Integrand size = 22, antiderivative size = 172 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\frac {\left (\frac {b^2}{a^2}-\frac {d^2}{c^2}\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 a c^2 x^2}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3}-\frac {(b c-a d)^2 (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{5/2} c^{5/2}} \]
-1/3*(b*x+a)^(3/2)*(d*x+c)^(3/2)/a/c/x^3-1/8*(-a*d+b*c)^2*(a*d+b*c)*arctan h(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(5/2)/c^(5/2)+1/4*(a*d+b* c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/c^2/x^2+1/8*(b^2/a^2-d^2/c^2)*(b*x+a)^(1/ 2)*(d*x+c)^(1/2)/x
Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 b^2 c^2 x^2-2 a b c x (c+d x)+a^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )}{24 a^2 c^2 x^3}-\frac {(b c-a d)^2 (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 a^{5/2} c^{5/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(3*b^2*c^2*x^2 - 2*a*b*c*x*(c + d*x) + a^2*(- 8*c^2 - 2*c*d*x + 3*d^2*x^2)))/(24*a^2*c^2*x^3) - ((b*c - a*d)^2*(b*c + a* d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(8*a^(5/2)*c^ (5/2))
Time = 0.25 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {107, 105, 105, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(a d+b c) \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^3}dx}{2 a c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {(b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 a c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {(b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 a c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {(b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 a c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 c x^2}\right )}{2 a c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 a c x^3}\) |
-1/3*((a + b*x)^(3/2)*(c + d*x)^(3/2))/(a*c*x^3) - ((b*c + a*d)*(-1/2*(Sqr t[a + b*x]*(c + d*x)^(3/2))/(c*x^2) + ((b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[ c + d*x])/(a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S qrt[c + d*x])])/(a^(3/2)*Sqrt[c])))/(4*c)))/(2*a*c)
3.6.54.3.1 Defintions of rubi rules used
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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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. \(407\) vs. \(2(140)=280\).
Time = 1.50 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.37
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x +4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x +16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{2} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{3} \sqrt {a c}}\) | \(408\) |
-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^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^3*d^3*x^3-3*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^2*b*c*d^2*x^3-3*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*b^2*c^2*d*x^3+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)*b^3*c^3*x^3-6*(a*c) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*x^2+4*(a*c)^(1/2)*((b*x+a)*(d*x+c)) ^(1/2)*a*b*c*d*x^2-6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x^2+4*(a* c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d*x+4*(a*c)^(1/2)*((b*x+a)*(d*x+c)) ^(1/2)*a*b*c^2*x+16*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))/((b*x+a)* (d*x+c))^(1/2)/x^3/(a*c)^(1/2)
Time = 0.42 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.52 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\left [\frac {3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{3} c^{3} - {\left (3 \, a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} + 2 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{3} c^{3} x^{3}}, \frac {3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (8 \, a^{3} c^{3} - {\left (3 \, a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} + 2 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{3} c^{3} x^{3}}\right ] \]
[1/96*(3*(b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*sqrt(a*c)*x^3*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*(8*a^3*c^3 - (3*a*b^2*c^3 - 2*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 + 2*(a^2* b*c^3 + a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^3*x^3), 1/48*(3* (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*sqrt(-a*c)*x^3*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*(8*a^3*c^3 - (3*a*b^2*c^3 - 2*a^ 2*b*c^2*d + 3*a^3*c*d^2)*x^2 + 2*(a^2*b*c^3 + a^3*c^2*d)*x)*sqrt(b*x + a)* sqrt(d*x + c))/(a^3*c^3*x^3)]
\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{4}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\text {Exception raised: ValueError} \]
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. 2104 vs. \(2 (140) = 280\).
