Integrand size = 31, antiderivative size = 1049 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \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 )}{4 \sqrt {2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \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 {\sqrt {b^2-4 a c} \left (e^2 (b f-a g)+c d (-2 e f+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 (a+x (b+c x))}{-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 )}{2 \sqrt {2} e^2 \left (c d^2+e (-b d+a e)\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+x (b+c x)}}-\frac {\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g} \left (3 a^2 e^4 g^2+c^2 d^3 g (4 e f-d g)+b^2 e^3 f (-e f+4 d g)+2 a c e^2 \left (2 e^2 f^2-2 d e f g+3 d^2 g^2\right )-2 b e^2 g \left (3 c d^2 f+a e (e f+2 d g)\right )\right ) \sqrt {\frac {g \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {g \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {2 c e f-b e g+\sqrt {b^2-4 a c} e g}{2 c e f-2 c d g},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g}}\right ),\frac {2 c f+\left (-b+\sqrt {b^2-4 a c}\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} \sqrt {c} e^2 \left (c d^2+e (-b d+a e)\right ) (e f-d g)^3 \sqrt {a+x (b+c x)}} \]
[Out]
Time = 4.94 (sec) , antiderivative size = 1747, normalized size of antiderivative = 1.67, number of steps used = 25, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {938, 6874, 732, 430, 953, 857, 435, 948, 175, 552, 551} \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=-\frac {\sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} 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 d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \sqrt {2} e \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {\sqrt {b^2-4 a c} f \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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 d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{2 \sqrt {2} e \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {b^2-4 a c} d g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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 d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{2 \sqrt {2} e^2 \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right ) (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \sqrt {2} \sqrt {c} e^2 \left (c d^2-b e d+a e^2\right ) (e f-d g)^3 \sqrt {c x^2+b x+a}}+\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \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 )}{\sqrt {2} e^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f+c d g-b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^2 (e f-d g)^2 \sqrt {c x^2+b x+a}}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2} \]
[In]
[Out]
Rule 175
Rule 430
Rule 435
Rule 551
Rule 552
Rule 732
Rule 857
Rule 938
Rule 948
Rule 953
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {\int \frac {b f-3 a g+2 (c f-b g) x-c g x^2}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{4 (e f-d g)} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {\int \left (-\frac {c g}{e^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {-c d (2 e f+d g)+e (b e f+2 b d g-3 a e g)}{e^2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {2 (c e f+c d g-b e g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx}{4 (e f-d g)} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}-\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{4 e^2 (e f-d g)}+\frac {(c e f+c d g-b e g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{2 e^2 (e f-d g)}-\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{4 e^2 (e f-d g)} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \int \frac {-2 c d (e f-d g)+e (b e f-2 b d g+a e g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}+\frac {\left ((c e f+c d g-b e g) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{2 e^2 (e f-d g) \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {b^2-4 a c} g \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{\sqrt {2} e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} g \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}} F\left (\sin ^{-1}\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 )}{\sqrt {2} e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \int \left (-\frac {c d g}{\sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {c e g x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {-c d (2 e f-3 d g)+e (b e f-2 b d g+a e g)}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac {\left ((c e f+c d g-b e g) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{e^2 (e f-d g) \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} g \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}} F\left (\sin ^{-1}\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 )}{\sqrt {2} e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {(c d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac {(c g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac {x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac {((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac {\left ((c e f+c d g-b e g) \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{e^2 (e f-d g) \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} g \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}} F\left (\sin ^{-1}\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 )}{\sqrt {2} e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {(c (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{8 e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}+\frac {(c f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{8 e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2}-\frac {\left ((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{8 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{2 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\left ((c e f+c d g-b e g) \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c (f+g x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{e^2 (e f-d g) \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+b x+c x^2}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} g \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}} F\left (\sin ^{-1}\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 )}{\sqrt {2} e^2 (e f-d g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} d g (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \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}} F\left (\sin ^{-1}\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 )}{2 \sqrt {2} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (c e f+c d g-b e g) \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {2} \sqrt {c} e^2 (e f-d g)^2 \sqrt {a+b x+c x^2}}+\frac {\left ((c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {a+b x+c x^2}}-\frac {\left (\sqrt {b^2-4 a c} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{4 \sqrt {2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (\sqrt {b^2-4 a c} f (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g)) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{2 \sqrt {2} e \left (c d^2-b d e+a e^2\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = \text {Too large to display} \\ \end{align*}
Result contains complex when optimal does not.
Time = 36.62 (sec) , antiderivative size = 36617, normalized size of antiderivative = 34.91 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Result too large to show} \]
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Time = 3.16 (sec) , antiderivative size = 1634, normalized size of antiderivative = 1.56
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1634\) |
default | \(\text {Expression too large to display}\) | \(57841\) |
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Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{3} \sqrt {f + g x}}\, dx \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3 \sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3} \,d x \]
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