Optimal. Leaf size=399 \[ -\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^6+b x^3}-2^{2/3} x \sqrt [4]{a x^6+b x^3}}{2^{2/3} x \sqrt [4]{a x^6+b x^3}+x^2 \left (-\sqrt [4]{a x^6+b x^3}\right )-\sqrt {2} x+2 \sqrt [6]{2}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {x^2 \sqrt [4]{a x^6+b x^3}-2^{2/3} x \sqrt [4]{a x^6+b x^3}}{2^{2/3} x \sqrt [4]{a x^6+b x^3}+x^2 \left (-\sqrt [4]{a x^6+b x^3}\right )+\sqrt {2} x-2 \sqrt [6]{2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-\sqrt {2} x^3 \sqrt [4]{a x^6+b x^3}-2\ 2^{5/6} x \sqrt [4]{a x^6+b x^3}+4 \sqrt [6]{2} x^2 \sqrt [4]{a x^6+b x^3}}{2\ 2^{2/3} x^3 \sqrt {a x^6+b x^3}+x^4 \left (-\sqrt {a x^6+b x^3}\right )-2 \sqrt [3]{2} x^2 \sqrt {a x^6+b x^3}-x^2+2\ 2^{2/3} x-2 \sqrt [3]{2}}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 2.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{b+a x^3}\right ) \int \frac {x^{17/4} \left (7 b+10 a x^3\right )}{\sqrt [4]{b+a x^3} \left (1+b x^7+a x^{10}\right )} \, dx}{\sqrt [4]{b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{20} \left (7 b+10 a x^{12}\right )}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {7 b x^{20}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )}+\frac {10 a x^{32}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ &=\frac {\left (40 a x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{32}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}+\frac {\left (28 b x^{3/4} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{b+a x^{12}} \left (1+b x^{28}+a x^{40}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x^3+a x^6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (7 b+10 a x^3\right )}{\sqrt [4]{b x^3+a x^6} \left (1+b x^7+a x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 20.10, size = 399, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-2^{2/3} x \sqrt [4]{b x^3+a x^6}+x^2 \sqrt [4]{b x^3+a x^6}}{2 \sqrt [6]{2}-\sqrt {2} x+2^{2/3} x \sqrt [4]{b x^3+a x^6}-x^2 \sqrt [4]{b x^3+a x^6}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {-2^{2/3} x \sqrt [4]{b x^3+a x^6}+x^2 \sqrt [4]{b x^3+a x^6}}{-2 \sqrt [6]{2}+\sqrt {2} x+2^{2/3} x \sqrt [4]{b x^3+a x^6}-x^2 \sqrt [4]{b x^3+a x^6}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {-2 2^{5/6} x \sqrt [4]{b x^3+a x^6}+4 \sqrt [6]{2} x^2 \sqrt [4]{b x^3+a x^6}-\sqrt {2} x^3 \sqrt [4]{b x^3+a x^6}}{-2 \sqrt [3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt [3]{2} x^2 \sqrt {b x^3+a x^6}+2\ 2^{2/3} x^3 \sqrt {b x^3+a x^6}-x^4 \sqrt {b x^3+a x^6}}\right ) \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 {{\left (10 \, a x^{3} + 7 \, b\right )} x^{5}}{{\left (a x^{10} + b x^{7} + 1\right )} {\left (a x^{6} + b x^{3}\right )}^{\frac {1}{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 \[\int \frac {x^{5} \left (10 a \,x^{3}+7 b \right )}{\left (a \,x^{6}+b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{10}+b \,x^{7}+1\right )}\, dx\]
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 {{\left (10 \, a x^{3} + 7 \, b\right )} x^{5}}{{\left (a x^{10} + b x^{7} + 1\right )} {\left (a x^{6} + b x^{3}\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 = -0.00 \begin {gather*} \int \frac {x^5\,\left (10\,a\,x^3+7\,b\right )}{{\left (a\,x^6+b\,x^3\right )}^{1/4}\,\left (a\,x^{10}+b\,x^7+1\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 {x^{5} \left (10 a x^{3} + 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{3} + b\right )} \left (a x^{10} + b x^{7} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________