Integrand size = 33, antiderivative size = 1301 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} \left (a B d \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )+A \left (2 b^3 d e^2+2 b c d \left (c d^2-3 a e^2\right )-4 a c e \left (-2 c d^2+a e^2\right )+b^2 \left (-4 c d^2 e+a e^3\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{2 a \left (-b^2+4 a c\right ) d \left (c d^2+e (-b d+a e)\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {e^{3/2} \left (A e \left (7 c d^2-e (4 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) \arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \left (c d^2-b d e+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} \left (a B d \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )-A \left (2 b^3 d e^2+2 b c d \left (c d^2-3 a e^2\right )-4 a c e \left (-2 c d^2+a e^2\right )+b^2 \left (-4 c d^2 e+a e^3\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}-\frac {b}{4 \sqrt {a} \sqrt {c}}\right )}{2 a^{3/4} \left (b^2-4 a c\right ) d \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (a \sqrt {c} e (B d-2 A e)+\sqrt {a} (B d-A e) (c d-b e)+A \sqrt {c} d (-c d+b e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}-\frac {b}{4 \sqrt {a} \sqrt {c}}\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) d \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+b x^2+c x^4}}-\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (A e \left (7 c d^2-e (4 b d-a e)\right )-B d \left (5 c d^2-e (2 b d+a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}} \]
[Out]
Time = 2.17 (sec) , antiderivative size = 2112, normalized size of antiderivative = 1.62, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1734, 1192, 1211, 1117, 1209, 1237, 1728, 1722, 1720, 1230} \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {(B d-A e) x \sqrt {c x^4+b x^2+a} e^3}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (e x^2+d\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e^2}{2 d \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+b x^2+a} e^2}{2 d \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \arctan \left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \arctan \left (\frac {\sqrt {c d^2-b e d+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+b x^2+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2-b e d+a e^2\right )^{5/2}}+\frac {\sqrt [4]{c} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^4+b x^2+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2-e (2 b d-a e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}-\frac {\sqrt [4]{c} \left (a^{3/2} B \sqrt {c} e^2+a (2 B c d-b B e-A c e) e+A (c d-b e)^2-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+b x^2+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}}+\frac {\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) x \sqrt {c x^4+b x^2+a}}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \left (-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right )\right ) x^2+a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 \sqrt {c x^4+b x^2+a}} \]
[In]
[Out]
Rule 1117
Rule 1192
Rule 1209
Rule 1211
Rule 1230
Rule 1237
Rule 1720
Rule 1722
Rule 1728
Rule 1734
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) x^2}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}}+\frac {e (-B d+A e)}{\left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}}\right ) \, dx \\ & = \frac {\int \frac {a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )-c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) x^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx}{\left (c d^2-b d e+a e^2\right )^2}-\frac {(e (B d-A e)) \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}} \, dx}{c d^2-b d e+a e^2}+\frac {\left (e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{\left (c d^2-b d e+a e^2\right )^2} \\ & = \frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}-\frac {\int \frac {a c \left (A b^2 e^2+2 c \left (A c d^2+2 a B d e-a A e^2\right )-b \left (B c d^2+2 A c d e+a B e^2\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac {(e (B d-A e)) \int \frac {-2 c d^2+e (2 b d-a e)+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (\sqrt {c} e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {a} e^2 \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2} \\ & = \frac {x \left (a b c \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right )+\left (b^2-2 a c\right ) \left (a B e (2 c d-b e)+A \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right )\right )-c \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right ) x^2\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right )^2 \left (d+e x^2\right )}+\frac {e^{3/2} \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )^{5/2}}+\frac {\sqrt [4]{c} e \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (A e (2 c d-b e)-B \left (c d^2-a e^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {(B d-A e) \int \frac {\sqrt {a} c^{3/2} d e^2+c e \left (-2 c d^2+e (2 b d-a e)\right )+\left (2 c^2 d e^2-c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx}{2 c d \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {a} \sqrt {c} e^2 (B d-A e)\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{2 d \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {c} \left (a^{3/2} B \sqrt {c} e^2+A (c d-b e)^2+a e (2 B c d-b B e-A c e)-\sqrt {a} \sqrt {c} \left (B c d^2-A e (2 c d-b e)\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (\sqrt {c} \left (a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+A \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right )\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2} \\ & = \text {Too large to display} \\ \end{align*}
Result contains complex when optimal does not.
Time = 15.22 (sec) , antiderivative size = 1116, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d \left (-a \left (b^2-4 a c\right ) e^3 (B d-A e) x \left (a+b x^2+c x^4\right )+2 d x \left (d+e x^2\right ) \left (a B \left (-b^3 e^2+b^2 c e \left (2 d-e x^2\right )+b c \left (3 a e^2-c d \left (d-2 e x^2\right )\right )-2 c^2 \left (c d^2 x^2+a e \left (2 d-e x^2\right )\right )\right )+A \left (b^4 e^2+b^3 c e \left (-2 d+e x^2\right )+2 a c^2 \left (a e^2-c d \left (d-2 e x^2\right )\right )+b^2 c \left (-4 a e^2+c d \left (d-2 e x^2\right )\right )+b c^2 \left (c d^2 x^2-3 a e \left (-2 d+e x^2\right )\right )\right )\right )\right )-i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (d+e x^2\right ) \left (\left (-b+\sqrt {b^2-4 a c}\right ) d \left (a B d \left (-4 c^2 d^2-3 b^2 e^2+4 c e (b d+2 a e)\right )+A \left (2 b^3 d e^2+2 b c d \left (c d^2-3 a e^2\right )-4 a c e \left (-2 c d^2+a e^2\right )+b^2 \left (-4 c d^2 e+a e^3\right )\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-d \left (a B d \left (3 b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^2-4 c^2 d \left (\sqrt {b^2-4 a c} d-6 a e\right )+2 c \left (-3 b+2 \sqrt {b^2-4 a c}\right ) e (b d+2 a e)\right )+A \left (-2 b^4 d e^2+b^2 \left (-2 c^2 d^3+a \sqrt {b^2-4 a c} e^3-4 c d e \left (\sqrt {b^2-4 a c} d-3 a e\right )\right )-4 a c \left (-2 c^2 d^3+a \sqrt {b^2-4 a c} e^3-2 c d e \left (\sqrt {b^2-4 a c} d-2 a e\right )\right )+b^3 e \left (4 c d^2+e \left (2 \sqrt {b^2-4 a c} d-a e\right )\right )+2 b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-8 a e\right )+a e^2 \left (-3 \sqrt {b^2-4 a c} d+2 a e\right )\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-2 a \left (-b^2+4 a c\right ) e \left (A e \left (7 c d^2+e (-4 b d+a e)\right )+B \left (-5 c d^3+d e (2 b d+a e)\right )\right ) \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{8 a \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (c d^3+d e (-b d+a e)\right )^2 \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(8275\) vs. \(2(1252)=2504\).
Time = 3.61 (sec) , antiderivative size = 8276, normalized size of antiderivative = 6.36
method | result | size |
default | \(\text {Expression too large to display}\) | \(8276\) |
elliptic | \(\text {Expression too large to display}\) | \(9725\) |
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Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x^2+d\right )}^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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