Optimal. Leaf size=89 \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right )}{2 \sqrt{b}} \]
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Rubi [A] time = 0.0599722, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {1885, 224, 221, 275, 217, 206} \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1885
Rule 224
Rule 221
Rule 275
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{c+d x}{\sqrt{-a+b x^4}} \, dx &=\int \left (\frac{c}{\sqrt{-a+b x^4}}+\frac{d x}{\sqrt{-a+b x^4}}\right ) \, dx\\ &=c \int \frac{1}{\sqrt{-a+b x^4}} \, dx+d \int \frac{x}{\sqrt{-a+b x^4}} \, dx\\ &=\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+b x^2}} \, dx,x,x^2\right )+\frac{\left (c \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{\sqrt{-a+b x^4}}\\ &=\frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{-a+b x^4}}+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{-a+b x^4}}\right )\\ &=\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a+b x^4}}\right )}{2 \sqrt{b}}+\frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{-a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0420012, size = 83, normalized size = 0.93 \[ \frac{c x \sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b x^4}{a}\right )}{\sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 95, normalized size = 1.1 \begin{align*}{\frac{d}{2}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}-a} \right ){\frac{1}{\sqrt{b}}}}+{c\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x + c}{\sqrt{b x^{4} - a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d x + c}{\sqrt{b x^{4} - a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.96492, size = 92, normalized size = 1.03 \begin{align*} d \left (\begin{cases} \frac{\operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \frac{\left |{b x^{4}}\right |}{\left |{a}\right |} > 1 \\- \frac{i \operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases}\right ) - \frac{i c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x + c}{\sqrt{b x^{4} - a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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