\(\int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx\) [2534]

Optimal result

Integrand size = 22, antiderivative size = 1465 \[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=-\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {5 \sqrt {c} (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{8 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}+\frac {\sqrt [4]{-b^2+4 a c} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{32 \sqrt [4]{c} \sqrt {e} \left (c d^2-b d e+a e^2\right )^{9/4} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt [4]{-b^2+4 a c} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{32 \sqrt [4]{c} \sqrt {e} \left (c d^2-b d e+a e^2\right )^{9/4} \sqrt [4]{a+b x+c x^2}}-\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{16 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{32 \sqrt {2} \sqrt {c} e \left (c d^2-b d e+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}+\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{32 \sqrt {2} \sqrt {c} e \left (c d^2-b d e+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{a+b x+c x^2}} \]

[Out]

-1/2*e*(c*x^2+b*x+a)^(3/4)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2-5/8*e*(-b*e+2*c*d)*(c*x^2+b*x+a)^(3/4)/(a*e^2-b*d*e+c
*d^2)^2/(e*x+d)+1/32*(4*a*c-b^2)^(1/4)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^
2))^(1/4)*arctan(1/2*(4*a*c-b^2)^(1/4)*(1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4)*e^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^
(1/4)*2^(1/2))/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(9/4)/(c*x^2+b*x+a)^(1/4)/e^(1/2)-1/32*(4*a*c-b^2)^(1/4)*(12*c^2*d^
2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*arctanh(1/2*(4*a*c-b^2)^(1/4)*(1-(2*c*x
+b)^2/(-4*a*c+b^2))^(1/4)*e^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4)*2^(1/2))/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(9/4)
/(c*x^2+b*x+a)^(1/4)/e^(1/2)-1/64*(-b*e+2*c*d)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(-c*(c*x^2+b*x+a)/(-
4*a*c+b^2))^(1/4)*EllipticPi((1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4),-1/2*e*(4*a*c-b^2)^(1/2)/c^(1/2)/(a*e^2-b*d*e+
c*d^2)^(1/2),I)*(4*a*c-b^2)^(1/2)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)/e/(a*e^2-b*d*e+c*d^2)^(5/2)/(2*c*x+b)/(c*x^
2+b*x+a)^(1/4)*2^(1/2)/c^(1/2)+1/64*(-b*e+2*c*d)*(12*c^2*d^2+5*b^2*e^2-4*c*e*(2*a*e+3*b*d))*(-c*(c*x^2+b*x+a)/
(-4*a*c+b^2))^(1/4)*EllipticPi((1-(2*c*x+b)^2/(-4*a*c+b^2))^(1/4),1/2*e*(4*a*c-b^2)^(1/2)/c^(1/2)/(a*e^2-b*d*e
+c*d^2)^(1/2),I)*(4*a*c-b^2)^(1/2)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)/e/(a*e^2-b*d*e+c*d^2)^(5/2)/(2*c*x+b)/(c*x
^2+b*x+a)^(1/4)*2^(1/2)/c^(1/2)+5/8*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)*c^(1/2)/(a*e^2-b*d*e+c*d^2)^2/(
-4*a*c+b^2)^(1/2)/(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))-5/16*c^(1/4)*(-4*a*c+b^2)^(3/4)*(-b*e+2
*c*d)*(cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*(c*
x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a
*c+b^2)^(1/4))),1/2*2^(1/2))*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))*((2*c*x+b)^2/(-4*a*c+b^2)/(1
+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^2/(2*c*x+b)*2^(1/2)+5/32*c^(1/
4)*(-4*a*c+b^2)^(3/4)*(-b*e+2*c*d)*(cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))^2)^(
1/2)/cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(c*x
^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4))),1/2*2^(1/2))*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))
*((2*c*x+b)^2/(-4*a*c+b^2)/(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^2
/(2*c*x+b)*2^(1/2)

Rubi [A] (warning: unable to verify)

