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

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=-\frac {(A-B x) \sqrt {a+b x^2}}{x}+A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\sqrt {a} B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {827, 858, 223, 212, 272, 65, 214} \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\sqrt {a+b x^2} (A-B x)}{x}-\sqrt {a} B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]

[In]

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

[Out]

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

Rule 65

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 272

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 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B x) \sqrt {a+b x^2}}{x}-\frac {1}{2} \int \frac {-2 a B-2 A b x}{x \sqrt {a+b x^2}} \, dx \\ & = -\frac {(A-B x) \sqrt {a+b x^2}}{x}+(A b) \int \frac {1}{\sqrt {a+b x^2}} \, dx+(a B) \int \frac {1}{x \sqrt {a+b x^2}} \, dx \\ & = -\frac {(A-B x) \sqrt {a+b x^2}}{x}+(A b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} (a B) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = -\frac {(A-B x) \sqrt {a+b x^2}}{x}+A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {(a B) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b} \\ & = -\frac {(A-B x) \sqrt {a+b x^2}}{x}+A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\sqrt {a} B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=\frac {(-A+B x) \sqrt {a+b x^2}}{x}+2 \sqrt {a} B \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-A \sqrt {b} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {A \sqrt {b \,x^{2}+a}}{x}+A \sqrt {b}\, \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )+\sqrt {b \,x^{2}+a}\, B -B \sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\)	\(78\)
default	\(B \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )\)	\(103\)

[In]

int((B*x+A)*(b*x^2+a)^(1/2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 333, normalized size of antiderivative = 4.44 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=\left [\frac {A \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + B \sqrt {a} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, \sqrt {b x^{2} + a} {\left (B x - A\right )}}{2 \, x}, -\frac {2 \, A \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - B \sqrt {a} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, \sqrt {b x^{2} + a} {\left (B x - A\right )}}{2 \, x}, \frac {2 \, B \sqrt {-a} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + A \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, \sqrt {b x^{2} + a} {\left (B x - A\right )}}{2 \, x}, -\frac {A \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - B \sqrt {-a} x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - \sqrt {b x^{2} + a} {\left (B x - A\right )}}{x}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=- \frac {A \sqrt {a}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + A \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {A b x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - B \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=A \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - B \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \sqrt {b x^{2} + a} B - \frac {\sqrt {b x^{2} + a} A}{x} \]

[In]

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

[Out]

A*sqrt(b)*arcsinh(b*x/sqrt(a*b)) - B*sqrt(a)*arcsinh(a/(sqrt(a*b)*abs(x))) + sqrt(b*x^2 + a)*B - sqrt(b*x^2 +
a)*A/x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=\frac {2 \, B a \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - A \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \sqrt {b x^{2} + a} B + \frac {2 \, A a \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} \]

[In]

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

[Out]

2*B*a*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - A*sqrt(b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a
))) + sqrt(b*x^2 + a)*B + 2*A*a*sqrt(b)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)

Mupad [B] (verification not implemented)

Time = 6.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{x^2} \, dx=B\,\sqrt {b\,x^2+a}-\frac {A\,\sqrt {b\,x^2+a}}{x}-B\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )-\frac {A\,\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,x\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b\,x^2}{a}+1}} \]

[In]

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

[Out]

B*(a + b*x^2)^(1/2) - (A*(a + b*x^2)^(1/2))/x - B*a^(1/2)*atanh((a + b*x^2)^(1/2)/a^(1/2)) - (A*b^(1/2)*asin((
b^(1/2)*x*1i)/a^(1/2))*(a + b*x^2)^(1/2)*1i)/(a^(1/2)*((b*x^2)/a + 1)^(1/2))