Integrand size = 28, antiderivative size = 387 \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {a+c x^2}}{(e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{(e f-d g) \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \]
2*g^2*(c*x^2+a)^(1/2)/(-d*g+e*f)/(a*g^2+c*f^2)/(g*x+f)^(1/2)+2*g*EllipticE (1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^( 1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(g*x+f)^(1/2)*(1+c*x^2/a)^(1/2)/(-d*g+e*f )/(a*g^2+c*f^2)/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2))) ^(1/2)-2*e*EllipticPi(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),2*e/(e+d* c^(1/2)/(-a)^(1/2)),2^(1/2)*(g*(-a)^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)) *(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/(-d*g+ e*f)/(e+d*c^(1/2)/(-a)^(1/2))/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 i \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x) \left (\sqrt {c} (e f-d g) E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\left (i \sqrt {a} e g+\sqrt {c} (-2 e f+d g)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+e \left (\sqrt {c} f-i \sqrt {a} g\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g)^2 \sqrt {a+c x^2}} \]
((2*I)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g) /Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*(Sqrt[c]*(e*f - d*g)*EllipticE[I*Arc Sinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[ a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + (I*Sqrt[a]*e*g + Sqrt[c]*(-2*e*f + d*g) )*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sq rt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + e*(Sqrt[c]*f - I*Sqrt[ a]*g)*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*Ar cSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt [a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/((Sqrt[c]*f - I*Sqrt[a]*g)*Sqrt[-f - ( I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)^2*Sqrt[a + c*x^2])
Leaf count is larger than twice the leaf count of optimal. \(1176\) vs. \(2(387)=774\).
Time = 2.93 (sec) , antiderivative size = 1176, normalized size of antiderivative = 3.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {740, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+c x^2} (d+e x) (f+g x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 740 |
| \(\displaystyle \int \left (\frac {e}{\sqrt {a+c x^2} (d+e x) \sqrt {f+g x} (e f-d g)}-\frac {g}{\sqrt {a+c x^2} (f+g x)^{3/2} (e f-d g)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {c x^2+a} g^2}{(e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {2 \sqrt {c} \sqrt {f+g x} \sqrt {c x^2+a} g^2}{(e f-d g) \left (c f^2+a g^2\right )^{3/2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}-\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {e} \sqrt {e f-d g} \sqrt {c x^2+a}}\right )}{\sqrt {c d^2+a e^2} (e f-d g)^{3/2}}+\frac {2 \sqrt [4]{c} \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{(e f-d g) \sqrt [4]{c f^2+a g^2} \sqrt {c x^2+a}}-\frac {\sqrt [4]{c} \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{(e f-d g) \sqrt [4]{c f^2+a g^2} \sqrt {c x^2+a}}+\frac {\sqrt [4]{c} e \sqrt [4]{c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{(e f-d g) \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a} g}-\frac {e \sqrt [4]{c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right )^2 \sqrt {\frac {g^2 \left (c x^2+a\right )}{\left (c f^2+a g^2\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c f^2+a g^2} e+\sqrt {c} (e f-d g)\right )^2}{4 \sqrt {c} e (e f-d g) \sqrt {c f^2+a g^2}},2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} (e f-d g)^2 \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a} g}\) |
(2*g^2*Sqrt[a + c*x^2])/((e*f - d*g)*(c*f^2 + a*g^2)*Sqrt[f + g*x]) - (2*S qrt[c]*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((e*f - d*g)*(c*f^2 + a*g^2)^(3/ 2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])) - (e^(3/2)*ArcTanh[(Sqrt [c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt[a + c*x^2])]) /(Sqrt[c*d^2 + a*e^2]*(e*f - d*g)^(3/2)) + (2*c^(1/4)*Sqrt[(g^2*(a + c*x^2 ))/((c*f^2 + a*g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*EllipticE[2*ArcTan[(c^(1/4)*Sqrt [f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2 ])/((e*f - d*g)*(c*f^2 + a*g^2)^(1/4)*Sqrt[a + c*x^2]) - (c^(1/4)*Sqrt[(g^ 2*(a + c*x^2))/((c*f^2 + a*g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^ 2])^2)]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*EllipticF[2*ArcTan[( c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/((e*f - d*g)*(c*f^2 + a*g^2)^(1/4)*Sqrt[a + c*x^2]) + (c^(1 /4)*e*(c*f^2 + a*g^2)^(1/4)*(Sqrt[c]*(e*f - d*g) - e*Sqrt[c*f^2 + a*g^2])* Sqrt[(g^2*(a + c*x^2))/((c*f^2 + a*g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^ 2 + a*g^2])^2)]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*EllipticF[2* ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sq rt[c*f^2 + a*g^2])/2])/(g*(e*f - d*g)*(a*e^2*g + c*d*(2*e*f - d*g))*Sqrt[a + c*x^2]) - (e*(c*f^2 + a*g^2)^(1/4)*(Sqrt[c]*(e*f - d*g) - e*Sqrt[c*f^2 + a*g^2])^2*Sqrt[(g^2*(a + c*x^2))/((c*f^2 + a*g^2)*(1 + (Sqrt[c]*(f + ...
3.7.52.3.1 Defintions of rubi rules used
Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^ 2]), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && Inte gerQ[n + 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(928\) vs. \(2(324)=648\).
Time = 2.53 (sec) , antiderivative size = 929, normalized size of antiderivative = 2.40
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c g \,x^{2}+a g \right ) g}{\left (a \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \sqrt {\left (x +\frac {f}{g}\right ) \left (c g \,x^{2}+a g \right )}}+\frac {2 c g f \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\left (a \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 c \,g^{2} \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\left (a \,g^{2}+c \,f^{2}\right ) \left (d g -e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \Pi \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\left (d g -e f \right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(929\) |
| default | \(\text {Expression too large to display}\) | \(2011\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(-2*(c*g*x^2+a*g)/ (a*g^2+c*f^2)*g/(d*g-e*f)/((x+f/g)*(c*g*x^2+a*g))^(1/2)+2/(a*g^2+c*f^2)*c* g*f/(d*g-e*f)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*(( x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+( -a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*EllipticF(((x+f/g) /(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c)) ^(1/2))+2/(a*g^2+c*f^2)*c*g^2/(d*g-e*f)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g -(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*( (x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f )^(1/2)*((-f/g-(-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1 /2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*El lipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g- (-a*c)^(1/2)/c))^(1/2)))-2/(d*g-e*f)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(- a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+ (-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^( 1/2)/(-f/g+d/e)*EllipticPi(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),(-f/g+(-a* c)^(1/2)/c)/(-f/g+d/e),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2) ))
Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]