3.566 \(\int \frac{A+B x^2}{x^6 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 b \sqrt{a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac{\sqrt{a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac{A \sqrt{a+b x^2}}{5 a x^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0346146, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ -\frac{2 b \sqrt{a+b x^2} (4 A b-5 a B)}{15 a^3 x}+\frac{\sqrt{a+b x^2} (4 A b-5 a B)}{15 a^2 x^3}-\frac{A \sqrt{a+b x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 264

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

Rubi steps

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

Mathematica [A]  time = 0.0191678, size = 62, normalized size = 0.74 \[ -\frac{\sqrt{a+b x^2} \left (a^2 \left (3 A+5 B x^2\right )-2 a b x^2 \left (2 A+5 B x^2\right )+8 A b^2 x^4\right )}{15 a^3 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 59, normalized size = 0.7 \begin{align*} -{\frac{8\,A{b}^{2}{x}^{4}-10\,B{x}^{4}ab-4\,aAb{x}^{2}+5\,B{x}^{2}{a}^{2}+3\,A{a}^{2}}{15\,{x}^{5}{a}^{3}}\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)^(1/2),x)

[Out]

-1/15*(b*x^2+a)^(1/2)*(8*A*b^2*x^4-10*B*a*b*x^4-4*A*a*b*x^2+5*B*a^2*x^2+3*A*a^2)/x^5/a^3

________________________________________________________________________________________

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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.71192, size = 130, normalized size = 1.55 \begin{align*} \frac{{\left (2 \,{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} -{\left (5 \, B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, a^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 2.34485, size = 355, normalized size = 4.23 \begin{align*} - \frac{3 A a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{2 A a^{3} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{3 A a^{2} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{12 A a b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{8 A b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a x^{2}} + \frac{2 B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**6/(b*x**2+a)**(1/2),x)

[Out]

-3*A*a**4*b**(9/2)*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 2*A*a**3
*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 3*A*a**2*b*
*(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 12*A*a*b**(15/
2)*x**6*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 8*A*b**(17/2)*x**8*
sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - B*sqrt(b)*sqrt(a/(b*x**2) +
1)/(3*a*x**2) + 2*B*b**(3/2)*sqrt(a/(b*x**2) + 1)/(3*a**2)

________________________________________________________________________________________

Giac [B]  time = 1.14161, size = 238, normalized size = 2.83 \begin{align*} \frac{4 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B b^{\frac{3}{2}} - 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a b^{\frac{3}{2}} + 40 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A b^{\frac{5}{2}} + 25 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{2} b^{\frac{3}{2}} - 20 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{5}{2}} - 5 \, B a^{3} b^{\frac{3}{2}} + 4 \, A a^{2} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

4/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*b^(3/2) - 35*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a*b^(3/2) + 40*(sqrt
(b)*x - sqrt(b*x^2 + a))^4*A*b^(5/2) + 25*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*b^(3/2) - 20*(sqrt(b)*x - sqrt
(b*x^2 + a))^2*A*a*b^(5/2) - 5*B*a^3*b^(3/2) + 4*A*a^2*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5