Integrand size = 31, antiderivative size = 631 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {8 e^2 (c e f-3 c d g+b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g^2}+\frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}+\frac {\sqrt {2} \sqrt {b^2-4 a c} e \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (4 b e^3 g^2 (b f-a g)+c^2 \left (8 e^3 f^3-30 d e^2 f^2 g+45 d^2 e f g^2-15 d^3 g^3\right )-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]
-8/15*e^2*(b*e*g-3*c*d*g+c*e*f)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/g^2+ 2/5*e^2*(e*x+d)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/g+1/15*e*(8*b^2*e^2*g^ 2+c*e*g*(-9*a*e*g-30*b*d*g+7*b*e*f)+c^2*(45*d^2*g^2-30*d*e*f*g+8*e^2*f^2)) *EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^( 1/2),(-2*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^( 1/2)*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2 )/c^3/g^3/(c*x^2+b*x+a)^(1/2)/(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2)))) ^(1/2)-2/15*(4*b*e^3*g^2*(-a*g+b*f)+c^2*(-15*d^3*g^3+45*d^2*e*f*g^2-30*d*e ^2*f^2*g+8*e^3*f^3)-c*e^2*g*(a*g*(-15*d*g+7*e*f)-3*b*f*(-5*d*g+e*f)))*Elli pticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2), (-2*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)* (-4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(g*x+f)/(2*c*f -g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^3/g^3/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2 )
Result contains complex when optimal does not.
Time = 34.49 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {4 e g^2 \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) (a+x (b+c x))}{\sqrt {f+g x}}+4 c e^2 g^2 \sqrt {f+g x} (a+x (b+c x)) (-4 b e g+c (-4 e f+15 d g+3 e g x))-\frac {i (f+g x) \sqrt {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {2+\frac {4 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left (e \left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (8 b^2 e^2 g^2+c e g (7 b e f-30 b d g-9 a e g)+c^2 \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )-\left (-8 b^3 e^3 g^3+b^2 e^2 g^2 \left (c e f+30 c d g+8 e \sqrt {\left (b^2-4 a c\right ) g^2}\right )+b c e g \left (-45 c d^2 g^2+e \left (17 a e g^2+\sqrt {\left (b^2-4 a c\right ) g^2} (7 e f-30 d g)\right )\right )+c \left (-a e^2 g^2 \left (4 c e f+30 c d g+9 e \sqrt {\left (b^2-4 a c\right ) g^2}\right )+c \left (30 c d^3 g^3+e \sqrt {\left (b^2-4 a c\right ) g^2} \left (8 e^2 f^2-30 d e f g+45 d^2 g^2\right )\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{\sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}}{30 c^3 g^4 \sqrt {a+x (b+c x)}} \]
((4*e*g^2*(8*b^2*e^2*g^2 + c*e*g*(7*b*e*f - 30*b*d*g - 9*a*e*g) + c^2*(8*e ^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*(a + x*(b + c*x)))/Sqrt[f + g*x] + 4*c* e^2*g^2*Sqrt[f + g*x]*(a + x*(b + c*x))*(-4*b*e*g + c*(-4*e*f + 15*d*g + 3 *e*g*x)) - (I*(f + g*x)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[2 + (4*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*(e*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(8*b^2*e^2*g^2 + c*e*g*(7*b*e*f - 30*b*d*g - 9*a*e*g) + c^2*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*EllipticE[I*ArcSi nh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c )*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c* f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] - (-8*b^3*e^3*g^3 + b^2*e^2*g^2*(c*e* f + 30*c*d*g + 8*e*Sqrt[(b^2 - 4*a*c)*g^2]) + b*c*e*g*(-45*c*d^2*g^2 + e*( 17*a*e*g^2 + Sqrt[(b^2 - 4*a*c)*g^2]*(7*e*f - 30*d*g))) + c*(-(a*e^2*g^2*( 4*c*e*f + 30*c*d*g + 9*e*Sqrt[(b^2 - 4*a*c)*g^2])) + c*(30*c*d^3*g^3 + e*S qrt[(b^2 - 4*a*c)*g^2]*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))))*EllipticF[ I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2] )/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/Sqrt[(c*f^2 + g*(-(b*f) + a* g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/(30*c^3*g^4*Sqrt[a + x*(b + c*x)])
Time = 1.20 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1278, 2184, 27, 1269, 1172, 321, 327}
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.
