Integrand size = 22, antiderivative size = 144 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=2 d \sqrt {a+b x} \sqrt {c+d x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}-\frac {\sqrt {c} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \] Output:
2*d*(b*x+a)^(1/2)*(d*x+c)^(1/2)-(b*x+a)^(1/2)*(d*x+c)^(3/2)/x-c^(1/2)*(3*a *d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(1/2)+d^(1/ 2)*(a*d+3*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2 )
Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \left (-c^2+d^2 x^2\right )}{x \sqrt {c+d x}}-\frac {\sqrt {c} (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+\frac {\sqrt {d} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \] Input:
Integrate[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^2,x]
Output:
(Sqrt[a + b*x]*(-c^2 + d^2*x^2))/(x*Sqrt[c + d*x]) - (Sqrt[c]*(b*c + 3*a*d )*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] + (Sqr t[d]*(3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x]) ])/Sqrt[b]
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {108, 27, 171, 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.
\(\displaystyle \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \int \frac {\sqrt {c+d x} (b c+3 a d+4 b d x)}{2 x \sqrt {a+b x}}dx-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {c+d x} (b c+3 a d+4 b d x)}{x \sqrt {a+b x}}dx-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {b (c (b c+3 a d)+d (3 b c+a d) x)}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}+4 d \sqrt {a+b x} \sqrt {c+d x}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\int \frac {c (b c+3 a d)+d (3 b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx+4 d \sqrt {a+b x} \sqrt {c+d x}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {1}{2} \left (d (a d+3 b c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+c (3 a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+4 d \sqrt {a+b x} \sqrt {c+d x}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {1}{2} \left (2 d (a d+3 b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+c (3 a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+4 d \sqrt {a+b x} \sqrt {c+d x}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{2} \left (2 d (a d+3 b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 c (3 a d+b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+4 d \sqrt {a+b x} \sqrt {c+d x}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \sqrt {d} (a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {2 \sqrt {c} (3 a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}+4 d \sqrt {a+b x} \sqrt {c+d x}\right )-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{x}\) |
Input:
Int[(Sqrt[a + b*x]*(c + d*x)^(3/2))/x^2,x]
Output:
-((Sqrt[a + b*x]*(c + d*x)^(3/2))/x) + (4*d*Sqrt[a + b*x]*Sqrt[c + d*x] - (2*Sqrt[c]*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] + (2*Sqrt[d]*(3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x] )/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b])/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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
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. \(297\) vs. \(2(112)=224\).
Time = 0.22 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a c d x \sqrt {d b}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b \,c^{2} x \sqrt {d b}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,d^{2} x \sqrt {a c}-3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b c d x \sqrt {a c}-2 d x \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 c \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\right )}{2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {d b}\, \sqrt {a c}}\) | \(298\) |
Input:
int((b*x+a)^(1/2)*(d*x+c)^(3/2)/x^2,x,method=_RETURNVERBOSE)
Output:
-1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a) *(d*x+c))^(1/2)+2*a*c)/x)*a*c*d*x*(d*b)^(1/2)+ln((a*d*x+b*c*x+2*(a*c)^(1/2 )*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b*c^2*x*(d*b)^(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^2*x*(a*c)^(1 /2)-3*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*d*x*(a*c)^(1/2)-2*d*x*(d*b)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c) )^(1/2)+2*c*(d*b)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/((b*x+a)*(d*x +c))^(1/2)/x/(d*b)^(1/2)/(a*c)^(1/2)
Time = 0.44 (sec) , antiderivative size = 893, normalized size of antiderivative = 6.20 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(3/2)/x^2,x, algorithm="fricas")
Output:
[1/4*((3*b*c + a*d)*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d /b) + 8*(b^2*c*d + a*b*d^2)*x) + (b*c + 3*a*d)*x*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sq rt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sq rt(b*x + a)*sqrt(d*x + c)*(d*x - c))/x, -1/4*(2*(3*b*c + a*d)*x*sqrt(-d/b) *arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/( b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) - (b*c + 3*a*d)*x*sqrt(c/a)*log((8 *a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2 *d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^ 2) - 4*sqrt(b*x + a)*sqrt(d*x + c)*(d*x - c))/x, 1/4*(2*(b*c + 3*a*d)*x*sq rt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sq rt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + (3*b*c + a*d)*x*sqrt(d /b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2 *c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)* x) + 4*sqrt(b*x + a)*sqrt(d*x + c)*(d*x - c))/x, 1/2*((b*c + 3*a*d)*x*sqrt (-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt (-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) - (3*b*c + a*d)*x*sqrt(-d/ b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b) /(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) + 2*sqrt(b*x + a)*sqrt(d*x + ...
\[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \] Input:
integrate((b*x+a)**(1/2)*(d*x+c)**(3/2)/x**2,x)
Output:
Integral(sqrt(a + b*x)*(c + d*x)**(3/2)/x**2, x)
Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(3/2)/x^2,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
Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (112) = 224\).
Time = 0.33 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.62 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\frac {1}{2} \, b {\left (\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} d {\left | b \right |}}{b^{3}} - \frac {2 \, {\left (\sqrt {b d} b c^{2} {\left | b \right |} + 3 \, \sqrt {b d} a c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b^{2}} - \frac {4 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a c d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} b} - \frac {{\left (3 \, \sqrt {b d} b c {\left | b \right |} + \sqrt {b d} a d {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b^{3}}\right )} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(3/2)/x^2,x, algorithm="giac")
Output:
1/2*b*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*d*abs(b)/b^3 - 2*(sqrt(b*d)*b*c^2*abs(b) + 3*sqrt(b*d)*a*c*d*abs(b))*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2 )/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b^2) - 4*(sqrt(b*d)*b^3*c^3*abs(b) - 2*sqrt(b*d)*a*b^2*c^2*d*abs(b) + sqrt(b*d)*a^2*b*c*d^2*abs(b) - sqrt(b*d) *(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b*c^2*a bs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*c*d*abs(b))/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b* d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt( b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt( b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*b) - (3*sqrt( b*d)*b*c*abs(b) + sqrt(b*d)*a*d*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqr t(b^2*c + (b*x + a)*b*d - a*b*d))^2)/b^3)
Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \] Input:
int(((a + b*x)^(1/2)*(c + d*x)^(3/2))/x^2,x)
Output:
int(((a + b*x)^(1/2)*(c + d*x)^(3/2))/x^2, x)
Time = 0.48 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^2} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c +2 \sqrt {d x +c}\, \sqrt {b x +a}\, a b d x +3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b d x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c x +3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b d x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c x -3 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a b d x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b^{2} c x +2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d x +6 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c x}{2 a b x} \] Input:
int((b*x+a)^(1/2)*(d*x+c)^(3/2)/x^2,x)
Output:
( - 2*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c + 2*sqrt(c + d*x)*sqrt(a + b*x)*a* b*d*x + 3*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))*a*b*d*x + sqrt (c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + s qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**2*c*x + 3*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))*a*b*d*x + 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**2*c*x - 3*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)*a*b*d*x - sqrt(c)*sqrt(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*s qrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*b**2*c*x + 2*sqrt(d)*sqrt(b)*log ((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*d*x + 6*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/s qrt(a*d - b*c))*a*b*c*x)/(2*a*b*x)