Optimal. Leaf size=110 \[ -\frac {2 \sqrt {\frac {\sqrt {-c} (f+g x)}{\sqrt {-c} f+g}} \Pi \left (\frac {2 e}{\sqrt {-c} d+e};\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c} x}}{\sqrt {2}}\right )|\frac {2 g}{\sqrt {-c} f+g}\right )}{\left (\sqrt {-c} d+e\right ) \sqrt {f+g x}} \]
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Rubi [A] time = 0.30, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {932, 168, 538, 537} \[ -\frac {2 \sqrt {\frac {\sqrt {-c} (f+g x)}{\sqrt {-c} f+g}} \Pi \left (\frac {2 e}{\sqrt {-c} d+e};\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c} x}}{\sqrt {2}}\right )|\frac {2 g}{\sqrt {-c} f+g}\right )}{\left (\sqrt {-c} d+e\right ) \sqrt {f+g x}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 537
Rule 538
Rule 932
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx &=\int \frac {1}{\sqrt {1-\sqrt {-c} x} \sqrt {1+\sqrt {-c} x} (d+e x) \sqrt {f+g x}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\sqrt {-c} d+e-e x^2\right ) \sqrt {f+\frac {g}{\sqrt {-c}}-\frac {g x^2}{\sqrt {-c}}}} \, dx,x,\sqrt {1-\sqrt {-c} x}\right )\right )\\ &=-\frac {\left (2 \sqrt {1+\frac {g \left (-1+\sqrt {-c} x\right )}{\sqrt {-c} f+g}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\sqrt {-c} d+e-e x^2\right ) \sqrt {1-\frac {g x^2}{\sqrt {-c} \left (f+\frac {g}{\sqrt {-c}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c} x}\right )}{\sqrt {f+g x}}\\ &=-\frac {2 \sqrt {1-\frac {g \left (1-\sqrt {-c} x\right )}{\sqrt {-c} f+g}} \Pi \left (\frac {2 e}{\sqrt {-c} d+e};\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c} x}}{\sqrt {2}}\right )|\frac {2 g}{\sqrt {-c} f+g}\right )}{\left (\sqrt {-c} d+e\right ) \sqrt {f+g x}}\\ \end {align*}
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Mathematica [C] time = 0.95, size = 261, normalized size = 2.37 \[ -\frac {2 i (f+g x) \sqrt {\frac {g \left (x+\frac {i}{\sqrt {c}}\right )}{f+g x}} \sqrt {-\frac {-g x+\frac {i g}{\sqrt {c}}}{f+g x}} \left (F\left (i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i g}{\sqrt {c} f+i g}\right )-\Pi \left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i g\right )};i \sinh ^{-1}\left (\frac {\sqrt {-f-\frac {i g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i g}{\sqrt {c} f+i g}\right )\right )}{\sqrt {c x^2+1} \sqrt {-f-\frac {i g}{\sqrt {c}}} (e f-d g)} \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + 1} {\left (e x + d\right )} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 215, normalized size = 1.95 \[ \frac {2 \left (\sqrt {-c}\, f +g \right ) \sqrt {-\frac {\left (\sqrt {-c}\, x -1\right ) g}{\sqrt {-c}\, f +g}}\, \sqrt {-\frac {\left (\sqrt {-c}\, x +1\right ) g}{\sqrt {-c}\, f -g}}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {-c}}{\sqrt {-c}\, f +g}}\, \sqrt {c \,x^{2}+1}\, \sqrt {g x +f}\, \EllipticPi \left (\sqrt {\frac {\left (g x +f \right ) \sqrt {-c}}{\sqrt {-c}\, f +g}}, -\frac {\left (\sqrt {-c}\, f +g \right ) e}{\sqrt {-c}\, \left (d g -e f \right )}, \sqrt {\frac {\sqrt {-c}\, f +g}{\sqrt {-c}\, f -g}}\right )}{\sqrt {-c}\, \left (d g -e f \right ) \left (c g \,x^{3}+c f \,x^{2}+g x +f \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + 1} {\left (e x + d\right )} \sqrt {g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+1}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right ) \sqrt {f + g x} \sqrt {c x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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