Optimal. Leaf size=429 \[ \frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {4 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}}} \]
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Rubi [A] time = 0.71, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6290, 1574, 958, 719, 419, 933, 168, 538, 537, 844, 424} \[ \frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {4 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 933
Rule 958
Rule 1574
Rule 6290
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {(2 b) \int \frac {(d+e x)^{3/2}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {(d+e x)^{3/2}}{x \sqrt {\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \left (\frac {2 d e}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}}+\frac {d^2}{x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}}+\frac {e^2 x}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {\left (4 b d \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d^2 \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b e \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b d \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d^2 \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}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (8 b \sqrt {-c^2} d \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 )}{3 c^3 \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 )}{3 e}+\frac {8 b \sqrt {-c^2} d \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 )}{3 c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d^2 \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 )}{3 c e \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 )}{3 c^3 \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} d \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 )}{3 c^3 \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 )}{3 e}+\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 )}{3 c^3 \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} d \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 )}{3 c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d^2 \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 )}{3 c e \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 )}{3 e}+\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 )}{3 c^3 \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} d \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 )}{3 c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^2 \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 )}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] time = 14.08, size = 926, normalized size = 2.16 \[ \frac {2 a (d+e x)^{3/2}}{3 e}+\frac {b \left (-\frac {(c d+c e x) \left (-\frac {2}{3} c x \text {csch}^{-1}(c x)-\frac {2 c d \text {csch}^{-1}(c x)}{3 e}-\frac {4}{3} \sqrt {1+\frac {1}{c^2 x^2}}\right )}{\sqrt {d+e x}}-\frac {2 (c d+c e x) \left (-\frac {\sqrt {2} c d e \sqrt {i c x+1} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2} \sqrt {\frac {e (1-i c x)}{i c d+e}}}+\frac {i \sqrt {2} (c d-i e) \left (c^2 d^2+e^2\right ) \sqrt {i c x+1} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}-\frac {2 e \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (\frac {c x \left (c d \sqrt {2 i c x+2} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (c x-i)}{c d+i e}} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e F\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2 i c x+2} \sqrt {-\frac {e (c x+i)}{c d-i e}} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2+2\right )}\right )}{3 e \sqrt {\frac {d}{x}+e} \sqrt {c x} \sqrt {d+e x}}\right )}{c^2} \]
Antiderivative was successfully verified.
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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 \[ \int \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 840, normalized size = 1.96 \[ \frac {\frac {2 a \left (e x +d \right )^{\frac {3}{2}}}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \mathrm {arccsch}\left (c x \right )}{3}+\frac {2 \sqrt {-\frac {i \left (e x +d \right ) c e +\left (e x +d \right ) c^{2} d -c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i \left (e x +d \right ) c e -\left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (i \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -2 \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-i \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e +\EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}-\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}+\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} x^{2} e^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e} \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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