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

Optimal. Leaf size=71 \[ -\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{B \sqrt{a+c x^2}}{2 x^2}-\frac{B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

-(B*Sqrt[a + c*x^2])/(2*x^2) - (A*(a + c*x^2)^(3/2))/(3*a*x^3) - (B*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqr
t[a])

________________________________________________________________________________________

Rubi [A]  time = 0.0407413, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {807, 266, 47, 63, 208} \[ -\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{B \sqrt{a+c x^2}}{2 x^2}-\frac{B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*Sqrt[a + c*x^2])/(2*x^2) - (A*(a + c*x^2)^(3/2))/(3*a*x^3) - (B*c*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqr
t[a])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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) \sqrt{a+c x^2}}{x^4} \, dx &=-\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}+B \int \frac{\sqrt{a+c x^2}}{x^3} \, dx\\ &=-\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{B \sqrt{a+c x^2}}{2 x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}+\frac{1}{4} (B c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )\\ &=-\frac{B \sqrt{a+c x^2}}{2 x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )\\ &=-\frac{B \sqrt{a+c x^2}}{2 x^2}-\frac{A \left (a+c x^2\right )^{3/2}}{3 a x^3}-\frac{B c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0374725, size = 85, normalized size = 1.2 \[ \frac{-\left (a+c x^2\right ) \left (2 a A+3 a B x+2 A c x^2\right )-3 a B c x^3 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^2}{a}+1}\right )}{6 a x^3 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a + c*x^2)*(2*a*A + 3*a*B*x + 2*A*c*x^2)) - 3*a*B*c*x^3*Sqrt[1 + (c*x^2)/a]*ArcTanh[Sqrt[1 + (c*x^2)/a]])/
(6*a*x^3*Sqrt[a + c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 84, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,a{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Bc}{2\,a}\sqrt{c{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(1/2)/x^4,x)

[Out]

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

________________________________________________________________________________________

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+A)*(c*x^2+a)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.65789, size = 344, normalized size = 4.85 \begin{align*} \left [\frac{3 \, B \sqrt{a} c x^{3} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (2 \, A c x^{2} + 3 \, B a x + 2 \, A a\right )} \sqrt{c x^{2} + a}}{12 \, a x^{3}}, \frac{3 \, B \sqrt{-a} c x^{3} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (2 \, A c x^{2} + 3 \, B a x + 2 \, A a\right )} \sqrt{c x^{2} + a}}{6 \, a x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*B*sqrt(a)*c*x^3*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2*A*c*x^2 + 3*B*a*x + 2*A*a)
*sqrt(c*x^2 + a))/(a*x^3), 1/6*(3*B*sqrt(-a)*c*x^3*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (2*A*c*x^2 + 3*B*a*x + 2
*A*a)*sqrt(c*x^2 + a))/(a*x^3)]

________________________________________________________________________________________

Sympy [A]  time = 3.95676, size = 92, normalized size = 1.3 \begin{align*} - \frac{A \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{3 x^{2}} - \frac{A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{3 a} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{B c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{2 \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.14185, size = 193, normalized size = 2.72 \begin{align*} \frac{B c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B c + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A c^{\frac{3}{2}} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{2} c + 2 \, A a^{2} c^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) + 1/3*(3*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*c + 6*
(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*c^(3/2) - 3*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^2*c + 2*A*a^2*c^(3/2))/((sqrt(
c)*x - sqrt(c*x^2 + a))^2 - a)^3