Optimal. Leaf size=278 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}} \]
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Rubi [A] time = 0.282689, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {746, 399, 490, 1218, 444, 63, 298, 205, 208} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\sqrt{b} \sqrt [4]{a^2 d+b^2 c}}-\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{b x \sqrt{a^2 d+b^2 c}} \]
Antiderivative was successfully verified.
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Rule 746
Rule 399
Rule 490
Rule 1218
Rule 444
Rule 63
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \sqrt [4]{c+d x^2}} \, dx &=a \int \frac{1}{\left (a^2-b^2 x^2\right ) \sqrt [4]{c+d x^2}} \, dx-b \int \frac{x}{\left (a^2-b^2 x^2\right ) \sqrt [4]{c+d x^2}} \, dx\\ &=-\left (\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{\left (a^2-b^2 x\right ) \sqrt [4]{c+d x}} \, dx,x,x^2\right )\right )+\frac{\left (2 a \sqrt{-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (b^2 c+a^2 d-b^2 x^4\right ) \sqrt{1-\frac{x^4}{c}}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{x}\\ &=-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{a^2+\frac{b^2 c}{d}-\frac{b^2 x^4}{d}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{d}+\frac{\left (a \sqrt{-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{b^2 c+a^2 d}-b x^2\right ) \sqrt{1-\frac{x^4}{c}}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{b x}-\frac{\left (a \sqrt{-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{b^2 c+a^2 d}+b x^2\right ) \sqrt{1-\frac{x^4}{c}}} \, dx,x,\sqrt [4]{c+d x^2}\right )}{b x}\\ &=-\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{b^2 c+a^2 d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c+d x^2}}{\sqrt [4]{c}}\right )\right |-1\right )}{b \sqrt{b^2 c+a^2 d} x}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{b^2 c+a^2 d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c+d x^2}}{\sqrt [4]{c}}\right )\right |-1\right )}{b \sqrt{b^2 c+a^2 d} x}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2 c+a^2 d}-b x^2} \, dx,x,\sqrt [4]{c+d x^2}\right )+\operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2 c+a^2 d}+b x^2} \, dx,x,\sqrt [4]{c+d x^2}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{b^2 c+a^2 d}}\right )}{\sqrt{b} \sqrt [4]{b^2 c+a^2 d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{b^2 c+a^2 d}}\right )}{\sqrt{b} \sqrt [4]{b^2 c+a^2 d}}-\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{b^2 c+a^2 d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c+d x^2}}{\sqrt [4]{c}}\right )\right |-1\right )}{b \sqrt{b^2 c+a^2 d} x}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{b^2 c+a^2 d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{c+d x^2}}{\sqrt [4]{c}}\right )\right |-1\right )}{b \sqrt{b^2 c+a^2 d} x}\\ \end{align*}
Mathematica [C] time = 0.102812, size = 126, normalized size = 0.45 \[ -\frac{2 \sqrt [4]{\frac{b \left (x-\sqrt{-\frac{c}{d}}\right )}{a+b x}} \sqrt [4]{\frac{b \left (\sqrt{-\frac{c}{d}}+x\right )}{a+b x}} F_1\left (\frac{1}{2};\frac{1}{4},\frac{1}{4};\frac{3}{2};\frac{a-b \sqrt{-\frac{c}{d}}}{a+b x},\frac{a+b \sqrt{-\frac{c}{d}}}{a+b x}\right )}{b \sqrt [4]{c+d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.567, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a}{\frac{1}{\sqrt [4]{d{x}^{2}+c}}}}\, dx \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{1}{{\left (d x^{2} + c\right )}^{\frac{1}{4}}{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right ) \sqrt [4]{c + d x^{2}}}\, dx \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{1}{{\left (d x^{2} + c\right )}^{\frac{1}{4}}{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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