Integrand size = 26, antiderivative size = 175 \[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e}}-\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e}} \] Output:
-arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x +c)^(1/2))/(-a)^(3/4)/(b^(1/2)*c-(-a)^(1/2)*d)^(1/2)/e^(1/2)-arctanh((b^(1 /2)*c+(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(- a)^(3/4)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)/e^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=-\frac {4 d^{3/2} \sqrt {x} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {\log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}}{-b c^3+3 b c^2 \text {$\#$1}+8 a d^2 \text {$\#$1}-3 b c \text {$\#$1}^2+b \text {$\#$1}^3}\&\right ]}{\sqrt {e x}} \] Input:
Integrate[1/(Sqrt[e*x]*Sqrt[c + d*x]*(a + b*x^2)),x]
Output:
(-4*d^(3/2)*Sqrt[x]*RootSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*# 1^2 - 4*b*c*#1^3 + b*#1^4 & , (Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1)/(-(b*c^3) + 3*b*c^2*#1 + 8*a*d^2*#1 - 3*b*c*#1^2 + b*#1^3) & ])/Sqrt[e*x]
Time = 0.57 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {615, 2009}
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.
\(\displaystyle \int \frac {1}{\sqrt {e x} \left (a+b x^2\right ) \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \int \left (\frac {\sqrt {-a}}{2 a \sqrt {e x} \left (\sqrt {-a}-\sqrt {b} x\right ) \sqrt {c+d x}}+\frac {\sqrt {-a}}{2 a \sqrt {e x} \left (\sqrt {-a}+\sqrt {b} x\right ) \sqrt {c+d x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {e} \sqrt {\sqrt {b} c-\sqrt {-a} d}}-\frac {\text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{3/4} \sqrt {e} \sqrt {\sqrt {-a} d+\sqrt {b} c}}\) |
Input:
Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(a + b*x^2)),x]
Output:
-(ArcTan[(Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt [c + d*x])]/((-a)^(3/4)*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*Sqrt[e])) - ArcTanh[( Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x]) ]/((-a)^(3/4)*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*Sqrt[e])
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(505\) vs. \(2(127)=254\).
Time = 0.48 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.89
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, x \sqrt {-a b}\, e \left (\sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a \,d^{2}+\sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) b \,c^{2}-\ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a \,d^{2}-\ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b \,c^{2}\right )}{2 \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a \left (\sqrt {-a b}\, d +b c \right ) \left (b c -\sqrt {-a b}\, d \right ) \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\) | \(506\) |
Input:
int(1/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2*(d*x+c)^(1/2)*x*(-a*b)^(1/2)*e*((e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*ln( (-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(-e*(a*d+c*(-a*b)^(1/ 2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2)))*a*d^2+(e*(-a*d+c*(-a* b)^(1/2))/b)^(1/2)*ln((-2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2) *(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b-c*e*(-a*b)^(1/2))/(b*x+(-a*b)^(1/2))) *b*c^2-ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e*x)^(1/2)*(e*(-a*d+c*( -a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b)^(1/2)))*(-e*(a*d+c*( -a*b)^(1/2))/b)^(1/2)*a*d^2-ln((2*(-a*b)^(1/2)*d*e*x+b*c*e*x+2*((d*x+c)*e* x)^(1/2)*(e*(-a*d+c*(-a*b)^(1/2))/b)^(1/2)*b+c*e*(-a*b)^(1/2))/(b*x-(-a*b) ^(1/2)))*(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*b*c^2)/(e*x)^(1/2)/((d*x+c)*e*x )^(1/2)/a/((-a*b)^(1/2)*d+b*c)/(b*c-(-a*b)^(1/2)*d)/(e*(-a*d+c*(-a*b)^(1/2 ))/b)^(1/2)/(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1051 vs. \(2 (127) = 254\).
