Optimal. Leaf size=679 \[ -\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {8 b d^3 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {8 b c d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {8 b c d \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \left (c^2 x^2+1\right ) \sqrt {d+e x}}{15 c^3 x \sqrt {\frac {1}{c^2 x^2}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.52, antiderivative size = 679, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 15, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.790, 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.
[In]
[Out]
Rule 12
Rule 43
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 743
Rule 844
Rule 932
Rule 958
Rule 6310
Rule 6721
Rule 6742
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 = 1.68, size = 418, normalized size = 0.62 \[ \frac {1}{15} \left (\frac {2 a \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}+\frac {4 b x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}{c}+\frac {4 i b \sqrt {-\frac {e (c x-i)}{c d+i e}} \sqrt {-\frac {e (c x+i)}{c d-i e}} \left (\left (c^2 d^2-2 i c d e+e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )-2 c^2 d^2 \Pi \left (1-\frac {i e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+2 c d (c d+i e) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )\right )}{c^3 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {-\frac {c}{c d-i e}}}+\frac {2 b \text {csch}^{-1}(c x) \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.08, size = 1964, normalized size = 2.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________