Optimal. Leaf size=141 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{7/4}} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {494, 481, 298, 203, 206} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 298
Rule 481
Rule 494
Rubi steps
\begin {align*} \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (b+a x^4\right )} \, dx &=-\left (b \operatorname {Subst}\left (\int \frac {x^6}{\left (1-a x^4\right ) \left (b-2 a b x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{a}-\frac {b \operatorname {Subst}\left (\int \frac {x^2}{b-2 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{3/2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2\ 2^{3/4} a^{7/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 67, normalized size = 0.48 \begin {gather*} \frac {x^7 \left (\frac {b-a x^4}{b}\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};\frac {a x^4}{b},-\frac {a x^4}{b}\right )}{7 b \left (a x^4-b\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.81, size = 141, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{7/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{7/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2\ 2^{3/4} a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 318, normalized size = 2.26 \begin {gather*} \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\left (\frac {1}{8}\right )^{\frac {3}{4}} a^{5} \sqrt {\frac {2 \, \sqrt {\frac {1}{2}} a^{4} \sqrt {\frac {1}{a^{7}}} x^{2} + \sqrt {a x^{4} - b}}{x^{2}}} \frac {1}{a^{7}}^{\frac {3}{4}} x - \left (\frac {1}{8}\right )^{\frac {3}{4}} {\left (a x^{4} - b\right )}^{\frac {1}{4}} a^{5} \frac {1}{a^{7}}^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{8}\right )^{\frac {1}{4}} \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x - {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{a^{7}}^{\frac {1}{4}} \arctan \left (\frac {a^{5} \frac {1}{a^{7}}^{\frac {3}{4}} x \sqrt {\frac {a^{4} \sqrt {\frac {1}{a^{7}}} x^{2} + \sqrt {a x^{4} - b}}{x^{2}}} - {\left (a x^{4} - b\right )}^{\frac {1}{4}} a^{5} \frac {1}{a^{7}}^{\frac {3}{4}}}{x}\right ) + \frac {1}{4} \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (\frac {a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \frac {1}{a^{7}}^{\frac {1}{4}} \log \left (-\frac {a^{2} \frac {1}{a^{7}}^{\frac {1}{4}} x - {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) \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^{6}}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (a \,x^{4}-b \right )^{\frac {3}{4}} \left (a \,x^{4}+b \right )}\, dx \end {gather*}
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^{6}}{{\left (a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {3}{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 {x^6}{\left (a\,x^4+b\right )\,{\left (a\,x^4-b\right )}^{3/4}} \,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^{6}}{\left (a x^{4} - b\right )^{\frac {3}{4}} \left (a x^{4} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________