Optimal. Leaf size=41 \[ \frac {2}{5} \tan ^{-1}\left (x \sqrt [4]{a x^6+b x}\right )-\frac {2}{5} \tanh ^{-1}\left (x \sqrt [4]{a x^6+b x}\right ) \]
________________________________________________________________________________________
Rubi [C] time = 1.44, antiderivative size = 187, normalized size of antiderivative = 4.56, number of steps used = 11, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2056, 6728, 466, 465, 511, 510} \begin {gather*} \frac {8 a x^4 \sqrt [4]{\frac {a x^5}{b}+1} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};-\frac {2 a x^5}{b-\sqrt {b^2+4 a}},-\frac {a x^5}{b}\right )}{15 \left (b-\sqrt {4 a+b^2}\right ) \sqrt [4]{a x^6+b x}}+\frac {8 a x^4 \sqrt [4]{\frac {a x^5}{b}+1} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};-\frac {2 a x^5}{b+\sqrt {b^2+4 a}},-\frac {a x^5}{b}\right )}{15 \left (\sqrt {4 a+b^2}+b\right ) \sqrt [4]{a x^6+b x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 465
Rule 466
Rule 510
Rule 511
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \int \frac {x^{11/4} \left (b+2 a x^5\right )}{\sqrt [4]{b+a x^5} \left (-1+b x^5+a x^{10}\right )} \, dx}{\sqrt [4]{b x+a x^6}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \int \left (\frac {2 a x^{11/4}}{\sqrt [4]{b+a x^5} \left (b-\sqrt {4 a+b^2}+2 a x^5\right )}+\frac {2 a x^{11/4}}{\sqrt [4]{b+a x^5} \left (b+\sqrt {4 a+b^2}+2 a x^5\right )}\right ) \, dx}{\sqrt [4]{b x+a x^6}}\\ &=\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \int \frac {x^{11/4}}{\sqrt [4]{b+a x^5} \left (b-\sqrt {4 a+b^2}+2 a x^5\right )} \, dx}{\sqrt [4]{b x+a x^6}}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \int \frac {x^{11/4}}{\sqrt [4]{b+a x^5} \left (b+\sqrt {4 a+b^2}+2 a x^5\right )} \, dx}{\sqrt [4]{b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{20}} \left (b-\sqrt {4 a+b^2}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{b+a x^{20}} \left (b+\sqrt {4 a+b^2}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^4} \left (b-\sqrt {4 a+b^2}+2 a x^4\right )} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^4} \left (b+\sqrt {4 a+b^2}+2 a x^4\right )} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{1+\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b-\sqrt {4 a+b^2}+2 a x^4\right ) \sqrt [4]{1+\frac {a x^4}{b}}} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{1+\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+\sqrt {4 a+b^2}+2 a x^4\right ) \sqrt [4]{1+\frac {a x^4}{b}}} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{b x+a x^6}}\\ &=\frac {8 a x^4 \sqrt [4]{1+\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};-\frac {2 a x^5}{b-\sqrt {4 a+b^2}},-\frac {a x^5}{b}\right )}{15 \left (b-\sqrt {4 a+b^2}\right ) \sqrt [4]{b x+a x^6}}+\frac {8 a x^4 \sqrt [4]{1+\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};-\frac {2 a x^5}{b+\sqrt {4 a+b^2}},-\frac {a x^5}{b}\right )}{15 \left (b+\sqrt {4 a+b^2}\right ) \sqrt [4]{b x+a x^6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 19.86, size = 41, normalized size = 1.00 \begin {gather*} \frac {2}{5} \tan ^{-1}\left (x \sqrt [4]{a x^6+b x}\right )-\frac {2}{5} \tanh ^{-1}\left (x \sqrt [4]{a x^6+b x}\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 (2 \, a x^{5} + b\right )} x^{3}}{{\left (a x^{10} + b x^{5} - 1\right )} {\left (a x^{6} + b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (2 a \,x^{5}+b \right )}{\left (a \,x^{6}+b x \right )^{\frac {1}{4}} \left (a \,x^{10}+b \,x^{5}-1\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 {{\left (2 \, a x^{5} + b\right )} x^{3}}{{\left (a x^{10} + b x^{5} - 1\right )} {\left (a x^{6} + b x\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 = -0.02 \begin {gather*} \int \frac {x^3\,\left (2\,a\,x^5+b\right )}{{\left (a\,x^6+b\,x\right )}^{1/4}\,\left (a\,x^{10}+b\,x^5-1\right )} \,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]
________________________________________________________________________________________