Optimal. Leaf size=1432 \[ -\frac {\sqrt [4]{b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+e x+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}} \left (1+\frac {\sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}{\left (1+\frac {\sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )|\frac {1}{4} \left (2+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {2 c^2 d+b \left (b+\sqrt {b^2-4 a c}\right ) f-c \left (b e+\sqrt {b^2-4 a c} e+2 a f\right )}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2} \sqrt {1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 3.90, antiderivative size = 1432, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1006, 949,
1117} \begin {gather*} -\frac {\sqrt [4]{d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt {b^2-4 a c}\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (f x^2+e x+d\right )}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}} \left (\frac {\sqrt {f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right ) \sqrt {\frac {\frac {\left (4 d c^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e c+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}{\left (\frac {\sqrt {f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {b+2 c x+\sqrt {b^2-4 a c}}}\right )|\frac {1}{4} \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt {d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {2 d c^2-\left (b e+\sqrt {b^2-4 a c} e+2 a f\right ) c+b \left (b+\sqrt {b^2-4 a c}\right ) f}}+2\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{f b^2-c e b+2 c^2 d-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {c x^2+b x+a} \sqrt {f x^2+e x+d} \sqrt {\frac {\left (4 d c^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e c+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (4 f a^2-2 \left (b+\sqrt {b^2-4 a c}\right ) e a+\left (b+\sqrt {b^2-4 a c}\right )^2 d\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )^2}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (d b^2+\left (\sqrt {b^2-4 a c} d-a e\right ) b-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 949
Rule 1006
Rule 1117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}} \, dx &=\frac {\left (\sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}\right ) \int \frac {1}{\sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {d+e x+f x^2}} \, dx}{\sqrt {a+b x+c x^2}}\\ &=-\frac {\left (2 \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+e x+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {\left (4 c \left (b+\sqrt {b^2-4 a c}\right ) d-4 a c e-\left (b+\sqrt {b^2-4 a c}\right )^2 e+4 a \left (b+\sqrt {b^2-4 a c}\right ) f\right ) x^2}{\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) x^4}{\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f}}} \, dx,x,\frac {\sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2}}\\ &=-\frac {\sqrt [4]{b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )^{3/2} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x} \sqrt {\frac {\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right )^2 \left (d+e x+f x^2\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}} \left (1+\frac {\sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \sqrt {\frac {1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}{\left (1+\frac {\sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x}}{\sqrt [4]{b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}}\right )|\frac {1}{4} \left (2+\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f)}{\sqrt {b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )} \sqrt {2 c^2 d+b \left (b+\sqrt {b^2-4 a c}\right ) f-c \left (b e+\sqrt {b^2-4 a c} e+2 a f\right )}}\right )\right )}{\left (4 a c-\left (b+\sqrt {b^2-4 a c}\right )^2\right ) \sqrt [4]{2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {a+b x+c x^2} \sqrt {d+e x+f x^2} \sqrt {1-\frac {\left (b+\sqrt {b^2-4 a c}\right ) (2 c d-b e+2 a f) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )}{\left (b^2 d+b \left (\sqrt {b^2-4 a c} d-a e\right )-a \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}+\frac {\left (4 c^2 d-2 c \left (b+\sqrt {b^2-4 a c}\right ) e+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) \left (2 a+\left (b+\sqrt {b^2-4 a c}\right ) x\right )^2}{\left (\left (b+\sqrt {b^2-4 a c}\right )^2 d-2 a \left (b+\sqrt {b^2-4 a c}\right ) e+4 a^2 f\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.92, size = 670, normalized size = 0.47 \begin {gather*} -\frac {\left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {-\frac {c \sqrt {b^2-4 a c} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {-\frac {c \left (4 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}-2 \sqrt {b^2-4 a c} f x+2 c \sqrt {e^2-4 d f} x-e \left (\sqrt {b^2-4 a c}+2 c x\right )+b \left (-e+\sqrt {e^2-4 d f}+2 f x\right )\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}}\right )|\frac {2 c d-b e+2 a f-\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}{2 c d-b e+2 a f+\sqrt {b^2-4 a c} \sqrt {e^2-4 d f}}\right )}{\left (\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )\right ) \sqrt {\frac {c \sqrt {b^2-4 a c} \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}} \sqrt {a+x (b+c x)} \sqrt {d+x (e+f x)}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.29, size = 906, normalized size = 0.63
method | result | size |
elliptic | \(\frac {2 \sqrt {\left (f \,x^{2}+e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \sqrt {\frac {\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\, \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )^{2} \sqrt {\frac {\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\, \sqrt {\frac {\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}{\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}, \sqrt {\frac {\left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{\left (-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}}\right )}{\sqrt {f \,x^{2}+e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (-\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}-\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \sqrt {c f \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right ) \left (x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}\) | \(905\) |
default | \(-\frac {8 \left (-2 b \,f^{2} x^{2}+2 c e f \,x^{2}-2 c f \,x^{2} \sqrt {-4 d f +e^{2}}-2 \sqrt {-4 a c +b^{2}}\, f^{2} x^{2}-2 b e f x -2 b f x \sqrt {-4 d f +e^{2}}+8 c d f x -2 \sqrt {-4 a c +b^{2}}\, e f x -2 f x \sqrt {-4 a c +b^{2}}\, \sqrt {-4 d f +e^{2}}+2 b d f -e^{2} b -b e \sqrt {-4 d f +e^{2}}+2 c d e +2 c d \sqrt {-4 d f +e^{2}}+2 \sqrt {-4 a c +b^{2}}\, d f -\sqrt {-4 a c +b^{2}}\, e^{2}-e \sqrt {-4 a c +b^{2}}\, \sqrt {-4 d f +e^{2}}\right ) \EllipticF \left (\sqrt {-\frac {\left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right )}{\left (f \sqrt {-4 a c +b^{2}}+c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}, \sqrt {\frac {\left (f \sqrt {-4 a c +b^{2}}+c \sqrt {-4 d f +e^{2}}-b f +c e \right ) \left (f \sqrt {-4 a c +b^{2}}+c \sqrt {-4 d f +e^{2}}+b f -c e \right )}{\left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}-b f +c e \right ) \left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}+b f -c e \right )}}\right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right ) f}{\left (f \sqrt {-4 a c +b^{2}}+c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) f}{\left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}-b f +c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}\, \sqrt {-\frac {\left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right )}{\left (f \sqrt {-4 a c +b^{2}}+c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right )}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {f \,x^{2}+e x +d}}{\sqrt {\frac {\left (-2 f x +\sqrt {-4 d f +e^{2}}-e \right ) \left (2 f x +\sqrt {-4 d f +e^{2}}+e \right ) \left (-b -2 c x +\sqrt {-4 a c +b^{2}}\right ) \left (b +2 c x +\sqrt {-4 a c +b^{2}}\right )}{c f}}\, \sqrt {-4 d f +e^{2}}\, \left (f \sqrt {-4 a c +b^{2}}-c \sqrt {-4 d f +e^{2}}+b f -c e \right ) \sqrt {\left (f \,x^{2}+e x +d \right ) \left (c \,x^{2}+b x +a \right )}}\) | \(906\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a + b x + c x^{2}} \sqrt {d + e x + f x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2+b\,x+a}\,\sqrt {f\,x^2+e\,x+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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