Optimal. Leaf size=530 \[ -\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac{32 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \left (1-c^2 x^2\right ) \sqrt{d+e x}}{15 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}} \]
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Rubi [A] time = 1.90737, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {43, 5247, 12, 6721, 6742, 719, 424, 944, 419, 932, 168, 538, 537, 833, 844} \[ \frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{32 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \left (1-c^2 x^2\right ) \sqrt{d+e x}}{15 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5247
Rule 12
Rule 6721
Rule 6742
Rule 719
Rule 424
Rule 944
Rule 419
Rule 932
Rule 168
Rule 538
Rule 537
Rule 833
Rule 844
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt{d+e x}} \, dx &=\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{b \int \frac{2 \sqrt{d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{15 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{(2 b) \int \frac{\sqrt{d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{\sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{15 c e^3}\\ &=\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{x \sqrt{1-c^2 x^2}} \, dx}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \left (-\frac{4 d e \sqrt{d+e x}}{\sqrt{1-c^2 x^2}}+\frac{8 d^2 \sqrt{d+e x}}{x \sqrt{1-c^2 x^2}}+\frac{3 e^2 x \sqrt{d+e x}}{\sqrt{1-c^2 x^2}}\right ) \, dx}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{\left (16 b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{x \sqrt{1-c^2 x^2}} \, dx}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (8 b d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{15 c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{x \sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{5 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{\left (16 b d^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (16 b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (4 b \sqrt{1-c^2 x^2}\right ) \int \frac{-\frac{e}{2}-\frac{1}{2} c^2 d x}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (16 b d \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{16 b d \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{\left (16 b d^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b d \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{15 c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b (c d-e) (c d+e) \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (32 b d^2 \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{16 b d \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^2 \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{d+\frac{e}{c}-\frac{e x^2}{c}}} \, dx,x,\sqrt{1-c x}\right )}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}+\frac{\left (4 b (c d-e) (c d+e) \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{15 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{4 b d \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^2 \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d^3 \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{c \left (d+\frac{e}{c}\right )}}} \, dx,x,\sqrt{1-c x}\right )}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b \sqrt{d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{2 d^2 \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac{4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{4 b d \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^2 \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{32 b d^3 \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 13.5729, size = 784, normalized size = 1.48 \[ \frac{b \left (-\frac{2 \sqrt{c x} \sqrt{\frac{d}{x}+e} \left (\frac{2 \sqrt{1-c^2 x^2} \left (7 c^2 d^2 e+e^3\right ) \sqrt{\frac{c d+c e x}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}-\frac{6 c d e \cos \left (2 \csc ^{-1}(c x)\right ) \left (c^2 d x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left ((c d+e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right )|\frac{c d-e}{c d+e}\right )-e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right ),\frac{c d-e}{c d+e}\right )\right )}{\sqrt{\frac{e (c x+1)}{e-c d}}}+\left (c^2 x^2-1\right ) (c d+c e x)+c e x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )\right )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c x} \left (c^2 x^2-2\right ) \sqrt{\frac{d}{x}+e}}+\frac{2 \sqrt{1-c^2 x^2} \left (8 c^3 d^3+3 c d e^2\right ) \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}\right )}{15 e^3 \sqrt{d+e x}}-\frac{c x \left (\frac{d}{x}+e\right ) \left (-\frac{16 c^2 d^2 \csc ^{-1}(c x)}{15 e^3}+\frac{4 c d \sqrt{1-\frac{1}{c^2 x^2}}}{5 e^2}-\frac{4 c x \left (e \sqrt{1-\frac{1}{c^2 x^2}}-2 c d \csc ^{-1}(c x)\right )}{15 e^2}-\frac{2 c^2 x^2 \csc ^{-1}(c x)}{5 e}\right )}{\sqrt{d+e x}}\right )}{c^3}-\frac{a d^3 \sqrt{\frac{e x}{d}+1} B_{-\frac{e x}{d}}\left (3,\frac{1}{2}\right )}{e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.276, size = 862, normalized size = 1.6 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{2}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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