| \(\displaystyle \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1278 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\int \frac {-5 c g d^3+b e^2 f d+4 e^2 (c e f-3 c d g+b e g) x^2+a e^2 (2 e f+d g)+e (c d (2 e f-15 d g)+e (3 b e f+2 b d g+3 a e g)) x}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{5 c g}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\frac {2 \int -\frac {g \left (4 b^2 f g e^3+b \left (4 a e g^2+c f (4 e f-15 d g)\right ) e^2+\left (\left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right ) c^2+e g (7 b e f-30 b d g-9 a e g) c+8 b^2 e^2 g^2\right ) x e+c g \left (15 c d^3 g-a e^2 (2 e f+15 d g)\right )\right )}{2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{3 c g^2}+\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{3 c g}}{5 c g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{3 c g}-\frac {\int \frac {4 b^2 f g e^3+b \left (4 a e g^2+c f (4 e f-15 d g)\right ) e^2+\left (\left (8 e^2 f^2-30 d e g f+45 d^2 g^2\right ) c^2+e g (7 b e f-30 b d g-9 a e g) c+8 b^2 e^2 g^2\right ) x e+c g \left (15 c d^3 g-a e^2 (2 e f+15 d g)\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{3 c g}}{5 c g}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{3 c g}-\frac {\frac {e \left (c e g (-9 a e g-30 b d g+7 b e f)+8 b^2 e^2 g^2+c^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {c x^2+b x+a}}dx}{g}-\frac {\left (-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))+4 b e^3 g^2 (b f-a g)+c^2 \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{g}}{3 c g}}{5 c g}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{3 c g}-\frac {\frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e g (-9 a e g-30 b d g+7 b e f)+8 b^2 e^2 g^2+c^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))+4 b e^3 g^2 (b f-a g)+c^2 \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{3 c g}}{5 c g}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{3 c g}-\frac {\frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e g (-9 a e g-30 b d g+7 b e f)+8 b^2 e^2 g^2+c^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))+4 b e^3 g^2 (b f-a g)+c^2 \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{3 c g}}{5 c g}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c g}-\frac {\frac {8 e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{3 c g}-\frac {\frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e g (-9 a e g-30 b d g+7 b e f)+8 b^2 e^2 g^2+c^2 \left (45 d^2 g^2-30 d e f g+8 e^2 f^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (-c e^2 g (a g (7 e f-15 d g)-3 b f (e f-5 d g))+4 b e^3 g^2 (b f-a g)+c^2 \left (-15 d^3 g^3+45 d^2 e f g^2-30 d e^2 f^2 g+8 e^3 f^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {a+b x+c x^2}}}{3 c g}}{5 c g}\) |
(2*e^2*(d + e*x)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(5*c*g) - ((8*e^2*(c *e*f - 3*c*d*g + b*e*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*c*g) - ((S qrt[2]*Sqrt[b^2 - 4*a*c]*e*(8*b^2*e^2*g^2 + c*e*g*(7*b*e*f - 30*b*d*g - 9* a*e*g) + c^2*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*Sqrt[f + g*x]*Sqrt[-(( c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2* c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(c*g*Sqrt[(c*(f + g*x))/(2*c*f - (b + S qrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c ]*(4*b*e^3*g^2*(b*f - a*g) + c^2*(8*e^3*f^3 - 30*d*e^2*f^2*g + 45*d^2*e*f* g^2 - 15*d^3*g^3) - c*e^2*g*(a*g*(7*e*f - 15*d*g) - 3*b*f*(e*f - 5*d*g)))* Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(c*g*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))/(3*c*g ))/(5*c*g)
3.10.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(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] && !IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2*(d + e*x)^(m - 2)*Sqrt[f + g *x]*(Sqrt[a + b*x + c*x^2]/(c*g*(2*m - 1))), x] - Simp[1/(c*g*(2*m - 1)) Int[((d + e*x)^(m - 3)/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*e^2* f + a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(2*b*d*g + e*(b* f + a*g)*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c *d*g + b*e*g)*(m - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && GeQ[m, 2]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 6.15 (sec) , antiderivative size = 985, normalized size of antiderivative = 1.56
