Optimal. Leaf size=679 \[ \frac{4 b c \sqrt{c^2 x^2+1} \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{8 b c d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{8 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{15 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b \left (c^2 x^2+1\right ) \sqrt{d+e x}}{15 c^3 x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{8 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]
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Rubi [A] time = 2.51914, antiderivative size = 679, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 15, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.79, Rules used = {43, 6310, 12, 6721, 6742, 743, 844, 719, 424, 419, 958, 932, 168, 538, 537} \[ -\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{4 b c \sqrt{c^2 x^2+1} \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{8 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{15 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{8 b c d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b \left (c^2 x^2+1\right ) \sqrt{d+e x}}{15 c^3 x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{8 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6310
Rule 12
Rule 6721
Rule 6742
Rule 743
Rule 844
Rule 719
Rule 424
Rule 419
Rule 958
Rule 932
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int x \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{b \int \frac{2 (d+e x)^{3/2} (-2 d+3 e x)}{15 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{(2 b) \int \frac{(d+e x)^{3/2} (-2 d+3 e x)}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{15 c e^2}\\ &=-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{(d+e x)^{3/2} (-2 d+3 e x)}{x \sqrt{1+c^2 x^2}} \, dx}{15 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (\frac{3 e (d+e x)^{3/2}}{\sqrt{1+c^2 x^2}}-\frac{2 d (d+e x)^{3/2}}{x \sqrt{1+c^2 x^2}}\right ) \, dx}{15 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{(d+e x)^{3/2}}{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{(d+e x)^{3/2}}{\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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \left (\frac{2 d e}{\sqrt{d+e x} \sqrt{1+c^2 x^2}}+\frac{d^2}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}}+\frac{e^2 x}{\sqrt{d+e x} \sqrt{1+c^2 x^2}}\right ) \, 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{1}{2} \left (3 c^2 d^2-e^2\right )+2 c^2 d e 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{4 b \sqrt{d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{15 c \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 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^2 \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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 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 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \left (-c^2 d^2-e^2\right ) \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 \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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{\left (4 b d^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-\sqrt{-c^2} x} \sqrt{1+\sqrt{-c^2} x} \sqrt{d+e x}} \, dx}{15 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 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 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 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 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (16 b \sqrt{-c^2} d \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{\left (16 b \sqrt{-c^2} d^2 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (4 b \sqrt{-c^2} \left (-c^2 d^2-e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c^5 e \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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{16 b \sqrt{-c^2} d \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{16 b \sqrt{-c^2} d^2 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 \sqrt{-c^2} e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (8 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}{\sqrt{-c^2}}-\frac{e x^2}{\sqrt{-c^2}}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{15 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 b \sqrt{-c^2} d \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{\left (8 b \sqrt{-c^2} d^2 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c^3 e \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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{8 b \sqrt{-c^2} d \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{8 b \sqrt{-c^2} d^2 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 \sqrt{-c^2} e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (8 b d^3 \sqrt{1+c^2 x^2} \sqrt{1+\frac{e \left (-1+\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{\sqrt{-c^2} \left (d+\frac{e}{\sqrt{-c^2}}\right )}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{15 c 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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{2 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{8 b \sqrt{-c^2} d \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{8 b \sqrt{-c^2} d^2 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c^3 \sqrt{-c^2} e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{8 b d^3 \sqrt{1+c^2 x^2} \sqrt{1-\frac{e \left (1-\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{15 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 1.60524, size = 418, normalized size = 0.62 \[ \frac{1}{15} \left (\frac{4 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left (\left (c^2 d^2-2 i c d e+e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right ),\frac{c d-i e}{c d+i e}\right )-2 c^2 d^2 \Pi \left (1-\frac{i e}{c d};i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )+2 c d (c d+i e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )\right )}{c^3 e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+\frac{2 a \sqrt{d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}{c}+\frac{2 b \text{csch}^{-1}(c x) \sqrt{d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.32, size = 1964, normalized size = 2.9 \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 \sqrt{e x + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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