Time = 2.24 (sec) , antiderivative size = 2104, normalized size of antiderivative = 12.23 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\text {Too large to display} \]
-1/24*(3*(sqrt(b*d)*b^4*c^3*abs(b) - sqrt(b*d)*a*b^3*c^2*d*abs(b) - sqrt(b *d)*a^2*b^2*c*d^2*abs(b) + sqrt(b*d)*a^3*b*d^3*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)*a^2*b*c^2) - 2*(3*sqrt(b*d)*b^14*c^ 8*abs(b) - 20*sqrt(b*d)*a*b^13*c^7*d*abs(b) + 60*sqrt(b*d)*a^2*b^12*c^6*d^ 2*abs(b) - 108*sqrt(b*d)*a^3*b^11*c^5*d^3*abs(b) + 130*sqrt(b*d)*a^4*b^10* c^4*d^4*abs(b) - 108*sqrt(b*d)*a^5*b^9*c^3*d^5*abs(b) + 60*sqrt(b*d)*a^6*b ^8*c^2*d^6*abs(b) - 20*sqrt(b*d)*a^7*b^7*c*d^7*abs(b) + 3*sqrt(b*d)*a^8*b^ 6*d^8*abs(b) - 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^12*c^7*abs(b) + 51*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a ) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^11*c^6*d*abs(b) - 63*sqrt(b *d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2* b^10*c^5*d^2*abs(b) + 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^9*c^4*d^3*abs(b) + 27*sqrt(b*d)*(sqrt(b*d )*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^8*c^3*d^4*a bs(b) - 63*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^7*c^2*d^5*abs(b) + 51*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a ) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^6*c*d^6*abs(b) - 15*sqrt( b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7 *b^5*d^7*abs(b) + 30*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + ...
Time = 96.32 (sec) , antiderivative size = 1459, normalized size of antiderivative = 8.48 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^4} \, dx=\text {Too large to display} \]
((((a + b*x)^(1/2) - a^(1/2))^4*((5*b^6*c^4)/64 + (5*a^4*b^2*d^4)/64 + (9* a^3*b^3*c*d^3)/32 - (3*a^2*b^4*c^2*d^2)/32 + (9*a*b^5*c^3*d)/32))/(a^3*c^3 *d^3*((c + d*x)^(1/2) - c^(1/2))^4) - (((a + b*x)^(1/2) - a^(1/2))^3*((17* b^6*c^3)/192 + (17*a^3*b^3*d^3)/192))/(a^(5/2)*c^(5/2)*d^3*((c + d*x)^(1/2 ) - c^(1/2))^3) - b^6/(192*a*c*d^3) - (((a + b*x)^(1/2) - a^(1/2))^5*((13* a^4*b^2*c*d^4)/32 - (a^5*b*d^5)/64 - (b^6*c^5)/64 + (a^2*b^4*c^3*d^2)/16 + (a^3*b^3*c^2*d^3)/16 + (13*a*b^5*c^4*d)/32))/(a^(7/2)*c^(7/2)*d^3*((c + d *x)^(1/2) - c^(1/2))^5) + (((a + b*x)^(1/2) - a^(1/2))^2*((b^6*c^2)/64 + ( a^2*b^4*d^2)/64 - (a*b^5*c*d)/16))/(a^2*c^2*d^3*((c + d*x)^(1/2) - c^(1/2) )^2) + (((a + b*x)^(1/2) - a^(1/2))^7*((3*a^5*d^5)/64 + (3*b^5*c^5)/64 + ( 3*a^2*b^3*c^3*d^2)/64 + (3*a^3*b^2*c^2*d^3)/64 - (15*a*b^4*c^4*d)/64 - (15 *a^4*b*c*d^4)/64))/(a^(7/2)*c^(7/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^7) + ( ((b^6*c)/64 + (a*b^5*d)/64)*((a + b*x)^(1/2) - a^(1/2)))/(a^(3/2)*c^(3/2)* d^3*((c + d*x)^(1/2) - c^(1/2))) + (((a + b*x)^(1/2) - a^(1/2))^6*((15*a^2 *b^4*c^4*d^2)/32 - (5*b^6*c^6)/192 - (5*a^6*d^6)/192 - (3*a^3*b^3*c^3*d^3) /8 + (15*a^4*b^2*c^2*d^4)/32 + (3*a*b^5*c^5*d)/32 + (3*a^5*b*c*d^5)/32))/( a^4*c^4*d^3*((c + d*x)^(1/2) - c^(1/2))^6) - (((a + b*x)^(1/2) - a^(1/2))^ 8*((a^4*d^4)/64 + (b^4*c^4)/64 + (7*a^2*b^2*c^2*d^2)/64 - (3*a*b^3*c^3*d)/ 32 - (3*a^3*b*c*d^3)/32))/(a^3*c^3*d*((c + d*x)^(1/2) - c^(1/2))^8))/(((a + b*x)^(1/2) - a^(1/2))^9/((c + d*x)^(1/2) - c^(1/2))^9 + (b^3*((a + b*...