Integrand size = 28, antiderivative size = 469 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \]
-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)*(g*x+f)^(1/2)*(1+c*x^2/a)^(1/2)/e/c ^(1/2)/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-2* g*(-d*g+e*f)*EllipticF(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)*(1+c*x^2/a)^(1/2)*((g*x+f)* c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)/e^2/c^(1/2)/(g*x+f)^(1/2)/(c*x^2+a )^(1/2)-2*(-d*g+e*f)^2*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)/e^2/(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 = 10.87 (sec) , antiderivative size = 927, normalized size of antiderivative = 1.98 \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}} \left (\frac {2 i \sqrt {a} f g \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e}-\frac {i \sqrt {a} d g^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e^2}+\frac {g \sqrt {\frac {g \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} f+\sqrt {a} g}} \left (i \sqrt {a}+\sqrt {c} x\right ) \left (\left (\sqrt {c} f+i \sqrt {a} g\right ) E\left (\arcsin \left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-i \sqrt {a} g \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c e \sqrt {\frac {g \left (\sqrt {a}-i \sqrt {c} x\right )}{i \sqrt {c} f+\sqrt {a} g}}}-\frac {\sqrt {a} f^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{i \sqrt {c} d+\sqrt {a} e}+\frac {2 \sqrt {a} d f g \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{i \sqrt {c} d e+\sqrt {a} e^2}-\frac {\sqrt {a} d^2 g^2 \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e},\arcsin \left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{e^2 \left (i \sqrt {c} d+\sqrt {a} e\right )}\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \]
(2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g)]*(((2*I)*Sqrt[a]*f*g *Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt [2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(Sqrt[c]*e) - (I*Sqrt[a]*d *g^2*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/ Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(Sqrt[c]*e^2) + (g*Sqr t[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*(I*Sqrt[a] + S qrt[c]*x)*((Sqrt[c]*f + I*Sqrt[a]*g)*EllipticE[ArcSin[Sqrt[(Sqrt[c]*(f + g *x))/(Sqrt[c]*f - I*Sqrt[a]*g)]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I *Sqrt[a]*g)] - I*Sqrt[a]*g*EllipticF[ArcSin[Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt [c]*f - I*Sqrt[a]*g)]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g )]))/(c*e*Sqrt[(g*(Sqrt[a] - I*Sqrt[c]*x))/(I*Sqrt[c]*f + Sqrt[a]*g)]) - ( Sqrt[a]*f^2*Sqrt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sq rt[a]*e), ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/( I*Sqrt[c]*f + Sqrt[a]*g)])/(I*Sqrt[c]*d + Sqrt[a]*e) + (2*Sqrt[a]*d*f*g*Sq rt[1 + (c*x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcS in[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g)])/(I*Sqrt[c]*d*e + Sqrt[a]*e^2) - (Sqrt[a]*d^2*g^2*Sqrt[1 + (c* x^2)/a]*EllipticPi[(2*Sqrt[a]*e)/(I*Sqrt[c]*d + Sqrt[a]*e), ArcSin[Sqrt[1 - (I*Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*g)/(I*Sqrt[c]*f + Sqrt[a]*g) ])/(e^2*(I*Sqrt[c]*d + Sqrt[a]*e))))/(Sqrt[f + g*x]*Sqrt[a + c*x^2])
Leaf count is larger than twice the leaf count of optimal. \(1299\) vs. \(2(469)=938\).
Time = 2.84 (sec) , antiderivative size = 1299, normalized size of antiderivative = 2.77, 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.
| \(\displaystyle \int \frac {(f+g x)^{3/2}}{\sqrt {a+c x^2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 740 |
| \(\displaystyle \int \left (\frac {(e f-d g)^2}{e^2 \sqrt {a+c x^2} (d+e x) \sqrt {f+g x}}+\frac {g (e f-d g)}{e^2 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {g \sqrt {f+g x}}{e \sqrt {a+c x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {f+g x} \sqrt {c x^2+a} g^2}{\sqrt {c} e \sqrt {c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}-\frac {(e f-d g)^{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 )}{e^{3/2} \sqrt {c d^2+a e^2}}-\frac {2 \left (c f^2+a g^2\right )^{3/4} \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 )}{c^{3/4} e \sqrt {c x^2+a}}+\frac {\left (c f^2+a g^2\right )^{3/4} \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 )}{c^{3/4} e \sqrt {c x^2+a}}+\frac {(e f-d g) \sqrt [4]{c f^2+a g^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 {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 )}{\sqrt [4]{c} e^2 \sqrt {c x^2+a}}+\frac {\sqrt [4]{c} (e f-d g)^2 \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^2 \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a} g}-\frac {(e f-d g) \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^2 \left (a g e^2+c d (2 e f-d g)\right ) \sqrt {c x^2+a} g}\) |
(2*g^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(Sqrt[c]*e*Sqrt[c*f^2 + a*g^2]*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])) - ((e*f - d*g)^(3/2)*ArcTanh[(Sq rt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt[a + c*x^2]) ])/(e^(3/2)*Sqrt[c*d^2 + a*e^2]) - (2*(c*f^2 + a*g^2)^(3/4)*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])/(c^(3/4)*e*Sqrt[a + c*x^2]) + ((e*f - d*g)*(c*f^2 + a*g^2)^(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)/Sq rt[c*f^2 + a*g^2])/2])/(c^(1/4)*e^2*Sqrt[a + c*x^2]) + ((c*f^2 + a*g^2)^(3 /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])*Elliptic F[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])/(c^(3/4)*e*Sqrt[a + c*x^2]) + (c^(1/4)*(e*f - d *g)^2*(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)...
3.7.44.3.1 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. \(851\) vs. \(2(382)=764\).
Time = 1.37 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.82
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 g \left (d g -2 e f \right ) \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 )}{e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 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 )}{e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \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 )}{e^{3} \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}}\) | \(852\) |
| default | \(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}+c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{g \sqrt {-a c}-c f}}\, \left (\sqrt {-a c}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) d \,g^{2}-\sqrt {-a c}\, F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) e f g -\sqrt {-a c}\, \Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) d \,g^{2}+\sqrt {-a c}\, \Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) e f g +F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) a e \,g^{2}-F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c d f g +2 F\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c e \,f^{2}-E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) a e \,g^{2}-E\left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c e \,f^{2}+\Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c d f g -\Pi \left (\sqrt {-\frac {\left (g x +f \right ) c}{g \sqrt {-a c}-c f}}, \frac {\left (g \sqrt {-a c}-c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {g \sqrt {-a c}-c f}{g \sqrt {-a c}+c f}}\right ) c e \,f^{2}\right )}{c \,e^{2} \left (c g \,x^{3}+c f \,x^{2}+a g x +f a \right )}\) | \(959\) |
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(-2*g*(d*g-2*e*f)/ e^2*(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* g^2/e*(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)*Ell ipticE(((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*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*(d^2*g^ 2-2*d*e*f*g+e^2*f^2)/e^3*(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. \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{\frac {3}{2}}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}} \,d x } \]
\[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{3/2}}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]