Integrand size = 18, antiderivative size = 212 \[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Time = 0.21 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5334, 1588, 958, 733, 430, 947, 174, 552, 551} \[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]
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Rule 174
Rule 430
Rule 551
Rule 552
Rule 733
Rule 947
Rule 958
Rule 1588
Rule 5334
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac {(2 b) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c e} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{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}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac {\left (2 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \text {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 )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {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 )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 16.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \left (a \sqrt {d+e x}+b \sqrt {d+e x} \sec ^{-1}(c x)+\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-\operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{e} \]
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Time = 5.21 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \operatorname {arcsec}\left (c x \right )-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e}\) | \(252\) |
default | \(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \operatorname {arcsec}\left (c x \right )-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e}\) | \(252\) |
parts | \(\frac {2 a \sqrt {e x +d}}{e}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arcsec}\left (c x \right )-\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e}\) | \(257\) |
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\[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\sqrt {d + e x}}\, dx \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{\sqrt {d+e\,x}} \,d x \]
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