Optimal. Leaf size=648 \[ -\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{4 b e \left (c^2 x^2+1\right )}{5 c d^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{d+e x}}-\frac{16 b c e \left (c^2 x^2+1\right )}{15 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right )^2 \sqrt{d+e x}}-\frac{4 b e \left (c^2 x^2+1\right )}{15 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) (d+e x)^{3/2}}-\frac{4 b c \sqrt{c^2 x^2+1} \left (7 c^2 d^2+3 e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 \sqrt{-c^2} e}{\sqrt{-c^2} e-c^2 d}\right )}{15 \sqrt{-c^2} d^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right )^2 \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}+\frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{5 c d^2 e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
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Rubi [A] time = 1.03544, antiderivative size = 785, normalized size of antiderivative = 1.21, number of steps used = 19, number of rules used = 14, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {6290, 1574, 958, 745, 835, 844, 719, 424, 419, 21, 933, 168, 538, 537} \[ -\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{4 b e \left (c^2 x^2+1\right )}{5 c d^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{d+e x}}-\frac{16 b c e \left (c^2 x^2+1\right )}{15 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right )^2 \sqrt{d+e x}}-\frac{4 b e \left (c^2 x^2+1\right )}{15 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) (d+e x)^{3/2}}-\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{d+e x}}+\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{5 c d^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}+\frac{16 b c \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{15 x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right )^2 \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}+\frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{5 c d^2 e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 6290
Rule 1574
Rule 958
Rule 745
Rule 835
Rule 844
Rule 719
Rule 424
Rule 419
Rule 21
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^{7/2}} \, dx &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{(2 b) \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^2 (d+e x)^{5/2}} \, dx}{5 c e}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x (d+e x)^{5/2} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \left (-\frac{e}{d (d+e x)^{5/2} \sqrt{\frac{1}{c^2}+x^2}}-\frac{e}{d^2 (d+e x)^{3/2} \sqrt{\frac{1}{c^2}+x^2}}+\frac{1}{d^2 x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}}\right ) \, dx}{5 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c d^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{(d+e x)^{5/2} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c d \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 c d^2 e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac{4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{\left (4 b c \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{-\frac{d}{2}-\frac{e x}{2}}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{-\frac{3 d}{2}+\frac{e x}{2}}{(d+e x)^{3/2} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 c d \left (d^2+\frac{e^2}{c^2}\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b \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}{5 c d^2 e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac{16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{\left (2 b c \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{\frac{1}{c^2}+x^2}} \, dx}{5 d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (8 b c \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\frac{1}{4} \left (3 d^2-\frac{e^2}{c^2}\right )+d e x}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 d \left (c^2 d^2+e^2\right ) \left (d^2+\frac{e^2}{c^2}\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \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 )}{5 c d^2 e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac{16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 c d \left (d^2+\frac{e^2}{c^2}\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (8 b c \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{\frac{1}{c^2}+x^2}} \, dx}{15 \left (c^2 d^2+e^2\right ) \left (d^2+\frac{e^2}{c^2}\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{-c^2} \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 \left (d-\frac{\sqrt{-c^2} e}{c^2}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d-\frac{\sqrt{-c^2} e}{c^2}}}}+\frac{\left (4 b \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 )}{5 c d^2 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac{16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{4 b \sqrt{-c^2} \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 )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}}}+\frac{4 b \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 )}{5 c d^2 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (16 b \sqrt{-c^2} \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 \left (d-\frac{\sqrt{-c^2} e}{c^2}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c \left (c^2 d^2+e^2\right ) \left (d^2+\frac{e^2}{c^2}\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d-\frac{\sqrt{-c^2} e}{c^2}}}}-\frac{\left (4 b \sqrt{-c^2} \sqrt{\frac{d+e x}{d-\frac{\sqrt{-c^2} e}{c^2}}} \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 \left (d-\frac{\sqrt{-c^2} e}{c^2}\right )}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{15 c^3 d \left (d^2+\frac{e^2}{c^2}\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac{16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac{16 b c \sqrt{-c^2} \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 \left (c^2 d^2+e^2\right )^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}}}+\frac{4 b \sqrt{-c^2} \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 )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}}}-\frac{4 b \sqrt{-c^2} \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}} \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 d \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \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 )}{5 c d^2 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 14.2953, size = 1217, normalized size = 1.88 \[ \frac{b \left (\frac{2 \left (\frac{d}{x}+e\right )^{7/2} (c x)^{7/2} \left (-\frac{\sqrt{2} \left (e^3+c^2 d^2 e\right ) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left (3 c^3 d^3-c d e^2\right ) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 \left (-3 e^3-7 c^2 d^2 e\right ) \cosh \left (2 \text{csch}^{-1}(c x)\right ) \left (\frac{c x \left (c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right )|\frac{c d-i e}{c d+i e}\right )-i e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right ),\frac{c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )\right )}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{c d \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2+2\right )}\right )}{15 c d e \left (c^2 d^2+e^2\right )^2 (d+e x)^{7/2}}-\frac{c^4 \left (\frac{d}{x}+e\right )^4 x^4 \left (-\frac{2 \text{csch}^{-1}(c x) e^2}{5 c^3 d^3 \left (\frac{d}{x}+e\right )^3}+\frac{2 \left (9 \text{csch}^{-1}(c x) e^3-2 c d \sqrt{1+\frac{1}{c^2 x^2}} e^2+9 c^2 d^2 \text{csch}^{-1}(c x) e\right )}{15 c^3 d^3 \left (c^2 d^2+e^2\right ) \left (\frac{d}{x}+e\right )^2}-\frac{2 \left (9 c^4 \text{csch}^{-1}(c x) d^4-16 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} d^3+18 c^2 e^2 \text{csch}^{-1}(c x) d^2-8 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} d+9 e^4 \text{csch}^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2+e^2\right )^2 \left (\frac{d}{x}+e\right )}-\frac{4 \left (7 c^2 d^2+3 e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}}}{15 c^2 d^2 \left (c^2 d^2+e^2\right )^2}+\frac{2 \text{csch}^{-1}(c x)}{5 c^3 d^3 e}\right )}{(d+e x)^{7/2}}\right )}{c}-\frac{2 a}{5 e (d+e x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.326, size = 3782, normalized size = 5.8 \begin{align*} \text{output 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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \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 \operatorname{arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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