Optimal. Leaf size=1494 \[ \frac {c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} (B d-A e) \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {a^{3/4} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.36, antiderivative size = 1494, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1735, 1193,
1212, 226, 1210, 1238, 1729, 1723, 1721, 1231} \begin {gather*} -\frac {(B d-A e) x \sqrt {c x^4+a} e^3}{2 d \left (c d^2+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+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e^2}{2 d \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+a} e^2}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2+a e^2\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (B c d^2-2 A c e d-a B e^2\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} (B d-A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {c x^4+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B e d-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt {c} \left (B c d^2-2 A c e d-a B e^2\right ) x \sqrt {c x^4+a}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {c x \left (A c d^2+2 a B e d-a A e^2+\left (B c d^2-2 A c e d-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 226
Rule 1193
Rule 1210
Rule 1212
Rule 1231
Rule 1238
Rule 1721
Rule 1723
Rule 1729
Rule 1735
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {c \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )^{3/2}}+\frac {e (-B d+A e)}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (-B c d^2+2 A c d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+c x^4}}\right ) \, dx\\ &=\frac {c \int \frac {A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{\left (c d^2+a e^2\right )^2}-\frac {(e (B d-A e)) \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx}{c d^2+a e^2}-\frac {\left (e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {c \int \frac {-A c d^2-2 a B d e+a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2}{\sqrt {a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )^2}+\frac {(e (B d-A e)) \int \frac {-2 c d^2-a e^2+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {c} e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {a} e^2 \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac {c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+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-a e^2\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+c x^4}} \, dx}{2 c d \left (c d^2+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+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {c} \left (B c d^2-2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt {a}}\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}\\ &=\frac {c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} e (B d-A e)\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )}+\frac {\left (\sqrt {a} e^2 (B d-A e) \left (3 c d^2+a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac {c x \left (A c d^2+2 a B d e-a A e^2+\left (B c d^2-2 A c d e-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} (B d-A e) \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^2 (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B d e-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (B c d^2-2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+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}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.85, size = 427, normalized size = 0.29 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d \left (a e^3 (-B d+A e) x \left (a+c x^4\right )+c d x \left (d+e x^2\right ) \left (-a A e^2+B c d^2 x^2+A c d \left (d-2 e x^2\right )+a B e \left (2 d-e x^2\right )\right )\right )-\left (d+e x^2\right ) \sqrt {1+\frac {c x^4}{a}} \left (-\sqrt {a} \sqrt {c} d \left (-B c d^3+2 A c d^2 e+2 a B d e^2-a A e^3\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (A c d^2+i \sqrt {a} \sqrt {c} d (B d-A e)+a e (2 B d-A e)\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+a e \left (-5 B c d^3+7 A c d^2 e+a B d e^2+a A e^3\right ) \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )\right )\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (c d^3+a d e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 1384, normalized size = 0.93
method | result | size |
default | \(\text {Expression too large to display}\) | \(1384\) |
elliptic | \(\text {Expression too large to display}\) | \(1664\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {A + B x^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{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 {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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