)^(1/2)+15/8*B*b^(1/2)*a^2*ln(x*
b^(1/2)+(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)^(5/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.250702, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt{b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (6 \, B b^{2} x^{6} + 3 \,{\left (9 \, B a b + 4 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 8 \,{\left (3 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{48 \, x^{3}}, \frac{15 \,{\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt{-b} x^{3} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) +{\left (6 \, B b^{2} x^{6} + 3 \,{\left (9 \, B a b + 4 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 8 \,{\left (3 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{24 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(15*(3*B*a^2 + 4*A*a*b)*sqrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(
b)*x - a) + 2*(6*B*b^2*x^6 + 3*(9*B*a*b + 4*A*b^2)*x^4 - 8*A*a^2 - 8*(3*B*a^2 +
7*A*a*b)*x^2)*sqrt(b*x^2 + a))/x^3, 1/24*(15*(3*B*a^2 + 4*A*a*b)*sqrt(-b)*x^3*ar
ctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b))) + (6*B*b^2*x^6 + 3*(9*B*a*b + 4*A*b^2)*x^4
- 8*A*a^2 - 8*(3*B*a^2 + 7*A*a*b)*x^2)*sqrt(b*x^2 + a))/x^3]

_______________________________________________________________________________________

Sympy [A]  time = 34.9128, size = 299, normalized size = 2.05 \[ - \frac{2 A a^{\frac{3}{2}} b}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{A \sqrt{a} b^{2} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} - \frac{2 A \sqrt{a} b^{2} x}{\sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{A a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} + \frac{5 A a b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} - \frac{B a^{\frac{5}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + B a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x^{2}}{a}} - \frac{7 B a^{\frac{3}{2}} b x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B \sqrt{a} b^{2} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 B a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8} + \frac{B b^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**4,x)

[Out]

-2*A*a**(3/2)*b/(x*sqrt(1 + b*x**2/a)) + A*sqrt(a)*b**2*x*sqrt(1 + b*x**2/a)/2 -
2*A*sqrt(a)*b**2*x/sqrt(1 + b*x**2/a) - A*a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*
x**2) - A*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/3 + 5*A*a*b**(3/2)*asinh(sqrt(b)*x/sqr
t(a))/2 - B*a**(5/2)/(x*sqrt(1 + b*x**2/a)) + B*a**(3/2)*b*x*sqrt(1 + b*x**2/a)
- 7*B*a**(3/2)*b*x/(8*sqrt(1 + b*x**2/a)) + 3*B*sqrt(a)*b**2*x**3/(8*sqrt(1 + b*
x**2/a)) + 15*B*a**2*sqrt(b)*asinh(sqrt(b)*x/sqrt(a))/8 + B*b**3*x**5/(4*sqrt(a)
*sqrt(1 + b*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.249823, size = 321, normalized size = 2.2 \[ \frac{1}{8} \,{\left (2 \, B b^{2} x^{2} + \frac{9 \, B a b^{3} + 4 \, A b^{4}}{b^{2}}\right )} \sqrt{b x^{2} + a} x - \frac{5}{16} \,{\left (3 \, B a^{2} \sqrt{b} + 4 \, A a b^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} \sqrt{b} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} \sqrt{b} - 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{3}{2}} + 3 \, B a^{5} \sqrt{b} + 7 \, A a^{4} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^4,x, algorithm="giac")

[Out]

1/8*(2*B*b^2*x^2 + (9*B*a*b^3 + 4*A*b^4)/b^2)*sqrt(b*x^2 + a)*x - 5/16*(3*B*a^2*
sqrt(b) + 4*A*a*b^(3/2))*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/3*(3*(sqrt(b)*x
- sqrt(b*x^2 + a))^4*B*a^3*sqrt(b) + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^
(3/2) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*sqrt(b) - 12*(sqrt(b)*x - sqrt(b
*x^2 + a))^2*A*a^3*b^(3/2) + 3*B*a^5*sqrt(b) + 7*A*a^4*b^(3/2))/((sqrt(b)*x - sq
rt(b*x^2 + a))^2 - a)^3