Time = 2.10 (sec) , antiderivative size = 1465, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {758, 848, 857, 637, 311, 226, 1210, 763, 762, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=-\frac {5 (2 c d-b e) \left (c x^2+b x+a\right )^{3/4} e}{8 \left (c d^2-b e d+a e^2\right )^2 (d+e x)}-\frac {\left (c x^2+b x+a\right )^{3/4} e}{2 \left (c d^2-b e d+a e^2\right ) (d+e x)^2}-\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}+\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{16 \sqrt {2} \left (c d^2-b e d+a e^2\right )^2 (b+2 c x)}+\frac {5 \sqrt {c} (2 c d-b e) (b+2 c x) \sqrt [4]{c x^2+b x+a}}{8 \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )}+\frac {\sqrt [4]{4 a c-b^2} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{32 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a} \sqrt {e}}-\frac {\sqrt [4]{4 a c-b^2} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{32 \sqrt [4]{c} \left (c d^2-b e d+a e^2\right )^{9/4} \sqrt [4]{c x^2+b x+a} \sqrt {e}}-\frac {\sqrt {4 a c-b^2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{32 \sqrt {2} \sqrt {c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a} e}+\frac {\sqrt {4 a c-b^2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{32 \sqrt {2} \sqrt {c} \left (c d^2-b e d+a e^2\right )^{5/2} (b+2 c x) \sqrt [4]{c x^2+b x+a} e} \]

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)^(1/4)),x]

[Out]

-1/2*(e*(a + b*x + c*x^2)^(3/4))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (5*e*(2*c*d - b*e)*(a + b*x + c*x^2)^
(3/4))/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (5*Sqrt[c]*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))
/(8*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + (
(-b^2 + 4*a*c)^(1/4)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))
^(1/4)*ArcTan[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 -
b*d*e + a*e^2)^(1/4))])/(32*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - ((-b^2 +
4*a*c)^(1/4)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*
ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e
+ a*e^2)^(1/4))])/(32*c^(1/4)*Sqrt[e]*(c*d^2 - b*d*e + a*e^2)^(9/4)*(a + b*x + c*x^2)^(1/4)) - (5*c^(1/4)*(b^
2 - 4*a*c)^(3/4)*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b
^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)
*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(8*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c*x)) + (5*c
^(1/4)*(b^2 - 4*a*c)^(3/4)*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[
2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(16*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)^2*(b + 2*c
*x)) - (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/(
b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))^(1/4)*EllipticPi[-1/2*(Sqrt[-b^2 + 4*a*c]*e)/(Sqrt[c]*S
qrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(b^2 - 4*a*c))^(1/4)], -1])/(32*Sqrt[2]*Sqrt[c]*e*(c*d^
2 - b*d*e + a*e^2)^(5/2)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4)) + (Sqrt[-b^2 + 4*a*c]*(2*c*d - b*e)*(12*c^2*d^2
+ 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/(b^2 - 4*a*c)]*(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))
)^(1/4)*EllipticPi[(Sqrt[-b^2 + 4*a*c]*e)/(2*Sqrt[c]*Sqrt[c*d^2 - b*d*e + a*e^2]), ArcSin[(1 - (b + 2*c*x)^2/(
b^2 - 4*a*c))^(1/4)], -1])/(32*Sqrt[2]*Sqrt[c]*e*(c*d^2 - b*d*e + a*e^2)^(5/2)*(b + 2*c*x)*(a + b*x + c*x^2)^(
1/4))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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]

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 408

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/x), Subst[I
nt[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 637

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[d*(Sqrt[(b + 2*c*x)
^2]/(b + 2*c*x)), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 760

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(1/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ
[c*d^2 + a*e^2, 0]

Rule 762

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol] :> Dist[1/(-4*(c/(b^2 - 4*a*c)))^
p, Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p/Simp[2*c*d - b*e + e*x, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b
, c, d, e, p}, x] && GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]

