Optimal. Leaf size=67 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {\log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ] \]
________________________________________________________________________________________
Rubi [B] time = 0.53, antiderivative size = 401, normalized size of antiderivative = 5.99, number of steps used = 12, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2056, 1270, 1529, 377, 212, 206, 203} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{5/8} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 377
Rule 1270
Rule 1529
Rule 2056
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {x^{3/2}}{\sqrt [4]{b+a x^2} \left (-b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^4} \left (-b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{2 \sqrt {a} \left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}}+\frac {1}{2 \sqrt {a} \left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{5/8} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.40, size = 66, normalized size = 0.99 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^2}{\left (b-a\,x^4\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________