Optimal. Leaf size=110 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\log \left (\sqrt [4]{a x^4+b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}} \]
________________________________________________________________________________________
Rubi [B] time = 0.58, antiderivative size = 285, normalized size of antiderivative = 2.59, number of steps used = 15, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6725, 240, 212, 206, 203, 1429, 377} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {x \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {x \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {x \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {x \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^4+b}}\right )}{2 \sqrt [4]{\sqrt {-a} \sqrt {b}+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 240
Rule 377
Rule 1429
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b+a x^8}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx &=\int \left (\frac {1}{\sqrt [4]{b+a x^4}}-\frac {2 b}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )}\right ) \, dx\\ &=-\left ((2 b) \int \frac {1}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx\right )+\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx\\ &=-\left (\left (\sqrt {-a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx\right )-\left (\sqrt {-a} \sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\left (\sqrt {-a} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\left (\sqrt {-a} \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+a x^8}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.26, size = 110, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt [4]{a}}+\frac {1}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.79, size = 1032, normalized size = 9.38
result too large to display
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 {a x^{8} - b}{{\left (a x^{8} + b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}-b}{\left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (a \,x^{8}+b \right )}\, dx\]
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 {a x^{8} - b}{{\left (a x^{8} + b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}\,{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 {b-a\,x^8}{{\left (a\,x^4+b\right )}^{1/4}\,\left (a\,x^8+b\right )} \,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 {a x^{8} - b}{\sqrt [4]{a x^{4} + b} \left (a x^{8} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________