Optimal. Leaf size=120 \[ \frac {\left (a^{5/4}+\sqrt [4]{a} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}-\frac {\left (a^{5/4}+\sqrt [4]{a} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}+\frac {\left (a x^4+b\right )^{5/4}}{5 b^2 x^5} \]
________________________________________________________________________________________
Rubi [A] time = 0.74, antiderivative size = 132, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {6725, 271, 264, 494, 298, 203, 206} \begin {gather*} \frac {a \sqrt [4]{a x^4+b}}{5 b^2 x}+\frac {\sqrt [4]{a} (a+b) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}-\frac {\sqrt [4]{a} (a+b) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}+\frac {\sqrt [4]{a x^4+b}}{5 b x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 264
Rule 271
Rule 298
Rule 494
Rule 6725
Rubi steps
\begin {align*} \int \frac {b+a x^8}{x^6 \left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx &=\int \left (-\frac {1}{x^6 \left (b+a x^4\right )^{3/4}}-\frac {a}{b x^2 \left (b+a x^4\right )^{3/4}}-\frac {a (a+b) x^2}{b \left (b-a x^4\right ) \left (b+a x^4\right )^{3/4}}\right ) \, dx\\ &=-\frac {a \int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx}{b}-\frac {(a (a+b)) \int \frac {x^2}{\left (b-a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx}{b}-\int \frac {1}{x^6 \left (b+a x^4\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{b+a x^4}}{5 b x^5}+\frac {a \sqrt [4]{b+a x^4}}{b^2 x}+\frac {(4 a) \int \frac {1}{x^2 \left (b+a x^4\right )^{3/4}} \, dx}{5 b}-\frac {(a (a+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 )}{b}\\ &=\frac {\sqrt [4]{b+a x^4}}{5 b x^5}+\frac {a \sqrt [4]{b+a x^4}}{5 b^2 x}-\frac {\left (\sqrt {a} (a+b)\right ) \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} b^2}+\frac {\left (\sqrt {a} (a+b)\right ) \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} b^2}\\ &=\frac {\sqrt [4]{b+a x^4}}{5 b x^5}+\frac {a \sqrt [4]{b+a x^4}}{5 b^2 x}+\frac {\sqrt [4]{a} (a+b) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2}-\frac {\sqrt [4]{a} (a+b) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.12, size = 65, normalized size = 0.54 \begin {gather*} \frac {3 \left (a x^4+b\right )^2-5 a x^8 (a+b) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 a x^4}{a x^4+b}\right )}{15 b^2 x^5 \left (a x^4+b\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.07, size = 120, normalized size = 1.00 \begin {gather*} \frac {\left (a^{5/4}+\sqrt [4]{a} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}-\frac {\left (a^{5/4}+\sqrt [4]{a} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} b^2}+\frac {\left (a x^4+b\right )^{5/4}}{5 b^2 x^5} \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 {a x^{8} + b}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \,x^{8}+b}{x^{6} \left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \right )^{\frac {3}{4}}}\, 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 {a x^{8} + b}{{\left (a x^{4} + b\right )}^{\frac {3}{4}} {\left (a x^{4} - b\right )} x^{6}}\,{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 {a\,x^8+b}{x^6\,{\left (a\,x^4+b\right )}^{3/4}\,\left (b-a\,x^4\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}{x^{6} \left (a x^{4} - b\right ) \left (a x^{4} + b\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________