Optimal. Leaf size=107 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{4\ 2^{3/4} a^{7/4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{4\ 2^{3/4} a^{7/4} b}+\frac {x^3}{6 a b \left (a x^4+b\right )^{3/4}} \]
________________________________________________________________________________________
Rubi [C] time = 0.09, antiderivative size = 44, normalized size of antiderivative = 0.41, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1479, 511, 510} \begin {gather*} -\frac {x^7 \, _2F_1\left (1,\frac {7}{4};\frac {11}{4};\frac {2 a x^4}{a x^4+b}\right )}{7 b \left (a x^4+b\right )^{7/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rule 1479
Rubi steps
\begin {align*} \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (-b^2+a^2 x^8\right )} \, dx &=\int \frac {x^6}{\left (-b+a x^4\right ) \left (b+a x^4\right )^{7/4}} \, dx\\ &=\frac {\left (1+\frac {a x^4}{b}\right )^{3/4} \int \frac {x^6}{\left (-b+a x^4\right ) \left (1+\frac {a x^4}{b}\right )^{7/4}} \, dx}{b \left (b+a x^4\right )^{3/4}}\\ &=-\frac {x^7 \, _2F_1\left (1,\frac {7}{4};\frac {11}{4};\frac {2 a x^4}{b+a x^4}\right )}{7 b \left (b+a x^4\right )^{7/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.13, size = 97, normalized size = 0.91 \begin {gather*} \frac {x^3 \left (\left (1-\frac {a x^4}{b}\right )^{3/4}-\left (\frac {a x^4}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {2 a x^4}{b-a x^4}\right )\right )}{6 a b \left (a x^4+b\right )^{3/4} \left (1-\frac {a x^4}{b}\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.65, size = 107, normalized size = 1.00 \begin {gather*} \frac {x^3}{6 a b \left (b+a x^4\right )^{3/4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4\ 2^{3/4} a^{7/4} b}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{4\ 2^{3/4} a^{7/4} b} \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^{6}}{{\left (a^{2} x^{8} - b^{2}\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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{6}}{\left (a \,x^{4}+b \right )^{\frac {3}{4}} \left (a^{2} x^{8}-b^{2}\right )}\, dx\]
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 {x^{6}}{{\left (a^{2} x^{8} - b^{2}\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 (b^2-a^2\,x^8\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(-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]
________________________________________________________________________________________