3.597 \(\int \frac{A+B x^2}{x^6 \left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt{a+b x^2}}-\frac{8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]

[Out]

-A/(5*a*x^5*(a + b*x^2)^(3/2)) + (8*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(3/2))
- (2*b*(8*A*b - 5*a*B))/(5*a^3*x*(a + b*x^2)^(3/2)) - (8*b^2*(8*A*b - 5*a*B)*x)/
(15*a^4*(a + b*x^2)^(3/2)) - (16*b^2*(8*A*b - 5*a*B)*x)/(15*a^5*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.176362, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt{a+b x^2}}-\frac{8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)),x]

[Out]

-A/(5*a*x^5*(a + b*x^2)^(3/2)) + (8*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(3/2))
- (2*b*(8*A*b - 5*a*B))/(5*a^3*x*(a + b*x^2)^(3/2)) - (8*b^2*(8*A*b - 5*a*B)*x)/
(15*a^4*(a + b*x^2)^(3/2)) - (16*b^2*(8*A*b - 5*a*B)*x)/(15*a^5*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 18.9269, size = 141, normalized size = 0.97 \[ - \frac{A}{5 a x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{8 A b - 5 B a}{15 a^{2} x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{2 b \left (8 A b - 5 B a\right )}{5 a^{3} x \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{8 b^{2} x \left (8 A b - 5 B a\right )}{15 a^{4} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{16 b^{2} x \left (8 A b - 5 B a\right )}{15 a^{5} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**6/(b*x**2+a)**(5/2),x)

[Out]

-A/(5*a*x**5*(a + b*x**2)**(3/2)) + (8*A*b - 5*B*a)/(15*a**2*x**3*(a + b*x**2)**
(3/2)) - 2*b*(8*A*b - 5*B*a)/(5*a**3*x*(a + b*x**2)**(3/2)) - 8*b**2*x*(8*A*b -
5*B*a)/(15*a**4*(a + b*x**2)**(3/2)) - 16*b**2*x*(8*A*b - 5*B*a)/(15*a**5*sqrt(a
+ b*x**2))

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Mathematica [A]  time = 0.113889, size = 105, normalized size = 0.72 \[ \frac{-a^4 \left (3 A+5 B x^2\right )+a^3 \left (8 A b x^2+30 b B x^4\right )+24 a^2 b^2 x^4 \left (5 B x^2-2 A\right )+16 a b^3 x^6 \left (5 B x^2-12 A\right )-128 A b^4 x^8}{15 a^5 x^5 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)),x]

[Out]

(-128*A*b^4*x^8 + 16*a*b^3*x^6*(-12*A + 5*B*x^2) + 24*a^2*b^2*x^4*(-2*A + 5*B*x^
2) - a^4*(3*A + 5*B*x^2) + a^3*(8*A*b*x^2 + 30*b*B*x^4))/(15*a^5*x^5*(a + b*x^2)
^(3/2))

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Maple [A]  time = 0.01, size = 107, normalized size = 0.7 \[ -{\frac{128\,A{b}^{4}{x}^{8}-80\,Ba{b}^{3}{x}^{8}+192\,Aa{b}^{3}{x}^{6}-120\,B{a}^{2}{b}^{2}{x}^{6}+48\,A{a}^{2}{b}^{2}{x}^{4}-30\,B{a}^{3}b{x}^{4}-8\,A{a}^{3}b{x}^{2}+5\,B{a}^{4}{x}^{2}+3\,A{a}^{4}}{15\,{x}^{5}{a}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^6/(b*x^2+a)^(5/2),x)

[Out]

-1/15*(128*A*b^4*x^8-80*B*a*b^3*x^8+192*A*a*b^3*x^6-120*B*a^2*b^2*x^6+48*A*a^2*b
^2*x^4-30*B*a^3*b*x^4-8*A*a^3*b*x^2+5*B*a^4*x^2+3*A*a^4)/(b*x^2+a)^(3/2)/x^5/a^5

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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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317007, size = 174, normalized size = 1.19 \[ \frac{{\left (16 \,{\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 24 \,{\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} - 3 \, A a^{4} + 6 \,{\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} -{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^6),x, algorithm="fricas")

[Out]

1/15*(16*(5*B*a*b^3 - 8*A*b^4)*x^8 + 24*(5*B*a^2*b^2 - 8*A*a*b^3)*x^6 - 3*A*a^4
+ 6*(5*B*a^3*b - 8*A*a^2*b^2)*x^4 - (5*B*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(
a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5)

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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)/x**6/(b*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246527, size = 454, normalized size = 3.11 \[ \frac{x{\left (\frac{{\left (8 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}\right )} x^{2}}{a^{9} b} + \frac{3 \,{\left (3 \, B a^{6} b^{3} - 4 \, A a^{5} b^{4}\right )}}{a^{9} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} - 45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 150 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} + 240 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{5}{2}} + 250 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} - 490 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 170 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} + 320 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{5}{2}} + 40 \, B a^{5} b^{\frac{3}{2}} - 73 \, A a^{4} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^6),x, algorithm="giac")

[Out]

1/3*x*((8*B*a^5*b^4 - 11*A*a^4*b^5)*x^2/(a^9*b) + 3*(3*B*a^6*b^3 - 4*A*a^5*b^4)/
(a^9*b))/(b*x^2 + a)^(3/2) - 2/15*(30*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2
) - 45*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 150*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*B*a^2*b^(3/2) + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a*b^(5/2) + 250*(sq
rt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) - 490*(sqrt(b)*x - sqrt(b*x^2 + a))^4
*A*a^2*b^(5/2) - 170*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) + 320*(sqrt(b
)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(5/2) + 40*B*a^5*b^(3/2) - 73*A*a^4*b^(5/2))/((
(sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^4)