Optimal. Leaf size=79 \[ \frac {2 \sqrt [4]{a^2 x^2+b^2}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )}{\sqrt {b}} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {446, 80, 63, 212, 206, 203} \begin {gather*} \frac {2 \sqrt [4]{a^2 x^2+b^2}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 80
Rule 203
Rule 206
Rule 212
Rule 446
Rubi steps
\begin {align*} \int \frac {2 b+a x^2}{x \left (b^2+a^2 x^2\right )^{3/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2 b+a x}{x \left (b^2+a^2 x\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac {2 \sqrt [4]{b^2+a^2 x^2}}{a}+b \operatorname {Subst}\left (\int \frac {1}{x \left (b^2+a^2 x\right )^{3/4}} \, dx,x,x^2\right )\\ &=\frac {2 \sqrt [4]{b^2+a^2 x^2}}{a}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^4}{a^2}} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right )}{a^2}\\ &=\frac {2 \sqrt [4]{b^2+a^2 x^2}}{a}-2 \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right )-2 \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt [4]{b^2+a^2 x^2}\right )\\ &=\frac {2 \sqrt [4]{b^2+a^2 x^2}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 79, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{a^2 x^2+b^2}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a^2 x^2+b^2}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 79, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{b^2+a^2 x^2}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{b^2+a^2 x^2}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 264, normalized size = 3.34 \begin {gather*} \left [-\frac {2 \, a \sqrt {b} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b}}\right ) - a \sqrt {b} \log \left (\frac {a^{2} x^{2} + 2 \, b^{2} - 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} b^{\frac {3}{2}} + 2 \, \sqrt {a^{2} x^{2} + b^{2}} b - 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}} \sqrt {b}}{x^{2}}\right ) - 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} b}{a b}, \frac {2 \, a \sqrt {-b} \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} \sqrt {-b}}{b}\right ) - a \sqrt {-b} \log \left (\frac {a^{2} x^{2} + 2 \, b^{2} - 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} \sqrt {-b} b - 2 \, \sqrt {a^{2} x^{2} + b^{2}} b + 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {3}{4}} \sqrt {-b}}{x^{2}}\right ) + 2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}} b}{a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 69, normalized size = 0.87 \begin {gather*} \frac {2 \, \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {2 \, \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b}}\right )}{\sqrt {b}} + \frac {2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{a} \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 \[\int \frac {a \,x^{2}+2 b}{x \left (a^{2} x^{2}+b^{2}\right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 92, normalized size = 1.16 \begin {gather*} -b {\left (\frac {2 \, \arctan \left (\frac {{\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}} - \frac {\log \left (-\frac {\sqrt {b} - {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{\sqrt {b} + {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}\right )}{b^{\frac {3}{2}}}\right )} + \frac {2 \, {\left (a^{2} x^{2} + b^{2}\right )}^{\frac {1}{4}}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.14, size = 65, normalized size = 0.82 \begin {gather*} \frac {2\,{\left (a^2\,x^2+b^2\right )}^{1/4}}{a}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a^2\,x^2+b^2\right )}^{1/4}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2\,\mathrm {atan}\left (\frac {{\left (a^2\,x^2+b^2\right )}^{1/4}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 7.86, size = 76, normalized size = 0.96 \begin {gather*} a \left (\begin {cases} \frac {x^{2}}{2 \left (b^{2}\right )^{\frac {3}{4}}} & \text {for}\: a^{2} = 0 \\\frac {2 \sqrt [4]{a^{2} x^{2} + b^{2}}}{a^{2}} & \text {otherwise} \end {cases}\right ) - \frac {b \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b^{2} e^{i \pi }}{a^{2} x^{2}}} \right )}}{a^{\frac {3}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________