Optimal. Leaf size=2452 \[ \text {result too large to display} \]
[Out]
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Rubi [A] time = 3.97, antiderivative size = 2452, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1721, 1179, 1198, 220, 1196, 1224, 1697, 1715, 1709, 1707, 1217} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 1179
Rule 1196
Rule 1198
Rule 1217
Rule 1224
Rule 1697
Rule 1707
Rule 1709
Rule 1715
Rule 1721
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {c \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2\right )}{\left (c d^2+a e^2\right )^3 \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 )^3 \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 )^2 \sqrt {a+c x^4}}+\frac {c e \left (-B c d^3+3 A c d^2 e+3 a B d e^2-a A e^3\right )}{\left (c d^2+a e^2\right )^3 \left (d+e x^2\right ) \sqrt {a+c x^4}}\right ) \, dx\\ &=\frac {c \int \frac {A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{\left (c d^2+a e^2\right )^3}-\frac {(e (B d-A e)) \int \frac {1}{\left (d+e x^2\right )^3 \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 )^2 \sqrt {a+c x^4}} \, dx}{\left (c d^2+a e^2\right )^2}-\frac {\left (c e \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3}\\ &=\frac {c x \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {e^3 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {c \int \frac {-A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2}{\sqrt {a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )^3}+\frac {(e (B d-A e)) \int \frac {-4 c d^2-3 a e^2+4 c d e x^2-c e^2 x^4}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx}{4 d \left (c d^2+a e^2\right )^2}+\frac {\left (e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \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 )^3}-\frac {\left (c^{3/2} e \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3}+\frac {\left (\sqrt {a} c e^2 \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3}\\ &=\frac {c x \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {3 e^3 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {e^3 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {c e^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^{7/2}}-\frac {c^{5/4} e \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} c^{3/4} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {(e (B d-A e)) \int \frac {8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4-4 c d e \left (4 c d^2+a e^2\right ) x^2-3 c e^2 \left (3 c d^2+a e^2\right ) x^4}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{8 d^2 \left (c d^2+a e^2\right )^3}+\frac {\left (B c d^2-2 A c d e-a B e^2\right ) \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 )^3}-\frac {\left (\sqrt {a} \sqrt {c} 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}}}{\sqrt {a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^3}+\frac {\left (c^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3}+\frac {\left (c \left (A c^2 d^3-a^2 B e^3-\sqrt {a} c^{3/2} d^2 (B d-3 A e)+3 a c d e (B d-A e)+a^{3/2} \sqrt {c} e^2 (3 B d-A e)\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 a \left (c d^2+a e^2\right )^3}\\ &=\frac {c x \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {3 e^3 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {e^3 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {c e^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^{7/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^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}} 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 )^3 \sqrt {a+c x^4}}+\frac {c^{5/4} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \sqrt {a+c x^4}}-\frac {c^{5/4} e \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \sqrt {a+c x^4}}+\frac {c^{3/4} \left (A c^2 d^3-a^2 B e^3-\sqrt {a} c^{3/2} d^2 (B d-3 A e)+3 a c d e (B d-A e)+a^{3/2} \sqrt {c} e^2 (3 B d-A e)\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^{5/4} \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} c^{3/4} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {(B d-A e) \int \frac {-3 \sqrt {a} c^{3/2} d e^2 \left (3 c d^2+a e^2\right )+c e \left (8 c^2 d^4+5 a c d^2 e^2+3 a^2 e^4\right )+\left (3 c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right ) \left (3 c d^2+a e^2\right )-4 c^2 d e^2 \left (4 c d^2+a e^2\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{8 c d^2 \left (c d^2+a e^2\right )^3}-\frac {\left (3 \sqrt {a} \sqrt {c} 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}}}{\sqrt {a+c x^4}} \, dx}{8 d^2 \left (c d^2+a e^2\right )^3}-\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}{d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {a} e^2 \left (3 c d^2+a e^2\right ) \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}{2 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3}\\ &=\frac {c x \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {3 \sqrt {c} e^2 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {c} e^2 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {3 e^3 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {e^3 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {e^{3/2} \left (3 c d^2+a e^2\right ) \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 )}{4 d^{3/2} \left (c d^2+a e^2\right )^{7/2}}-\frac {c e^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^{7/2}}-\frac {3 \sqrt [4]{a} \sqrt [4]{c} e^2 (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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^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}} 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 )^3 \sqrt {a+c x^4}}+\frac {c^{5/4} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \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} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {c^{5/4} e \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \sqrt {a+c x^4}}+\frac {c^{3/4} \left (A c^2 d^3-a^2 B e^3-\sqrt {a} c^{3/2} d^2 (B d-3 A e)+3 a c d e (B d-A e)+a^{3/2} \sqrt {c} e^2 (3 B d-A e)\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^{5/4} \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 c d^2+a e^2\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 )}{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 )^3 \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} c^{3/4} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} e (B d-A e) \left (4 c d^2-\sqrt {a} \sqrt {c} d e+3 a e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{4 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}+\frac {\left (3 \sqrt {a} e^2 (B d-A e) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{8 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3}\\ &=\frac {c x \left (A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {3 \sqrt {c} e^2 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {c} e^2 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {c^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {3 e^3 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {e^3 \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^3 \left (d+e x^2\right )}-\frac {e^{3/2} \left (3 c d^2+a e^2\right ) \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 )}{4 d^{3/2} \left (c d^2+a e^2\right )^{7/2}}-\frac {c e^{3/2} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^{7/2}}-\frac {3 e^{3/2} (B d-A e) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \left (c d^2+a e^2\right )^{7/2}}-\frac {3 \sqrt [4]{a} \sqrt [4]{c} e^2 (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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 d^2 \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} e^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}} 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 )^3 \sqrt {a+c x^4}}+\frac {c^{5/4} \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e (B d-A e) \left (4 c d^2-\sqrt {a} \sqrt {c} d e+3 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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} 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 {\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} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {c^{5/4} e \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 )^3 \sqrt {a+c x^4}}+\frac {c^{3/4} \left (A c^2 d^3-a^2 B e^3-\sqrt {a} c^{3/2} d^2 (B d-3 A e)+3 a c d e (B d-A e)+a^{3/2} \sqrt {c} e^2 (3 B d-A e)\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^{5/4} \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 c d^2+a e^2\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 )}{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 )^3 \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} c^{3/4} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (B c d^3-3 A c d^2 e-3 a B d e^2+a A e^3\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 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}+\frac {3 e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\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 )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3 \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 2.74, size = 630, normalized size = 0.26 \[ \frac {d x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (4 c d^2 \left (d+e x^2\right )^2 \left (B \left (-a^2 e^3+3 a c d e \left (d-e x^2\right )+c^2 d^3 x^2\right )+A c \left (a e^2 \left (e x^2-3 d\right )+c d^2 \left (d-3 e x^2\right )\right )\right )-2 a d e^3 \left (a+c x^4\right ) \left (a e^2+c d^2\right ) (B d-A e)+a e^3 \left (a+c x^4\right ) \left (d+e x^2\right ) \left (3 a A e^3+a B d e^2+17 A c d^2 e-13 B c d^3\right )\right )-\sqrt {\frac {c x^4}{a}+1} \left (d+e x^2\right )^2 \left (\sqrt {a} \sqrt {c} d E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right ) \left (3 A e \left (a^2 e^4+7 a c d^2 e^2-4 c^2 d^4\right )+B \left (a^2 d e^4-25 a c d^3 e^2+4 c^2 d^5\right )\right )+i \left (a e \left (3 A e \left (a^2 e^4+2 a c d^2 e^2+21 c^2 d^4\right )+B \left (a^2 d e^4+26 a c d^3 e^2-35 c^2 d^5\right )\right ) \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right ) \left (-2 i a^{3/2} \sqrt {c} d e^2 (3 B d-A e)-a^2 e^3 (3 A e+B d)+4 i \sqrt {a} c^{3/2} d^3 (B d-2 A e)+19 a c d^2 e (B d-A e)+4 A c^2 d^4\right )\right )\right )}{8 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt {a+c x^4} \left (d+e x^2\right )^2 \left (a d e^2+c d^3\right )^3} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 2326, normalized size = 0.95 \[ \text {result too large to display} \]
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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3}}\,{d x} \]
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 \[ \int \frac {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,{\left (e\,x^2+d\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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