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

Optimal. Leaf size=108 \[ \frac{8 b x (2 A b-a B)}{3 a^4 \sqrt{a+b x^2}}+\frac{4 b x (2 A b-a B)}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac{2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.145678, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{8 b x (2 A b-a B)}{3 a^4 \sqrt{a+b x^2}}+\frac{4 b x (2 A b-a B)}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac{2 A b-a B}{a^2 x \left (a+b x^2\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9613, size = 99, normalized size = 0.92 \[ - \frac{A}{3 a x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{2 A b - B a}{a^{2} x \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{4 b x \left (2 A b - B a\right )}{3 a^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{8 b x \left (2 A b - B a\right )}{3 a^{4} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0841386, size = 79, normalized size = 0.73 \[ \frac{-a^3 \left (A+3 B x^2\right )+6 a^2 b x^2 \left (A-2 B x^2\right )-8 a b^2 x^4 \left (B x^2-3 A\right )+16 A b^3 x^6}{3 a^4 x^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(16*A*b^3*x^6 + 6*a^2*b*x^2*(A - 2*B*x^2) - 8*a*b^2*x^4*(-3*A + B*x^2) - a^3*(A
+ 3*B*x^2))/(3*a^4*x^3*(a + b*x^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 82, normalized size = 0.8 \[ -{\frac{-16\,A{b}^{3}{x}^{6}+8\,Ba{b}^{2}{x}^{6}-24\,Aa{b}^{2}{x}^{4}+12\,B{a}^{2}b{x}^{4}-6\,A{a}^{2}b{x}^{2}+3\,B{a}^{3}{x}^{2}+A{a}^{3}}{3\,{x}^{3}{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-16*A*b^3*x^6+8*B*a*b^2*x^6-24*A*a*b^2*x^4+12*B*a^2*b*x^4-6*A*a^2*b*x^2+3*
B*a^3*x^2+A*a^3)/(b*x^2+a)^(3/2)/x^3/a^4

_______________________________________________________________________________________

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 = 0.254945, size = 136, normalized size = 1.26 \[ -\frac{{\left (8 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 12 \,{\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{4} + A a^{3} + 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} \]

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]

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

_______________________________________________________________________________________

Sympy [A]  time = 141.831, size = 524, normalized size = 4.85 \[ A \left (- \frac{a^{4} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{5 a^{3} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{30 a^{2} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{40 a b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{16 b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}}\right ) + B \left (- \frac{3 a^{2} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac{12 a b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}} - \frac{8 b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-a**4*b**(19/2)*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 +
9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 5*a**3*b**(21/2)*x**2*sqrt(a/(b*x**2) +
1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**
8) + 30*a**2*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**1
0*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 40*a*b**(25/2)*x**6*sqrt(a/(b*
x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b*
*12*x**8) + 16*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b*
*10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8)) + B*(-3*a**2*b**(9/2)*sqrt(a/
(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4) - 12*a*b**(11/
2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**4)
- 8*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4 + 6*a**4*b**5*x**2 + 3*a**
3*b**6*x**4))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.244826, size = 302, normalized size = 2.8 \[ -\frac{x{\left (\frac{{\left (5 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{2}}{a^{7} b} + \frac{3 \,{\left (2 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3}\right )}}{a^{7} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a \sqrt{b} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} \sqrt{b} + 18 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{3}{2}} + 3 \, B a^{3} \sqrt{b} - 8 \, A a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{3}} \]

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]

-1/3*x*((5*B*a^4*b^3 - 8*A*a^3*b^4)*x^2/(a^7*b) + 3*(2*B*a^5*b^2 - 3*A*a^4*b^3)/
(a^7*b))/(b*x^2 + a)^(3/2) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a*sqrt(b)
- 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^
2*B*a^2*sqrt(b) + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2) + 3*B*a^3*sqrt(
b) - 8*A*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^3)