Optimal. Leaf size=918 \[ -\frac {32 b \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {c^2 x^2+1} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right ) d^4}{105 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {32 b c \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {c^2 x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right ) d^3}{105 \left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) d^2}{3 e^3}-\frac {32 b c \sqrt {d+e x} \sqrt {c^2 x^2+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right ) d^2}{105 \left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {4 b \sqrt {d+e x} \left (c^2 x^2+1\right ) d}{105 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {4 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right ) d}{5 e^3}-\frac {4 b c \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {c^2 x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right ) d}{105 \left (-c^2\right )^{5/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (d+e x)^{3/2} \left (c^2 x^2+1\right )}{35 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^3}-\frac {4 b c \left (c^2 d^2-3 e^2\right ) \sqrt {d+e x} \sqrt {c^2 x^2+1} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{35 \left (-c^2\right )^{5/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}} \]
[Out]
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Rubi [A] time = 3.25, antiderivative size = 918, normalized size of antiderivative = 1.00, 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 12
Rule 43
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 743
Rule 833
Rule 844
Rule 932
Rule 958
Rule 6310
Rule 6721
Rule 6742
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*}
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Mathematica [C] time = 14.40, 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}} F\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}} F\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 F\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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 2515, normalized size = 2.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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