Optimal. Leaf size=2452 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 3.97126, antiderivative size = 2452, normalized size of antiderivative = 1., 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 1721
Rule 1179
Rule 1198
Rule 220
Rule 1196
Rule 1224
Rule 1697
Rule 1715
Rule 1709
Rule 1707
Rule 1217
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*}
Mathematica [C] time = 3.03304, size = 630, normalized size = 0.26 \[ \frac{d x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (4 c d^2 \left (d+e x^2\right )^2 \left (B \left (-a^2 e^3+3 a c d e \left (d-e x^2\right )+c^2 d^3 x^2\right )+A c \left (a e^2 \left (e x^2-3 d\right )+c d^2 \left (d-3 e x^2\right )\right )\right )-2 a d e^3 \left (a+c x^4\right ) \left (a e^2+c d^2\right ) (B d-A e)+a e^3 \left (a+c x^4\right ) \left (d+e x^2\right ) \left (3 a A e^3+a B d e^2+17 A c d^2 e-13 B c d^3\right )\right )-\sqrt{\frac{c x^4}{a}+1} \left (d+e x^2\right )^2 \left (i \left (\sqrt{c} d \left (\sqrt{c} d-i \sqrt{a} e\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}\right ),-1\right ) \left (-2 i a^{3/2} \sqrt{c} d e^2 (3 B d-A e)-a^2 e^3 (3 A e+B d)+4 i \sqrt{a} c^{3/2} d^3 (B d-2 A e)+19 a c d^2 e (B d-A e)+4 A c^2 d^4\right )+a e \left (3 A e \left (a^2 e^4+2 a c d^2 e^2+21 c^2 d^4\right )+B \left (a^2 d e^4+26 a c d^3 e^2-35 c^2 d^5\right )\right ) \Pi \left (-\frac{i \sqrt{a} e}{\sqrt{c} d};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )+\sqrt{a} \sqrt{c} d E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right ) \left (3 A e \left (a^2 e^4+7 a c d^2 e^2-4 c^2 d^4\right )+B \left (a^2 d e^4-25 a c d^3 e^2+4 c^2 d^5\right )\right )\right )}{8 a \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4} \left (d+e x^2\right )^2 \left (a d e^2+c d^3\right )^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.035, size = 2326, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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