Optimal. Leaf size=64 \[ \frac {x^5 \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {5}{6};2,\frac {1}{2};\frac {11}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{5 a^2 \sqrt {c+d x^6}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac {x^5 \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {5}{6};2,\frac {1}{2};\frac {11}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{5 a^2 \sqrt {c+d x^6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {\sqrt {1+\frac {d x^6}{c}} \int \frac {x^4}{\left (a+b x^6\right )^2 \sqrt {1+\frac {d x^6}{c}}} \, dx}{\sqrt {c+d x^6}}\\ &=\frac {x^5 \sqrt {1+\frac {d x^6}{c}} F_1\left (\frac {5}{6};2,\frac {1}{2};\frac {11}{6};-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{5 a^2 \sqrt {c+d x^6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.22, size = 169, normalized size = 2.64 \[ \frac {x^5 \left (-10 b d x^6 \left (a+b x^6\right ) \sqrt {\frac {d x^6}{c}+1} F_1\left (\frac {11}{6};\frac {1}{2},1;\frac {17}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+11 \left (a+b x^6\right ) \sqrt {\frac {d x^6}{c}+1} (b c-6 a d) F_1\left (\frac {5}{6};\frac {1}{2},1;\frac {11}{6};-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+55 a b \left (c+d x^6\right )\right )}{330 a^2 \left (a+b x^6\right ) \sqrt {c+d x^6} (b c-a d)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x^{6} + c} x^{4}}{b^{2} d x^{18} + {\left (b^{2} c + 2 \, a b d\right )} x^{12} + {\left (2 \, a b c + a^{2} d\right )} x^{6} + a^{2} c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^4}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________