Optimal. Leaf size=1605 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 9.67962, antiderivative size = 1605, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6728, 1725, 1217, 220, 1707, 1248, 725, 206} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6728
Rule 1725
Rule 1217
Rule 220
Rule 1707
Rule 1248
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (c+d x+e x^2\right ) \sqrt{a+b x^4}} \, dx &=\int \left (\frac{2 e}{\sqrt{d^2-4 c e} \left (d-\sqrt{d^2-4 c e}+2 e x\right ) \sqrt{a+b x^4}}-\frac{2 e}{\sqrt{d^2-4 c e} \left (d+\sqrt{d^2-4 c e}+2 e x\right ) \sqrt{a+b x^4}}\right ) \, dx\\ &=\frac{(2 e) \int \frac{1}{\left (d-\sqrt{d^2-4 c e}+2 e x\right ) \sqrt{a+b x^4}} \, dx}{\sqrt{d^2-4 c e}}-\frac{(2 e) \int \frac{1}{\left (d+\sqrt{d^2-4 c e}+2 e x\right ) \sqrt{a+b x^4}} \, dx}{\sqrt{d^2-4 c e}}\\ &=-\frac{\left (4 e^2\right ) \int \frac{x}{\left (\left (d-\sqrt{d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt{a+b x^4}} \, dx}{\sqrt{d^2-4 c e}}+\frac{\left (4 e^2\right ) \int \frac{x}{\left (\left (d+\sqrt{d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt{a+b x^4}} \, dx}{\sqrt{d^2-4 c e}}-\left (2 e \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right )\right ) \int \frac{1}{\left (\left (d-\sqrt{d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt{a+b x^4}} \, dx-\left (2 e \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right )\right ) \int \frac{1}{\left (\left (d+\sqrt{d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt{a+b x^4}} \, dx\\ &=-\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (d-\sqrt{d^2-4 c e}\right )^2-4 e^2 x\right ) \sqrt{a+b x^2}} \, dx,x,x^2\right )}{\sqrt{d^2-4 c e}}+\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (d+\sqrt{d^2-4 c e}\right )^2-4 e^2 x\right ) \sqrt{a+b x^2}} \, dx,x,x^2\right )}{\sqrt{d^2-4 c e}}-\frac{\left (\sqrt{b} e \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )}-\frac{\left (4 \sqrt{a} e^3 \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right )\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (\left (d-\sqrt{d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt{a+b x^4}} \, dx}{2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )}-\frac{\left (\sqrt{b} e \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )}-\frac{\left (4 \sqrt{a} e^3 \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right )\right ) \int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a}}}{\left (\left (d+\sqrt{d^2-4 c e}\right )^2-4 e^2 x^2\right ) \sqrt{a+b x^4}} \, dx}{2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )}\\ &=-\frac{e^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4-b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt{d^2-4 c e}\right ) \sqrt{a+b x^4}}\right )}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2+d^3 \sqrt{d^2-4 c e}-2 c d e \sqrt{d^2-4 c e}\right )}}+\frac{e^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4+b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt{d^2-4 c e}\right ) \sqrt{a+b x^4}}\right )}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2-d^3 \sqrt{d^2-4 c e}+2 c d e \sqrt{d^2-4 c e}\right )}}-\frac{\sqrt [4]{b} e \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} e \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}-\frac{\sqrt [4]{a} e \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (4 e^2-\frac{\sqrt{b} \left (d-\sqrt{d^2-4 c e}\right )^2}{\sqrt{a}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (\frac{\left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )\right )^2}{4 \sqrt{a} \sqrt{b} e^2 \left (d-\sqrt{d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{b} \left (d-\sqrt{d^2-4 c e}\right )^2 \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}-\frac{\sqrt [4]{a} e \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (4 e^2-\frac{\sqrt{b} \left (d+\sqrt{d^2-4 c e}\right )^2}{\sqrt{a}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (\frac{\left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )\right )^2}{4 \sqrt{a} \sqrt{b} e^2 \left (d+\sqrt{d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{b} \left (d+\sqrt{d^2-4 c e}\right )^2 \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}+\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{16 a e^4+b \left (d-\sqrt{d^2-4 c e}\right )^4-x^2} \, dx,x,\frac{-4 a e^2-b \left (d-\sqrt{d^2-4 c e}\right )^2 x^2}{\sqrt{a+b x^4}}\right )}{\sqrt{d^2-4 c e}}-\frac{\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{16 a e^4+b \left (d+\sqrt{d^2-4 c e}\right )^4-x^2} \, dx,x,\frac{-4 a e^2-b \left (d+\sqrt{d^2-4 c e}\right )^2 x^2}{\sqrt{a+b x^4}}\right )}{\sqrt{d^2-4 c e}}\\ &=-\frac{e^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4-b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d+\sqrt{d^2-4 c e}\right ) \sqrt{a+b x^4}}\right )}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2+d^3 \sqrt{d^2-4 c e}-2 c d e \sqrt{d^2-4 c e}\right )}}+\frac{e^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{-b