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{3} x \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{5 c g}+\frac {2 \left (3 d \,e^{2}-\frac {2 \left (2 b g +2 c f \right ) e^{3}}{5 c g}\right ) \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{3 c g}+\frac {2 \left (d^{3}-\frac {2 f a \,e^{3}}{5 c g}-\frac {2 \left (3 d \,e^{2}-\frac {2 \left (2 b g +2 c f \right ) e^{3}}{5 c g}\right ) \left (\frac {a g}{2}+\frac {b f}{2}\right )}{3 c g}\right ) \left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}+\frac {2 \left (3 d^{2} e -\frac {2 e^{3} \left (\frac {3 a g}{2}+\frac {3 b f}{2}\right )}{5 c g}-\frac {2 \left (3 d \,e^{2}-\frac {2 \left (2 b g +2 c f \right ) e^{3}}{5 c g}\right ) \left (b g +c f \right )}{3 c g}\right ) \left (\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\) | \(985\) |
| risch | \(\text {Expression too large to display}\) | \(2635\) |
| default | \(\text {Expression too large to display}\) | \(8755\) |
((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/5*e^3/c /g*x*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)+2/3*(3*d*e^2-2/5/c/g* (2*b*g+2*c*f)*e^3)/c/g*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)+2*( d^3-2/5*f*a/c/g*e^3-2/3*(3*d*e^2-2/5/c/g*(2*b*g+2*c*f)*e^3)/c/g*(1/2*a*g+1 /2*b*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*(b+(-4*a*c+b ^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1/2/c*(-b+(- 4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g+1/2*(b+( -4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/ 2)*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2 *(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2* (3*d^2*e-2/5*e^3/c/g*(3/2*a*g+3/2*b*f)-2/3*(3*d*e^2-2/5/c/g*(2*b*g+2*c*f)* e^3)/c/g*(b*g+c*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-1/2*( b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-f/g-1 /2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f /g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f *x+a*f)^(1/2)*((-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+f/g)/(f/ g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c )/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2 ))*EllipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2 *(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{3} e^{3} f^{3} - 3 \, {\left (10 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f^{2} g + 3 \, {\left (15 \, c^{3} d^{2} e - 5 \, b c^{2} d e^{2} + {\left (b^{2} c - a c^{2}\right )} e^{3}\right )} f g^{2} - {\left (45 \, c^{3} d^{3} - 45 \, b c^{2} d^{2} e + 15 \, {\left (2 \, b^{2} c - 3 \, a c^{2}\right )} d e^{2} - {\left (8 \, b^{3} - 21 \, a b c\right )} e^{3}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 3 \, {\left (8 \, c^{3} e^{3} f^{2} g - {\left (30 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} f g^{2} + {\left (45 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} e^{3}\right )} g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) - 3 \, {\left (3 \, c^{3} e^{3} g^{3} x - 4 \, c^{3} e^{3} f g^{2} + {\left (15 \, c^{3} d e^{2} - 4 \, b c^{2} e^{3}\right )} g^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{45 \, c^{4} g^{4}} \]
-2/45*((8*c^3*e^3*f^3 - 3*(10*c^3*d*e^2 - b*c^2*e^3)*f^2*g + 3*(15*c^3*d^2 *e - 5*b*c^2*d*e^2 + (b^2*c - a*c^2)*e^3)*f*g^2 - (45*c^3*d^3 - 45*b*c^2*d ^2*e + 15*(2*b^2*c - 3*a*c^2)*d*e^2 - (8*b^3 - 21*a*b*c)*e^3)*g^3)*sqrt(c* g)*weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^ 2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g)) + 3*(8*c^3*e^3 *f^2*g - (30*c^3*d*e^2 - 7*b*c^2*e^3)*f*g^2 + (45*c^3*d^2*e - 30*b*c^2*d*e ^2 + (8*b^2*c - 9*a*c^2)*e^3)*g^3)*sqrt(c*g)*weierstrassZeta(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), weierstra ssPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2 *c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g ^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)/(c*g))) - 3*(3*c^3*e^3*g^3*x - 4* c^3*e^3*f*g^2 + (15*c^3*d*e^2 - 4*b*c^2*e^3)*g^3)*sqrt(c*x^2 + b*x + a)*sq rt(g*x + f))/(c^4*g^4)
\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]
\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
\[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]