Optimal. Leaf size=91 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b \sqrt {d}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b \sqrt {b c-a d}} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 483, 217, 206, 377, 205} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b \sqrt {d}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b \sqrt {b c-a d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 206
Rule 217
Rule 377
Rule 465
Rule 483
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,x^3\right )}{3 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{3 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{3 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{3 b}\\ &=-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b \sqrt {b c-a d}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 90, normalized size = 0.99 \begin {gather*} \frac {\frac {\log \left (\sqrt {d} \sqrt {c+d x^6}+d x^3\right )}{\sqrt {d}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{\sqrt {b c-a d}}}{3 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.62, size = 144, normalized size = 1.58 \begin {gather*} \frac {\log \left (\sqrt {c+d x^6}+\sqrt {d} x^3\right )}{3 b \sqrt {d}}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {b \sqrt {d} x^6}{\sqrt {a} \sqrt {b c-a d}}+\frac {b x^3 \sqrt {c+d x^6}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{3 b \sqrt {b c-a d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 632, normalized size = 6.95 \begin {gather*} \left [\frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} - {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 2 \, \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, b d}, \frac {d \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} - 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{9} - {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) - 4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{12 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{9} + a c x^{3}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{6 \, b d}, \frac {d \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{9} + a c x^{3}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{6 \, b d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.23, size = 156, normalized size = 1.71 \begin {gather*} -\frac {{\left (a \sqrt {-d} \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right )\right )} \mathrm {sgn}\relax (x)}{3 \, \sqrt {a b c - a^{2} d} b \sqrt {-d}} + \frac {a \arctan \left (\frac {a \sqrt {d + \frac {c}{x^{6}}}}{\sqrt {a b c - a^{2} d}}\right )}{3 \, \sqrt {a b c - a^{2} d} b \mathrm {sgn}\relax (x)} - \frac {\arctan \left (\frac {\sqrt {d + \frac {c}{x^{6}}}}{\sqrt {-d}}\right )}{3 \, b \sqrt {-d} \mathrm {sgn}\relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}\, 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 {x^{8}}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c}}\,{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^8}{\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,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^{8}}{\left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________