d^4+4 b c d^2 e-2 b c^2 e^2-2 a e^4+b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )} x}{e \left (d-\sqrt{d^2-4 c e}\right ) \sqrt{a+b x^4}}\right )}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{-2 a e^4-b \left (d^4-4 c d^2 e+2 c^2 e^2-d^3 \sqrt{d^2-4 c e}+2 c d e \sqrt{d^2-4 c e}\right )}}-\frac{e^2 \tanh ^{-1}\left (\frac{4 a e^2+b \left (d-\sqrt{d^2-4 c e}\right )^2 x^2}{2 \sqrt{2} \sqrt{b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4-b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )} \sqrt{a+b x^4}}\right )}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4-b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )}}+\frac{e^2 \tanh ^{-1}\left (\frac{4 a e^2+b \left (d+\sqrt{d^2-4 c e}\right )^2 x^2}{2 \sqrt{2} \sqrt{b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4+b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )} \sqrt{a+b x^4}}\right )}{\sqrt{2} \sqrt{d^2-4 c e} \sqrt{b d^4-4 b c d^2 e+2 b c^2 e^2+2 a e^4+b d \sqrt{d^2-4 c e} \left (d^2-2 c e\right )}}-\frac{\sqrt [4]{b} e \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}-\frac{\sqrt [4]{b} e \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}-\frac{\sqrt [4]{a} e \left (1-\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (4 e^2-\frac{\sqrt{b} \left (d-\sqrt{d^2-4 c e}\right )^2}{\sqrt{a}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (\frac{\left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )\right )^2}{4 \sqrt{a} \sqrt{b} e^2 \left (d-\sqrt{d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{b} \left (d-\sqrt{d^2-4 c e}\right )^2 \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e-d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}-\frac{\sqrt [4]{a} e \left (1+\frac{d}{\sqrt{d^2-4 c e}}\right ) \left (4 e^2-\frac{\sqrt{b} \left (d+\sqrt{d^2-4 c e}\right )^2}{\sqrt{a}}\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \Pi \left (\frac{\left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )\right )^2}{4 \sqrt{a} \sqrt{b} e^2 \left (d+\sqrt{d^2-4 c e}\right )^2};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{b} \left (d+\sqrt{d^2-4 c e}\right )^2 \left (2 \sqrt{a} e^2+\sqrt{b} \left (d^2-2 c e+d \sqrt{d^2-4 c e}\right )\right ) \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 7.37276, size = 1416, normalized size = 0.88 \[ -\frac{i \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left (-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left (-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right ) d^2}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left (-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right ) \left (\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt{b x^4+a}}-\frac{i \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left (-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left (-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right ) d^2}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left (-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right ) \left (\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt{b x^4+a}}-\frac{\sqrt{2} e^2 \left (\frac{\tanh ^{-1}\left (\frac{2 a e^2+b \left (d^2-\sqrt{d^2-4 c e} d-2 c e\right ) x^2}{\sqrt{4 a e^4+b \left (2 d^4-2 \sqrt{d^2-4 c e} d^3-8 c e d^2+4 c e \sqrt{d^2-4 c e} d+4 c^2 e^2\right )} \sqrt{b x^4+a}}\right )}{2 \sqrt{2 a e^4+b \left (d^4-\sqrt{d^2-4 c e} d^3-4 c e d^2+2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right )}}-\frac{\tanh ^{-1}\left (\frac{2 a e^2+b \left (d^2+\sqrt{d^2-4 c e} d-2 c e\right ) x^2}{\sqrt{4 a e^4+2 b \left (d^4+\sqrt{d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right )} \sqrt{b x^4+a}}\right )}{2 \sqrt{2 a e^4+b \left (d^4+\sqrt{d^2-4 c e} d^3-4 c e d^2-2 c e \sqrt{d^2-4 c e} d+2 c^2 e^2\right )}}\right )}{\sqrt{d^2-4 c e}}-\frac{i \sqrt{d^4-4 c d^2 e} \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left (-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left (-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left (-d^2+2 c e-\sqrt{d^4-4 c d^2 e}\right ) \left (\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt{b x^4+a}}+\frac{i \sqrt{d^4-4 c d^2 e} \sqrt{1-\frac{i \sqrt{b} x^2}{\sqrt{a}}} \sqrt{\frac{i \sqrt{b} x^2}{\sqrt{a}}+1} \Pi \left (-\frac{2 i \sqrt{a} e^2}{\sqrt{b} \left (-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right )};\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} e \left (-d^2+2 c e+\sqrt{d^4-4 c d^2 e}\right ) \left (\frac{d^2-2 c e-\sqrt{d^4-4 c d^2 e}}{2 e^2}-\frac{d^2-2 c e+\sqrt{d^4-4 c d^2 e}}{2 e^2}\right ) \sqrt{b x^4+a}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.069, size = 1153, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a}{\left (e x^{2} + d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x^{4}} \left (c + d x + e x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a}{\left (e x^{2} + d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]