Optimal. Leaf size=66 \[ \sqrt [4]{a x^4+b}-\frac {1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )-\frac {1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 50, 63, 212, 206, 203} \begin {gather*} \sqrt [4]{a x^4+b}-\frac {1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )-\frac {1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{b+a x^4}}{x} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{b+a x}}{x} \, dx,x,x^4\right )\\ &=\sqrt [4]{b+a x^4}+\frac {1}{4} b \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^{3/4}} \, dx,x,x^4\right )\\ &=\sqrt [4]{b+a x^4}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^4}\right )}{a}\\ &=\sqrt [4]{b+a x^4}-\frac {1}{2} \sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )-\frac {1}{2} \sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^4}\right )\\ &=\sqrt [4]{b+a x^4}-\frac {1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )-\frac {1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 66, normalized size = 1.00 \begin {gather*} \sqrt [4]{a x^4+b}-\frac {1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right )-\frac {1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^4+b}}{\sqrt [4]{b}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.06, size = 66, normalized size = 1.00 \begin {gather*} \sqrt [4]{b+a x^4}-\frac {1}{2} \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right )-\frac {1}{2} \sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^4}}{\sqrt [4]{b}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 93, normalized size = 1.41 \begin {gather*} b^{\frac {1}{4}} \arctan \left (\frac {b^{\frac {3}{4}} \sqrt {\sqrt {a x^{4} + b} + \sqrt {b}} - {\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{\frac {3}{4}}}{b}\right ) - \frac {1}{4} \, b^{\frac {1}{4}} \log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}\right ) + \frac {1}{4} \, b^{\frac {1}{4}} \log \left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}\right ) + {\left (a x^{4} + b\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.87, size = 183, normalized size = 2.77 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \left (-b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-b\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right ) - \frac {1}{8} \, \sqrt {2} \left (-b\right )^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right ) + \frac {1}{8} \, \sqrt {2} \left (-b\right )^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} + \sqrt {-b}\right ) + {\left (a x^{4} + b\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 66, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, b^{\frac {1}{4}} \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) + \frac {1}{4} \, b^{\frac {1}{4}} \log \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right ) + {\left (a x^{4} + b\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.82, size = 48, normalized size = 0.73 \begin {gather*} {\left (a\,x^4+b\right )}^{1/4}-\frac {b^{1/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )}{2}-\frac {b^{1/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^4+b\right )}^{1/4}}{b^{1/4}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.93, size = 42, normalized size = 0.64 \begin {gather*} - \frac {\sqrt [4]{a} x \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________