Optimal. Leaf size=918 \[ -\frac{32 b \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \sqrt{c^2 x^2+1} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right ) d^4}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{32 b c \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^3}{105 \left (-c^2\right )^{3/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) d^2}{3 e^3}-\frac{32 b c \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^2}{105 \left (-c^2\right )^{3/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{d+e x} \left (c^2 x^2+1\right ) d}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{4 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right ) d}{5 e^3}-\frac{4 b c \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d}{105 \left (-c^2\right )^{5/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b (d+e x)^{3/2} \left (c^2 x^2+1\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{4 b c \left (c^2 d^2-3 e^2\right ) \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 \left (-c^2\right )^{5/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]
[Out]
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Rubi [A] time = 3.2528, antiderivative size = 918, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 16, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.762, Rules used = {43, 6310, 12, 6721, 6742, 743, 844, 719, 424, 419, 958, 932, 168, 538, 537, 833} \[ -\frac{32 b \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \sqrt{c^2 x^2+1} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right ) d^4}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{32 b c \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^3}{105 \left (-c^2\right )^{3/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) d^2}{3 e^3}-\frac{32 b c \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^2}{105 \left (-c^2\right )^{3/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{d+e x} \left (c^2 x^2+1\right ) d}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{4 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right ) d}{5 e^3}-\frac{4 b c \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d}{105 \left (-c^2\right )^{5/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b (d+e x)^{3/2} \left (c^2 x^2+1\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{4 b c \left (c^2 d^2-3 e^2\right ) \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 \left (-c^2\right )^{5/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]
Antiderivative was successfully verified.
[In]
[Out]
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
Rule 833
Rubi steps
\begin{align*} \int x^2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{b \int \frac{2 (d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{105 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{(2 b) \int \frac{(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{105 c e^3}\\ &=\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x \sqrt{1+c^2 x^2}} \, dx}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (-\frac{12 d e (d+e x)^{3/2}}{\sqrt{1+c^2 x^2}}+\frac{8 d^2 (d+e x)^{3/2}}{x \sqrt{1+c^2 x^2}}+\frac{15 e^2 x (d+e x)^{3/2}}{\sqrt{1+c^2 x^2}}\right ) \, dx}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{\left (16 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{(d+e x)^{3/2}}{x \sqrt{1+c^2 x^2}} \, dx}{105 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{(d+e x)^{3/2}}{\sqrt{1+c^2 x^2}} \, dx}{35 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 (d+e x)^{3/2}}{\sqrt{1+c^2 x^2}} \, dx}{7 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{16 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{4 b (d+e x)^{3/2} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{\left (16 b d^2 \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}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (16 b d \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}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{1+c^2 x^2}\right ) \int \frac{\left (-\frac{3 e}{2}+\frac{3}{2} c^2 d x\right ) \sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{4 b (d+e x)^{3/2} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{\left (16 b d^4 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (32 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{105 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (32 b d^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (8 b \sqrt{1+c^2 x^2}\right ) \int \frac{-3 c^2 d e+\frac{3}{4} c^2 \left (c^2 d^2-3 e^2\right ) x}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c^5 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (16 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{x}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c e \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 b d \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}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{4 b (d+e x)^{3/2} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}+\frac{\left (16 b d^4 \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}{105 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{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{105 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (16 b d^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \left (c^2 d^2-3 e^2\right ) \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{35 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b d \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}{35 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (64 b \sqrt{-c^2} d^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 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{105 c^3 e^2 \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 (64 b \sqrt{-c^2} d^3 \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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (16 b \sqrt{-c^2} d \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 )}{105 c^5 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{4 b (d+e x)^{3/2} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{64 b \sqrt{-c^2} d^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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{64 b \sqrt{-c^2} d^3 \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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{16 b d \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 )}{105 c^3 \sqrt{-c^2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d^4 \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 )}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (32 b \sqrt{-c^2} d^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 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{105 c^3 e^2 \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 (4 b \sqrt{-c^2} \left (c^2 d^2-3 e^2\right ) \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 )}{35 c^5 e^2 \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 (32 b \sqrt{-c^2} d^3 \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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (4 b \sqrt{-c^2} d \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 )}{35 c^5 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{4 b (d+e x)^{3/2} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{32 b \sqrt{-c^2} d^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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{4 b \sqrt{-c^2} \left (c^2 d^2-3 e^2\right ) \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 )}{35 c^5 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{32 b \sqrt{-c^2} d^3 \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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{16 b d \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 )}{105 c^3 \sqrt{-c^2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b \sqrt{-c^2} d \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 )}{35 c^5 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d^4 \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 )}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{4 b (d+e x)^{3/2} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^3}-\frac{32 b \sqrt{-c^2} d^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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{4 b \sqrt{-c^2} \left (c^2 d^2-3 e^2\right ) \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 )}{35 c^5 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{32 b \sqrt{-c^2} d^3 \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 )}{105 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{16 b d \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 )}{105 c^3 \sqrt{-c^2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b \sqrt{-c^2} d \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 )}{35 c^5 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{32 b d^4 \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 )}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 14.284, size = 1094, normalized size = 1.19 \[ \frac{b \left (-\frac{c \left (\frac{d}{x}+e\right ) x \left (-\frac{16 c^3 \text{csch}^{-1}(c x) d^3}{105 e^3}-\frac{2}{7} c^3 x^3 \text{csch}^{-1}(c x)-\frac{2 c^2 x^2 \left (2 \sqrt{1+\frac{1}{c^2 x^2}} e+c d \text{csch}^{-1}(c x)\right )}{35 e}-\frac{8 c x \left (c d e \sqrt{1+\frac{1}{c^2 x^2}}-c^2 d^2 \text{csch}^{-1}(c x)\right )}{105 e^2}+\frac{4 \left (5 c^2 d^2+9 e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}}}{105 e^2}\right )}{\sqrt{d+e x}}-\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (-\frac{\sqrt{2} \left (9 c^3 e d^3+c e^3 d\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 (8 c^4 d^4-5 c^2 e^2 d^2-9 e^4\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 (-5 c^3 e d^3-9 c e^3 d\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 )}{105 e^3 \sqrt{d+e x}}\right )}{c^4}-\frac{a d^3 \sqrt{d+e x} B_{-\frac{e x}{d}}\left (3,\frac{3}{2}\right )}{e^3 \sqrt{\frac{e x}{d}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.684, size = 2515, normalized size = 2.7 \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^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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