3.564 \(\int \frac{A+B x^2}{x^4 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0197902, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {453, 264} \[ \frac{\sqrt{a+b x^2} (2 A b-3 a B)}{3 a^2 x}-\frac{A \sqrt{a+b x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0126149, size = 39, normalized size = 0.74 \[ -\frac{\sqrt{a+b x^2} \left (a \left (A+3 B x^2\right )-2 A b x^2\right )}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 36, normalized size = 0.7 \begin{align*} -{\frac{-2\,Ab{x}^{2}+3\,Ba{x}^{2}+Aa}{3\,{x}^{3}{a}^{2}}\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)^(1/2),x)

[Out]

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

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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)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61191, size = 81, normalized size = 1.53 \begin{align*} -\frac{{\left ({\left (3 \, B a - 2 \, A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a}}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.68327, size = 70, normalized size = 1.32 \begin{align*} - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a x^{2}} + \frac{2 A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{2}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**4/(b*x**2+a)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.13835, size = 162, normalized size = 3.06 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B \sqrt{b} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a \sqrt{b} + 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A b^{\frac{3}{2}} + 3 \, B a^{2} \sqrt{b} - 2 \, A a b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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