Optimal. Leaf size=1049 \[ -\frac {\sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right ) (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \sqrt {2} e \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a} (c d (2 e f+d g)-e (b e f+2 b d g-3 a e g))}{4 \left (c d^2-b e d+a e^2\right ) (e f-d g)^2 (d+e x)}-\frac {\sqrt {b^2-4 a c} \left ((b f-a g) e^2+c d (d g-2 e f)\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} g}{\left (b+\sqrt {b^2-4 a c}\right ) g-2 c f}\right )}{2 \sqrt {2} e^2 \left (c d^2+e (a e-b d)\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+x (b+c x)}}-\frac {\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g} \left (3 a^2 g^2 e^4+b^2 f (4 d g-e f) e^3+2 a c \left (2 e^2 f^2-2 d e g f+3 d^2 g^2\right ) e^2-2 b g \left (3 c f d^2+a e (e f+2 d g)\right ) e^2+c^2 d^3 g (4 e f-d g)\right ) \sqrt {\frac {g \left (-b-2 c x+\sqrt {b^2-4 a c}\right )}{2 c f+\left (\sqrt {b^2-4 a c}-b\right ) g}} \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) g-2 c f}} \Pi \left (\frac {2 c e f-b e g+\sqrt {b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-b g+\sqrt {b^2-4 a c} g}}\right )|\frac {2 c f+\left (\sqrt {b^2-4 a c}-b\right ) g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{4 \sqrt {2} \sqrt {c} e^2 \left (c d^2+e (a e-b d)\right ) (e f-d g)^3 \sqrt {a+x (b+c x)}}-\frac {\sqrt {f+g x} \sqrt {c x^2+b x+a}}{2 (e f-d g) (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 8.18, antiderivative size = 1747, normalized size of antiderivative = 1.67, number of steps used = 25, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {924, 6742, 718, 419, 939, 843, 424, 934, 169, 538, 537} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 169
Rule 419
Rule 424
Rule 537
Rule 538
Rule 718
Rule 843
Rule 924
Rule 934
Rule 939
Rule 6742
Rubi steps
\begin {align*} \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 {\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}}\\ &=-\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 (\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 )}{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} 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} 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-\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 \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 {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 {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \operatorname {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 )}{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 {\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 (\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 )}{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} 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} 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-\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 \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 {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 {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 ) \operatorname {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 )}{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 {\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 (\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 )}{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} 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} 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-\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 \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 {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 {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (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 {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 )}{4 \sqrt {2} \sqrt {c} e^2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^3 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 16.68, size = 36616, normalized size = 34.91 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 57841, normalized size = 55.14 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + b x + a}}{{\left (e x + d\right )}^{3} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c\,x^2+b\,x+a}}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{3} \sqrt {f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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