Optimal. Leaf size=86 \[ -\frac {2 \sqrt {\frac {f^2 (c+d x)}{c f^2+d}} \Pi \left (\frac {2 b}{a f^2+b};\sin ^{-1}\left (\frac {\sqrt {1-f^2 x}}{\sqrt {2}}\right )|\frac {2 d}{c f^2+d}\right )}{\left (a f^2+b\right ) \sqrt {c+d x}} \]
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Rubi [A] time = 0.17, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {932, 168, 538, 537} \[ -\frac {2 \sqrt {\frac {f^2 (c+d x)}{c f^2+d}} \Pi \left (\frac {2 b}{a f^2+b};\sin ^{-1}\left (\frac {\sqrt {1-f^2 x}}{\sqrt {2}}\right )|\frac {2 d}{c f^2+d}\right )}{\left (a f^2+b\right ) \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 537
Rule 538
Rule 932
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^4 x^2}} \, dx &=\int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {1-f^2 x} \sqrt {1+f^2 x}} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (b+a f^2-b x^2\right ) \sqrt {c+\frac {d}{f^2}-\frac {d x^2}{f^2}}} \, dx,x,\sqrt {1-f^2 x}\right )\right )\\ &=-\frac {\left (2 \sqrt {\frac {f^2 (c+d x)}{d+c f^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (b+a f^2-b x^2\right ) \sqrt {1-\frac {d x^2}{\left (c+\frac {d}{f^2}\right ) f^2}}} \, dx,x,\sqrt {1-f^2 x}\right )}{\sqrt {c+d x}}\\ &=-\frac {2 \sqrt {\frac {f^2 (c+d x)}{d+c f^2}} \Pi \left (\frac {2 b}{b+a f^2};\sin ^{-1}\left (\frac {\sqrt {1-f^2 x}}{\sqrt {2}}\right )|\frac {2 d}{d+c f^2}\right )}{\left (b+a f^2\right ) \sqrt {c+d x}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 218, normalized size = 2.53 \[ \frac {2 i (c+d x) \sqrt {\frac {d \left (f^2 x-1\right )}{f^2 (c+d x)}} \sqrt {\frac {d \left (f^2 x+1\right )}{f^2 (c+d x)}} \left (\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {-c-\frac {d}{f^2}}}{\sqrt {c+d x}}\right ),\frac {c f^2-d}{c f^2+d}\right )-\Pi \left (\frac {(b c-a d) f^2}{b \left (c f^2+d\right )};i \sinh ^{-1}\left (\frac {\sqrt {-c-\frac {d}{f^2}}}{\sqrt {c+d x}}\right )|\frac {c f^2-d}{c f^2+d}\right )\right )}{\sqrt {1-f^4 x^2} \sqrt {-c-\frac {d}{f^2}} (a d-b c)} \]
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 \[ \int \frac {1}{\sqrt {-f^{4} x^{2} + 1} {\left (b x + a\right )} \sqrt {d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 205, normalized size = 2.38 \[ -\frac {2 \left (c \,f^{2}-d \right ) \sqrt {-\frac {\left (f^{2} x +1\right ) d}{c \,f^{2}-d}}\, \sqrt {-\frac {\left (f^{2} x -1\right ) d}{c \,f^{2}+d}}\, \sqrt {\frac {\left (d x +c \right ) f^{2}}{c \,f^{2}-d}}\, \sqrt {-f^{4} x^{2}+1}\, \sqrt {d x +c}\, \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f^{2}}{c \,f^{2}-d}}, -\frac {\left (c \,f^{2}-d \right ) b}{\left (a d -b c \right ) f^{2}}, \sqrt {\frac {c \,f^{2}-d}{c \,f^{2}+d}}\right )}{\left (a d -b c \right ) \left (d \,f^{4} x^{3}+c \,f^{4} x^{2}-d x -c \right ) f^{2}} \]
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 \[ \int \frac {1}{\sqrt {-f^{4} x^{2} + 1} {\left (b x + a\right )} \sqrt {d x + c}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-f^4\,x^2}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \left (f^{2} x - 1\right ) \left (f^{2} x + 1\right )} \left (a + b x\right ) \sqrt {c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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