Optimal. Leaf size=448 \[ \frac {\left (-1+\sqrt [4]{-1}\right ) \text {ArcTan}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}{(-1)^{3/4} \sqrt [4]{a^2+b} x^2+\sqrt {-b+a x^4}}\right )}{8 \left (a^2+b\right )^{5/8}}+\frac {i \left (i \sqrt {2}+\sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \text {ArcTan}\left (\frac {(-1)^{7/8} \left (-2+\sqrt {2}\right ) \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} \sqrt [4]{a^2+b} x^2+\sqrt {2-\sqrt {2}} \sqrt {-b+a x^4}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (\sqrt {2}-i \sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2+b} x^2-\sqrt [8]{-1} \sqrt {-b+a x^4}}{\sqrt {2-\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (-1+\sqrt [4]{-1}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} \sqrt [4]{a^2+b} x^2-\sqrt [8]{-1} \sqrt {-b+a x^4}}{\sqrt {2+\sqrt {2}} \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{8 \left (a^2+b\right )^{5/8}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.55, antiderivative size = 409, normalized size of antiderivative = 0.91, number of steps
used = 10, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1542, 385,
218, 214, 211} \begin {gather*} -\frac {\sqrt [4]{a-\sqrt {a^2+b}} \text {ArcTan}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \text {ArcTan}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 385
Rule 1542
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{-b+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx &=\int \left (\frac {1-\frac {a}{\sqrt {a^2+b}}}{\left (2 a-2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {1+\frac {a}{\sqrt {a^2+b}}}{\left (2 a+2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=\left (1-\frac {a}{\sqrt {a^2+b}}\right ) \int \frac {1}{\left (2 a-2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx+\left (1+\frac {a}{\sqrt {a^2+b}}\right ) \int \frac {1}{\left (2 a+2 \sqrt {a^2+b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\\ &=\left (1-\frac {a}{\sqrt {a^2+b}}\right ) \text {Subst}\left (\int \frac {1}{2 a-2 \sqrt {a^2+b}-\left (2 b+a \left (2 a-2 \sqrt {a^2+b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (1+\frac {a}{\sqrt {a^2+b}}\right ) \text {Subst}\left (\int \frac {1}{2 a+2 \sqrt {a^2+b}-\left (2 b+a \left (2 a+2 \sqrt {a^2+b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=-\frac {\sqrt {a-\sqrt {a^2+b}} \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+b}}-\sqrt {a^2+b-a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b}}-\frac {\sqrt {a-\sqrt {a^2+b}} \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+b}}+\sqrt {a^2+b-a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b}}+\frac {\sqrt {a+\sqrt {a^2+b}} \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+b}}-\sqrt {a^2+b+a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b}}+\frac {\sqrt {a+\sqrt {a^2+b}} \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+b}}+\sqrt {a^2+b+a \sqrt {a^2+b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b}}\\ &=-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+b}} x}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+b}} x}{\sqrt [4]{a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a^2+b+a \sqrt {a^2+b}}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+b}} x}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+b}} x}{\sqrt [4]{a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a^2+b+a \sqrt {a^2+b}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 10.56, size = 376, normalized size = 0.84 \begin {gather*} \frac {-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \text {ArcTan}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+b}} x}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \text {ArcTan}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+b}} x}{\sqrt [4]{a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a^2+b+a \sqrt {a^2+b}}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b-a \sqrt {a^2+b}} x}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+b+a \sqrt {a^2+b}} x}{\sqrt [4]{a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a^2+b+a \sqrt {a^2+b}}}}{4 \sqrt {a^2+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (x^{8}+2 a \,x^{4}-b \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
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]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (x^8+2\,a\,x^4-b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________