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

Optimal. Leaf size=86 \[ -\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0671464, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ -\frac{3 A b-2 a B}{2 a^2 \sqrt{a+b x^2}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{A}{2 a x^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.0184403, size = 57, normalized size = 0.66 \[ \frac{x^2 (2 a B-3 A b) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^2}{a}+1\right )-a A}{2 a^2 x^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a*A) + (-3*A*b + 2*a*B)*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^2)/a])/(2*a^2*x^2*Sqrt[a + b*x^2])

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Maple [A]  time = 0.009, size = 109, normalized size = 1.3 \begin{align*}{\frac{B}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Ab}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7149, size = 506, normalized size = 5.88 \begin{align*} \left [-\frac{{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{a} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (A a^{2} -{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{4 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, \frac{{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (A a^{2} -{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(((2*B*a*b - 3*A*b^2)*x^4 + (2*B*a^2 - 3*A*a*b)*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2
*a)/x^2) + 2*(A*a^2 - (2*B*a^2 - 3*A*a*b)*x^2)*sqrt(b*x^2 + a))/(a^3*b*x^4 + a^4*x^2), 1/2*(((2*B*a*b - 3*A*b^
2)*x^4 + (2*B*a^2 - 3*A*a*b)*x^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (A*a^2 - (2*B*a^2 - 3*A*a*b)*x^2
)*sqrt(b*x^2 + a))/(a^3*b*x^4 + a^4*x^2)]

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Sympy [B]  time = 20.9669, size = 262, normalized size = 3.05 \begin{align*} A \left (- \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}}\right ) + B \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{3} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} + \frac{a^{2} b x^{2} \log{\left (\frac{b x^{2}}{a} \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}} - \frac{2 a^{2} b x^{2} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2 a^{\frac{9}{2}} + 2 a^{\frac{7}{2}} b x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) - 3*sqrt(b)/(2*a**2*x*sqrt(a/(b*x**2) + 1)) + 3*b*asinh(sqrt(a)/
(sqrt(b)*x))/(2*a**(5/2))) + B*(2*a**3*sqrt(1 + b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**3*log(b*x**2/a
)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**3*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**
2*b*x**2*log(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**2*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2
) + 2*a**(7/2)*b*x**2))

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Giac [A]  time = 1.13625, size = 134, normalized size = 1.56 \begin{align*} \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a^{2}} + \frac{2 \,{\left (b x^{2} + a\right )} B a - 2 \, B a^{2} - 3 \,{\left (b x^{2} + a\right )} A b + 2 \, A a b}{2 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{2} + a} a\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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