3.581 \(\int \frac{A+B x^2}{x^6 (a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{8 b^2 x (6 A b-5 a B)}{15 a^4 \sqrt{a+b x^2}}-\frac{4 b (6 A b-5 a B)}{15 a^3 x \sqrt{a+b x^2}}+\frac{6 A b-5 a B}{15 a^2 x^3 \sqrt{a+b x^2}}-\frac{A}{5 a x^5 \sqrt{a+b x^2}} \]

[Out]

-A/(5*a*x^5*Sqrt[a + b*x^2]) + (6*A*b - 5*a*B)/(15*a^2*x^3*Sqrt[a + b*x^2]) - (4*b*(6*A*b - 5*a*B))/(15*a^3*x*
Sqrt[a + b*x^2]) - (8*b^2*(6*A*b - 5*a*B)*x)/(15*a^4*Sqrt[a + b*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0447022, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 191} \[ -\frac{8 b^2 x (6 A b-5 a B)}{15 a^4 \sqrt{a+b x^2}}-\frac{4 b (6 A b-5 a B)}{15 a^3 x \sqrt{a+b x^2}}+\frac{6 A b-5 a B}{15 a^2 x^3 \sqrt{a+b x^2}}-\frac{A}{5 a x^5 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x]

[Out]

-A/(5*a*x^5*Sqrt[a + b*x^2]) + (6*A*b - 5*a*B)/(15*a^2*x^3*Sqrt[a + b*x^2]) - (4*b*(6*A*b - 5*a*B))/(15*a^3*x*
Sqrt[a + b*x^2]) - (8*b^2*(6*A*b - 5*a*B)*x)/(15*a^4*Sqrt[a + b*x^2])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac{A}{5 a x^5 \sqrt{a+b x^2}}-\frac{(6 A b-5 a B) \int \frac{1}{x^4 \left (a+b x^2\right )^{3/2}} \, dx}{5 a}\\ &=-\frac{A}{5 a x^5 \sqrt{a+b x^2}}+\frac{6 A b-5 a B}{15 a^2 x^3 \sqrt{a+b x^2}}+\frac{(4 b (6 A b-5 a B)) \int \frac{1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=-\frac{A}{5 a x^5 \sqrt{a+b x^2}}+\frac{6 A b-5 a B}{15 a^2 x^3 \sqrt{a+b x^2}}-\frac{4 b (6 A b-5 a B)}{15 a^3 x \sqrt{a+b x^2}}-\frac{\left (8 b^2 (6 A b-5 a B)\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^3}\\ &=-\frac{A}{5 a x^5 \sqrt{a+b x^2}}+\frac{6 A b-5 a B}{15 a^2 x^3 \sqrt{a+b x^2}}-\frac{4 b (6 A b-5 a B)}{15 a^3 x \sqrt{a+b x^2}}-\frac{8 b^2 (6 A b-5 a B) x}{15 a^4 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0230303, size = 60, normalized size = 0.52 \[ \frac{x^2 \left (a^2-4 a b x^2-8 b^2 x^4\right ) (6 A b-5 a B)-3 a^3 A}{15 a^4 x^5 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x]

[Out]

(-3*a^3*A + (6*A*b - 5*a*B)*x^2*(a^2 - 4*a*b*x^2 - 8*b^2*x^4))/(15*a^4*x^5*Sqrt[a + b*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 83, normalized size = 0.7 \begin{align*} -{\frac{48\,A{b}^{3}{x}^{6}-40\,Ba{b}^{2}{x}^{6}+24\,Aa{b}^{2}{x}^{4}-20\,B{a}^{2}b{x}^{4}-6\,A{a}^{2}b{x}^{2}+5\,B{a}^{3}{x}^{2}+3\,A{a}^{3}}{15\,{x}^{5}{a}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x)

[Out]

-1/15*(48*A*b^3*x^6-40*B*a*b^2*x^6+24*A*a*b^2*x^4-20*B*a^2*b*x^4-6*A*a^2*b*x^2+5*B*a^3*x^2+3*A*a^3)/(b*x^2+a)^
(1/2)/x^5/a^4

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.57907, size = 194, normalized size = 1.69 \begin{align*} \frac{{\left (8 \,{\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{6} + 4 \,{\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} -{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 14.5231, size = 593, normalized size = 5.16 \begin{align*} A \left (- \frac{a^{5} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac{5 a^{3} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac{30 a^{2} b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac{40 a b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac{16 b^{\frac{29}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}}\right ) + B \left (- \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{12 a b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{8 b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**6/(b*x**2+a)**(3/2),x)

[Out]

A*(-a**5*b**(19/2)*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b
**12*x**10) - 5*a**3*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**1
1*x**8 + 5*a**4*b**12*x**10) - 30*a**2*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x
**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 40*a*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 +
15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 16*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(5*a**
7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10)) + B*(-a**3*b**(9/2)*sqrt(a/(b*x**
2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 3*a**2*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/
(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 12*a*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**5*b*
*4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 8*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a
**4*b**5*x**4 + 3*a**3*b**6*x**6))

________________________________________________________________________________________

Giac [B]  time = 1.15474, size = 397, normalized size = 3.45 \begin{align*} \frac{{\left (B a b^{2} - A b^{3}\right )} x}{\sqrt{b x^{2} + a} a^{4}} - \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} - 15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} + 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{5}{2}} + 160 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} - 240 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 110 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} + 150 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{5}{2}} + 25 \, B a^{5} b^{\frac{3}{2}} - 33 \, 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^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

(B*a*b^2 - A*b^3)*x/(sqrt(b*x^2 + a)*a^4) - 2/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2) - 15*(sqrt(b)
*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 90*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) + 90*(sqrt(b)*x - sqrt(b*
x^2 + a))^6*A*a*b^(5/2) + 160*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) - 240*(sqrt(b)*x - sqrt(b*x^2 + a)
)^4*A*a^2*b^(5/2) - 110*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*
a^3*b^(5/2) + 25*B*a^5*b^(3/2) - 33*A*a^4*b^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^3)