Optimal. Leaf size=496 \[ -\frac{4 b d \sqrt{1-c^2 x^2} \left (9 c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{105 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac{4 b \sqrt{1-c^2 x^2} \left (5 c^2 d^2-9 e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{105 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^4 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{105 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \left (1-c^2 x^2\right ) \sqrt{d+e x}}{105 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \left (1-c^2 x^2\right ) (d+e x)^{3/2}}{35 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.8273, antiderivative size = 693, normalized size of antiderivative = 1.4, 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, 5247, 12, 6721, 6742, 743, 844, 719, 424, 419, 958, 932, 168, 538, 537, 833} \[ \frac{2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac{32 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{105 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{105 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \left (c^2 d^2+3 e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{35 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^4 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{105 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{105 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \left (1-c^2 x^2\right ) \sqrt{d+e x}}{105 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \left (1-c^2 x^2\right ) (d+e x)^{3/2}}{35 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 5247
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 \csc ^{-1}(c x)\right ) \, dx &=\frac{2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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 (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}{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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-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-c x} \sqrt{1+c 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 (8 b \left (3 c^2 d e^2-\frac{3}{4} c^2 d \left (c^2 d^2+3 e^2\right )\right ) \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{105 c^5 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (64 b d^2 \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 )}{105 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 (64 b d^3 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (16 b d (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 )}{105 c^4 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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac{64 b d^2 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{64 b d^3 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{16 b d (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 )}{105 c^4 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}{c}-\frac{e x^2}{c}}} \, dx,x,\sqrt{1-c x}\right )}{105 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (32 b d^2 \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 )}{105 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 \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 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{35 c^4 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 (32 b d^3 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (16 b \left (3 c^2 d e^2-\frac{3}{4} c^2 d \left (c^2 d^2+3 e^2\right )\right ) \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 )}{105 c^6 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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac{32 b d^2 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \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-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{35 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^3 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{16 b d (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 )}{105 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b d \left (c^2 d^2-e^2\right ) \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 )}{35 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (32 b d^4 \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 )}{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 \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac{2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac{32 b d^2 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \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-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{35 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d^3 \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 )}{105 c^2 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{16 b d (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 )}{105 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b d \left (c^2 d^2-e^2\right ) \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 )}{35 c^4 e^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{32 b d^4 \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 )}{105 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 13.9094, size = 870, normalized size = 1.75 \[ \frac{b \left (-\frac{c \left (\frac{d}{x}+e\right ) x \left (-\frac{16 c^3 \csc ^{-1}(c x) d^3}{105 e^3}-\frac{2}{7} c^3 x^3 \csc ^{-1}(c x)-\frac{2 c^2 x^2 \left (2 \sqrt{1-\frac{1}{c^2 x^2}} e+c d \csc ^{-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 \csc ^{-1}(c x)\right )}{105 e^2}-\frac{4 \left (9 e^2-5 c^2 d^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{2 \left (9 c^3 d^3 e-c d e^3\right ) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \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}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left (8 c^4 d^4+5 c^2 e^2 d^2-9 e^4\right ) \sqrt{\frac{c d+c 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 )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left (9 c d e^3-5 c^3 d^3 e\right ) \cos \left (2 \csc ^{-1}(c x)\right ) \left (d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right ) c^2-\frac{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 ) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c 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 ) c+(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 [B] time = 0.285, size = 1222, normalized size = 2.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} \operatorname{arccsc}\left (c x\right ) + a x^{2}\right )} \sqrt{e x + d}, 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 \sqrt{e x + d}{\left (b \operatorname{arccsc}\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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