\(\int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx\) [44]

Optimal result

Integrand size = 35, antiderivative size = 156 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=-\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x (a+b x)}+\frac {a \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{a+b x}-\frac {b \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a+b x} \]

[Out]

-b*arctanh((d*x^2+c)^(1/2)/c^(1/2))*c^(1/2)*((b*x+a)^2)^(1/2)/(b*x+a)+a*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))*d^(
1/2)*((b*x+a)^2)^(1/2)/(b*x+a)-(-b*x+a)*((b*x+a)^2)^(1/2)*(d*x^2+c)^(1/2)/x/(b*x+a)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1015, 827, 858, 223, 212, 272, 65, 214} \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\frac {a \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{a+b x}-\frac {b \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a+b x}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a-b x) \sqrt {c+d x^2}}{x (a+b x)} \]

[In]

Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^2,x]

[Out]

-(((a - b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/(x*(a + b*x))) + (a*Sqrt[d]*Sqrt[a^2 + 2*a*b*x + b
^2*x^2]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(a + b*x) - (b*Sqrt[c]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[Sqr
t[c + d*x^2]/Sqrt[c]])/(a + b*x)

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]

Rule 1015

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :
> Dist[(a + b*x + c*x^2)^FracPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(g + h*x)^m*(b + 2*c*
x)^(2*p)*(d + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, f, g, h, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (2 a b+2 b^2 x\right ) \sqrt {c+d x^2}}{x^2} \, dx}{2 a b+2 b^2 x} \\ & = -\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {-4 b^2 c-4 a b d x}{x \sqrt {c+d x^2}} \, dx}{2 \left (2 a b+2 b^2 x\right )} \\ & = -\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x (a+b x)}+\frac {\left (2 b^2 c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{x \sqrt {c+d x^2}} \, dx}{2 a b+2 b^2 x}+\frac {\left (2 a b d \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 a b+2 b^2 x} \\ & = -\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x (a+b x)}+\frac {\left (b^2 c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a b+2 b^2 x}+\frac {\left (2 a b d \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a b+2 b^2 x} \\ & = -\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x (a+b x)}+\frac {a \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{a+b x}+\frac {\left (2 b^2 c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{d \left (2 a b+2 b^2 x\right )} \\ & = -\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x (a+b x)}+\frac {a \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{a+b x}-\frac {b \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a+b x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\frac {\sqrt {(a+b x)^2} \left ((-a+b x) \sqrt {c+d x^2}+2 b \sqrt {c} x \text {arctanh}\left (\frac {\sqrt {d} x-\sqrt {c+d x^2}}{\sqrt {c}}\right )-a \sqrt {d} x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{x (a+b x)} \]

[In]

Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {a \sqrt {d \,x^{2}+c}\, \sqrt {\left (b x +a \right )^{2}}}{x \left (b x +a \right )}+\frac {\left (a \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+\sqrt {d \,x^{2}+c}\, b -b \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\)	\(112\)
default	\(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (-a \,d^{\frac {3}{2}} x^{2} \sqrt {d \,x^{2}+c}+c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) \sqrt {d}\, b x +a \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {d}-\sqrt {d \,x^{2}+c}\, \sqrt {d}\, b c x -\ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) a c d x \right )}{c x \sqrt {d}}\)	\(120\)

[In]

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

[Out]

-a*(d*x^2+c)^(1/2)/x*((b*x+a)^2)^(1/2)/(b*x+a)+(a*d^(1/2)*ln(d^(1/2)*x+(d*x^2+c)^(1/2))+(d*x^2+c)^(1/2)*b-b*c^
(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x))*((b*x+a)^2)^(1/2)/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.13 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\left [\frac {a \sqrt {d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + b \sqrt {c} x \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {d x^{2} + c} {\left (b x - a\right )}}{2 \, x}, -\frac {2 \, a \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - b \sqrt {c} x \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {d x^{2} + c} {\left (b x - a\right )}}{2 \, x}, \frac {2 \, b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + a \sqrt {d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, \sqrt {d x^{2} + c} {\left (b x - a\right )}}{2 \, x}, -\frac {a \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - \sqrt {d x^{2} + c} {\left (b x - a\right )}}{x}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\int \frac {\sqrt {c + d x^{2}} \sqrt {\left (a + b x\right )^{2}}}{x^{2}}\, dx \]

[In]

integrate(((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2)/x**2,x)

[Out]

Integral(sqrt(c + d*x**2)*sqrt((a + b*x)**2)/x**2, x)

Maxima [F]

\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} \sqrt {{\left (b x + a\right )}^{2}}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2)/x^2, x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\frac {2 \, b c \arctan \left (-\frac {\sqrt {d} x - \sqrt {d x^{2} + c}}{\sqrt {-c}}\right ) \mathrm {sgn}\left (b x + a\right )}{\sqrt {-c}} - a \sqrt {d} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \sqrt {d x^{2} + c} b \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, a c \sqrt {d} \mathrm {sgn}\left (b x + a\right )}{{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c} \]

[In]

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

[Out]

2*b*c*arctan(-(sqrt(d)*x - sqrt(d*x^2 + c))/sqrt(-c))*sgn(b*x + a)/sqrt(-c) - a*sqrt(d)*log(abs(-sqrt(d)*x + s
qrt(d*x^2 + c)))*sgn(b*x + a) + sqrt(d*x^2 + c)*b*sgn(b*x + a) + 2*a*c*sqrt(d)*sgn(b*x + a)/((sqrt(d)*x - sqrt
(d*x^2 + c))^2 - c)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{x^2} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+c}}{x^2} \,d x \]

[In]

int((((a + b*x)^2)^(1/2)*(c + d*x^2)^(1/2))/x^2,x)

[Out]

int((((a + b*x)^2)^(1/2)*(c + d*x^2)^(1/2))/x^2, x)