Integrand size = 22, antiderivative size = 110 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\sqrt {a+b x} \sqrt {c+d x}-2 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}} \] Output:
(b*x+a)^(1/2)*(d*x+c)^(1/2)-2*a^(1/2)*c^(1/2)*arctanh(c^(1/2)*(b*x+a)^(1/2 )/a^(1/2)/(d*x+c)^(1/2))+(a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/( d*x+c)^(1/2))/b^(1/2)/d^(1/2)
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\sqrt {a+b x} \sqrt {c+d x}-2 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}} \] Input:
Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x,x]
Output:
Sqrt[a + b*x]*Sqrt[c + d*x] - 2*Sqrt[a]*Sqrt[c]*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])] + ((b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d* x])/(Sqrt[d]*Sqrt[a + b*x])])/(Sqrt[b]*Sqrt[d])
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {112, 27, 175, 66, 104, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx\) |
\(\Big \downarrow \) 112 |
\(\displaystyle \sqrt {a+b x} \sqrt {c+d x}-\int -\frac {2 a c+(b c+a d) x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {2 a c+(b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx+\sqrt {a+b x} \sqrt {c+d x}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {1}{2} \left ((a d+b c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+2 a c \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\right )+\sqrt {a+b x} \sqrt {c+d x}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {1}{2} \left (2 a c \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+2 (a d+b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\sqrt {a+b x} \sqrt {c+d x}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{2} \left (4 a c \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 (a d+b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\sqrt {a+b x} \sqrt {c+d x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {2 (a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}-4 \sqrt {a} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )+\sqrt {a+b x} \sqrt {c+d x}\) |
Input:
Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x,x]
Output:
Sqrt[a + b*x]*Sqrt[c + d*x] + (-4*Sqrt[a]*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + (2*(b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[d]))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(82)=164\).
Time = 0.22 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.90
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (\sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a d +\sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b c -2 a c \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) \sqrt {d b}+2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\right )}{2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}}\) | \(209\) |
Input:
int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*((a*c)^(1/2)*ln(1/2*(2*b*d*x+2*((b*x+a)*(d *x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*d+(a*c)^(1/2)*ln(1/2*(2*b *d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b*c-2*a*c *ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*(d*b)^(1/ 2)+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1 /2)/(d*b)^(1/2)/(a*c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (82) = 164\).
Time = 0.32 (sec) , antiderivative size = 822, normalized size of antiderivative = 7.47 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x,x, algorithm="fricas")
Output:
[1/4*(2*sqrt(a*c)*b*d*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a* b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*sqrt(d*x + c)*b*d + (b*c + a*d) *sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2) *x))/(b*d), 1/2*(sqrt(a*c)*b*d*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2 *d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c ) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 2*sqrt(b*x + a)*sqrt(d*x + c)*b*d - (b *c + a*d)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)))/(b*d) , 1/4*(4*sqrt(-a*c)*b*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt (b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 4*sqrt(b*x + a)*sqrt(d*x + c)*b*d + (b*c + a*d)*sqrt(b*d)*log(8*b^2*d^2*x ^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqr t(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x))/(b*d), 1/2*(2*sqrt(-a *c)*b*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d *x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*sqrt(b*x + a) *sqrt(d*x + c)*b*d - (b*c + a*d)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a* d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c* d + a*b*d^2)*x)))/(b*d)]
\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x}\, dx \] Input:
integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x,x)
Output:
Integral(sqrt(a + b*x)*sqrt(c + d*x)/x, x)
Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Time = 15.15 (sec) , antiderivative size = 4785, normalized size of antiderivative = 43.50 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\text {Too large to display} \] Input:
int(((a + b*x)^(1/2)*(c + d*x)^(1/2))/x,x)
Output:
(((2*a*d + 2*b*c)*((a + b*x)^(1/2) - a^(1/2))^3)/(d*((c + d*x)^(1/2) - c^( 1/2))^3) + ((2*b^2*c + 2*a*b*d)*((a + b*x)^(1/2) - a^(1/2)))/(d^2*((c + d* x)^(1/2) - c^(1/2))) - (8*a^(1/2)*b*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^2) /(d*((c + d*x)^(1/2) - c^(1/2))^2))/(((a + b*x)^(1/2) - a^(1/2))^4/((c + d *x)^(1/2) - c^(1/2))^4 + b^2/d^2 - (2*b*((a + b*x)^(1/2) - a^(1/2))^2)/(d* ((c + d*x)^(1/2) - c^(1/2))^2)) + a^(1/2)*c^(1/2)*log(((c^(1/2)*(a + b*x)^ (1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c^(1/2) - (a^(1/2)*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))))/((c + d*x)^(1/2) - c^(1/2))) - a ^(1/2)*c^(1/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + d*x)^(1/2) - c^(1/2)) ) + (atan((((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2)*(a*d + b*c)*(((b*d)^(1/2 )*(a*d + b*c)*((2*(a^3*b^9*c^8*d + 9*a^4*b^8*c^7*d^2 - 10*a^5*b^7*c^6*d^3 - 10*a^6*b^6*c^5*d^4 + 9*a^7*b^5*c^4*d^5 + a^8*b^4*c^3*d^6))/(a^5*c^5*d^10 ) + ((b*d)^(1/2)*((2*(2*a^(5/2)*b^9*c^(15/2)*d - 2*a^(9/2)*b^7*c^(11/2)*d^ 3 - 2*a^(11/2)*b^6*c^(9/2)*d^4 + 2*a^(15/2)*b^4*c^(5/2)*d^6))/(a^5*c^5*d^1 0) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^3*b^9*c^9*d - 15*a^4*b^8*c^8*d^2 + 36*a^5*b^7*c^7*d^3 - 50*a^6*b^6*c^6*d^4 + 36*a^7*b^5*c^5*d^5 - 15*a^8*b^ 4*c^4*d^6 + 4*a^9*b^3*c^3*d^7))/(a^6*c^6*d^10*((c + d*x)^(1/2) - c^(1/2))) )*(a*d + b*c))/(b*d) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b^9*c^(19 /2)*d - 19*a^(9/2)*b^8*c^(17/2)*d^2 + 64*a^(11/2)*b^7*c^(15/2)*d^3 - 98*a^ (13/2)*b^6*c^(13/2)*d^4 + 64*a^(15/2)*b^5*c^(11/2)*d^5 - 19*a^(17/2)*b^...
Time = 0.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x} \, dx=\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, b d +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b d +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b d -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b d +\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a d +\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b c}{b d} \] Input:
int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x,x)
Output:
(sqrt(c + d*x)*sqrt(a + b*x)*b*d + sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*s qrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt (c + d*x))*b*d + sqrt(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a ) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b*d - sqrt (c)*sqrt(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)* sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*b*d + sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt (a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*d + sqrt(d)*sqrt(b)* log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b*c)/ (b*d)