Rule 763

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol] :> Dist[(a + b*x + c*x^2)^p/((-c)
*((a + b*x + c*x^2)/(b^2 - 4*a*c)))^p, Int[((-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4
*a*c)))^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&  !GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\int \frac {\frac {1}{4} (-8 c d+5 b e)+\frac {c e x}{2}}{(d+e x)^2 \sqrt [4]{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\int \frac {\frac {1}{16} \left (32 c^2 d^2+5 b^2 e^2-2 c e (11 b d+4 a e)\right )+\frac {5}{8} c e (2 c d-b e) x}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {(5 c (2 c d-b e)) \int \frac {1}{\sqrt [4]{a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \int \frac {1}{(d+e x) \sqrt [4]{a+b x+c x^2}} \, dx}{32 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (5 c (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{4 \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {\left (\left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{(d+e x) \sqrt [4]{-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{32 \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (5 \sqrt {c} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{8 \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac {\left (5 \sqrt {c} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1-\frac {2 \sqrt {c} x^2}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{8 \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {\left (\left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\left (-\frac {c (2 c d-b e)}{b^2-4 a c}+e x\right ) \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}}} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{16 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {5 \sqrt {c} (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{8 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}-\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{16 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac {\left (e \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{16 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac {\left (c (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x^2\right )} \, dx,x,-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )}{16 \sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {5 \sqrt {c} (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{8 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}-\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{16 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}-\frac {\left (e \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) x}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 x\right )} \, dx,x,\left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2\right )}{32 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac {\left (c (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}-e^2 x^4\right )} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{8 \sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}} \\ & = -\frac {e \left (a+b x+c x^2\right )^{3/4}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/4}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {5 \sqrt {c} (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{8 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}-\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {5 \sqrt [4]{c} \left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{16 \sqrt {2} \left (c d^2-b d e+a e^2\right )^2 (b+2 c x)}+\frac {\left (c^2 e \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{-\frac {c^2 e^2}{b^2-4 a c}+\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}+\frac {c^2 e^2 x^4}{b^2-4 a c}} \, dx,x,\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right )}{8 \sqrt {2} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt [4]{a+b x+c x^2}}-\frac {\left (c \sqrt {-b^2+4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}-\sqrt {-b^2+4 a c} e x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{16 \sqrt {2} \left (b^2-4 a c\right ) e \left (c d^2-b d e+a e^2\right )^2 \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}}+\frac {\left (c \sqrt {-b^2+4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}+\sqrt {-b^2+4 a c} e x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right )^2}{c^2}}\right )}{16 \sqrt {2} \left (b^2-4 a c\right ) e \left (c d^2-b d e+a e^2\right )^2 \left (-\frac {b c}{b^2-4 a c}-\frac {2 c^2 x}{b^2-4 a c}\right ) \sqrt [4]{a+b x+c x^2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 12.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.13 \[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=-\frac {\sqrt {2} \sqrt [4]{\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \sqrt [4]{\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}} \operatorname {AppellF1}\left (\frac {5}{2},\frac {1}{4},\frac {1}{4},\frac {7}{2},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{5 e (d+e x)^2 \sqrt [4]{a+x (b+c x)}} \]

[In]

Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(1/4)),x]

[Out]

-1/5*(Sqrt[2]*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(1/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(
c*(d + e*x)))^(1/4)*AppellF1[5/2, 1/4, 1/4, 7/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d -
b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(e*(d + e*x)^2*(a + x*(b + c*x))^(1/4))

Maple [F]

\[\int \frac {1}{\left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x\]

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^(1/4),x)

[Out]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^(1/4),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{3} \sqrt [4]{a + b x + c x^{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x+a)**(1/4),x)

[Out]

Integral(1/((d + e*x)**3*(a + b*x + c*x**2)**(1/4)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3), x)

Giac [F]

\[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {1}{4}} {\left (e x + d\right )}^{3}} \,d x } \]

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(1/4)*(e*x + d)^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt [4]{a+b x+c x^2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{1/4}} \,d x \]

[In]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(1/4)),x)

[Out]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(1/4)), x)