Time = 0.11 (sec) , antiderivative size = 1051, normalized size of antiderivative = 6.01 \[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/2*sqrt(((a*b*c^2 + a^2*d^2)*e*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d ^2 + a^5*d^4)*e^2)) + d)/((a*b*c^2 + a^2*d^2)*e))*log((((a^2*b*c^2 + a^3*d ^2)*e^2*x*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)*e^2)) - a *d*e*x)*sqrt(((a*b*c^2 + a^2*d^2)*e*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^ 2*d^2 + a^5*d^4)*e^2)) + d)/((a*b*c^2 + a^2*d^2)*e)) + sqrt(d*x + c)*sqrt( e*x))/x) + 1/2*sqrt(((a*b*c^2 + a^2*d^2)*e*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a ^4*b*c^2*d^2 + a^5*d^4)*e^2)) + d)/((a*b*c^2 + a^2*d^2)*e))*log(-(((a^2*b* c^2 + a^3*d^2)*e^2*x*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4 )*e^2)) - a*d*e*x)*sqrt(((a*b*c^2 + a^2*d^2)*e*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)*e^2)) + d)/((a*b*c^2 + a^2*d^2)*e)) - sqrt(d*x + c)*sqrt(e*x))/x) + 1/2*sqrt(-((a*b*c^2 + a^2*d^2)*e*sqrt(-b*c^2/((a^3*b ^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)*e^2)) - d)/((a*b*c^2 + a^2*d^2)*e))*lo g((((a^2*b*c^2 + a^3*d^2)*e^2*x*sqrt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d^ 2 + a^5*d^4)*e^2)) + a*d*e*x)*sqrt(-((a*b*c^2 + a^2*d^2)*e*sqrt(-b*c^2/((a ^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)*e^2)) - d)/((a*b*c^2 + a^2*d^2)*e) ) + sqrt(d*x + c)*sqrt(e*x))/x) - 1/2*sqrt(-((a*b*c^2 + a^2*d^2)*e*sqrt(-b *c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)*e^2)) - d)/((a*b*c^2 + a^2 *d^2)*e))*log(-(((a^2*b*c^2 + a^3*d^2)*e^2*x*sqrt(-b*c^2/((a^3*b^2*c^4 + 2 *a^4*b*c^2*d^2 + a^5*d^4)*e^2)) + a*d*e*x)*sqrt(-((a*b*c^2 + a^2*d^2)*e*sq rt(-b*c^2/((a^3*b^2*c^4 + 2*a^4*b*c^2*d^2 + a^5*d^4)*e^2)) - d)/((a*b*c...
\[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int \frac {1}{\sqrt {e x} \left (a + b x^{2}\right ) \sqrt {c + d x}}\, dx \] Input:
integrate(1/(e*x)**(1/2)/(d*x+c)**(1/2)/(b*x**2+a),x)
Output:
Integral(1/(sqrt(e*x)*(a + b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x + c} \sqrt {e x}} \,d x } \] Input:
integrate(1/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)*sqrt(d*x + c)*sqrt(e*x)), x)
Exception generated. \[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[1,0]%%%},[0,2,0]%%%}+%%%{%%%{1,[0,1]%%%},[0,0,2]%%% } / %%%{%
Time = 20.68 (sec) , antiderivative size = 3512, normalized size of antiderivative = 20.07 \[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/((e*x)^(1/2)*(a + b*x^2)*(c + d*x)^(1/2)),x)
Output:
atan(((a^9*d^8*e*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4* a^3*b*c^2*e))^(1/2)*48i)/((c + d*x)^(1/2) - c^(1/2)) - (a^11*d^9*e^2*(e*x) ^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(3/2)*40 0i)/((c + d*x)^(1/2) - c^(1/2)) + (a^13*d^10*e^3*(e*x)^(1/2)*((a^2*d - c*( -a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(5/2)*832i)/((c + d*x)^(1/2) - c^(1/2)) + (a^8*b^5*c^10*e^3*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4 *a^4*d^2*e + 4*a^3*b*c^2*e))^(5/2)*4096i)/((c + d*x)^(1/2) - c^(1/2)) + (a ^5*b^4*c^8*e*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3* b*c^2*e))^(1/2)*256i)/((c + d*x)^(1/2) - c^(1/2)) + (a^8*b*c^2*d^6*e*(e*x) ^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(1/2)*40 0i)/((c + d*x)^(1/2) - c^(1/2)) + (a^6*b^3*c^6*d^2*e*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(1/2)*816i)/((c + d*x)^( 1/2) - c^(1/2)) + (a^7*b^2*c^4*d^4*e*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2 ))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(1/2)*912i)/((c + d*x)^(1/2) - c^(1/2)) - (a^7*b^4*c^8*d*e^2*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(3/2)*2048i)/((c + d*x)^(1/2) - c^(1/2)) - (a^10*b*c^2*d ^7*e^2*(e*x)^(1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2* e))^(3/2)*3248i)/((c + d*x)^(1/2) - c^(1/2)) + (a^12*b*c^2*d^8*e^3*(e*x)^( 1/2)*((a^2*d - c*(-a^3*b)^(1/2))/(4*a^4*d^2*e + 4*a^3*b*c^2*e))^(5/2)*7360 i)/((c + d*x)^(1/2) - c^(1/2)) - (a^8*b^3*c^6*d^3*e^2*(e*x)^(1/2)*((a^2...
\[ \int \frac {1}{\sqrt {e x} \sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\int \frac {1}{\sqrt {x}\, \sqrt {d x +c}\, a +\sqrt {x}\, \sqrt {d x +c}\, b \,x^{2}}d x}{\sqrt {e}} \] Input:
int(1/(e*x)^(1/2)/(d*x+c)^(1/2)/(b*x^2+a),x)
Output:
int(1/(sqrt(x)*sqrt(c + d*x)*a + sqrt(x)*sqrt(c + d*x)*b*x**2),x)/sqrt(e)