3.579 \(\int \frac{A+B x^2}{x^4 (a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{2 b x (4 A b-3 a B)}{3 a^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0311549, antiderivative size = 82, 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, 191} \[ \frac{2 b x (4 A b-3 a B)}{3 a^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)),x]

[Out]

-A/(3*a*x^3*Sqrt[a + b*x^2]) + (4*A*b - 3*a*B)/(3*a^2*x*Sqrt[a + b*x^2]) + (2*b*(4*A*b - 3*a*B)*x)/(3*a^3*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^4 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac{A}{3 a x^3 \sqrt{a+b x^2}}-\frac{(4 A b-3 a B) \int \frac{1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{A}{3 a x^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}+\frac{(2 b (4 A b-3 a B)) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a^2}\\ &=-\frac{A}{3 a x^3 \sqrt{a+b x^2}}+\frac{4 A b-3 a B}{3 a^2 x \sqrt{a+b x^2}}+\frac{2 b (4 A b-3 a B) x}{3 a^3 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0176366, size = 61, normalized size = 0.74 \[ \frac{\left (a+2 b x^2\right ) (4 A b-3 a B)}{3 a^3 x \sqrt{a+b x^2}}-\frac{A}{3 a x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 58, normalized size = 0.7 \begin{align*} -{\frac{-8\,A{b}^{2}{x}^{4}+6\,B{x}^{4}ab-4\,aAb{x}^{2}+3\,B{x}^{2}{a}^{2}+A{a}^{2}}{3\,{x}^{3}{a}^{3}}{\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^4/(b*x^2+a)^(3/2),x)

[Out]

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

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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^4/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55447, size = 143, normalized size = 1.74 \begin{align*} -\frac{{\left (2 \,{\left (3 \, B a b - 4 \, A b^{2}\right )} x^{4} + A a^{2} +{\left (3 \, B a^{2} - 4 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 8.17542, size = 284, normalized size = 3.46 \begin{align*} A \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 ) + B \left (- \frac{1}{a \sqrt{b} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{2 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x^{2}} + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**4/(b*x**2+a)**(3/2),x)

[Out]

A*(-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**(1
1/2)*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)) + B*(-1/(a*sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)
) - 2*sqrt(b)/(a**2*sqrt(a/(b*x**2) + 1)))

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Giac [B]  time = 1.13369, size = 244, normalized size = 2.98 \begin{align*} -\frac{{\left (B a b - A b^{2}\right )} x}{\sqrt{b x^{2} + a} a^{3}} + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a \sqrt{b} - 3 \,{\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} + 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a b^{\frac{3}{2}} + 3 \, B a^{3} \sqrt{b} - 5 \, 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^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

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