Optimal. Leaf size=123 \[ \frac {x^3 \sqrt {c+d x^6}}{6 b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{6 b^2 d^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {476, 490, 537,
223, 212, 385, 211} \begin {gather*} \frac {a^{3/2} \text {ArcTan}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {b c-a d}}-\frac {(2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{6 b^2 d^{3/2}}+\frac {x^3 \sqrt {c+d x^6}}{6 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 212
Rule 223
Rule 385
Rule 476
Rule 490
Rule 537
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )\\ &=\frac {x^3 \sqrt {c+d x^6}}{6 b d}-\frac {\text {Subst}\left (\int \frac {a c+(b c+2 a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 b d}\\ &=\frac {x^3 \sqrt {c+d x^6}}{6 b d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{3 b^2}-\frac {(b c+2 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 b^2 d}\\ &=\frac {x^3 \sqrt {c+d x^6}}{6 b d}+\frac {a^2 \text {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^2}-\frac {(b c+2 a d) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{6 b^2 d}\\ &=\frac {x^3 \sqrt {c+d x^6}}{6 b d}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {b c-a d}}-\frac {(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{6 b^2 d^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.95, size = 140, normalized size = 1.14 \begin {gather*} \frac {\sqrt {d} \left (b x^3 \sqrt {c+d x^6}+\frac {2 a^{3/2} d \tan ^{-1}\left (\frac {a \sqrt {d}+b \sqrt {d} x^6+b x^3 \sqrt {c+d x^6}}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}\right )-(b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {d} x^3}\right )}{6 b^2 d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{14}}{\left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 5.63, size = 739, normalized size = 6.01 \begin {gather*} \left [\frac {2 \, \sqrt {d x^{6} + c} b d x^{3} + a d^{2} \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 ) + {\left (b c + 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{6} + 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, b^{2} d^{2}}, \frac {2 \, \sqrt {d x^{6} + c} b d x^{3} + a d^{2} \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 \, {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{12 \, b^{2} d^{2}}, \frac {2 \, \sqrt {d x^{6} + c} b d x^{3} - 2 \, a d^{2} \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 ) + {\left (b c + 2 \, a d\right )} \sqrt {d} \log \left (-2 \, d x^{6} + 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, b^{2} d^{2}}, \frac {\sqrt {d x^{6} + c} b d x^{3} - a d^{2} \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 ) + {\left (b c + 2 \, a d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right )}{6 \, b^{2} d^{2}}\right ] \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^{14}}{\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]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^{14}}{\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]
________________________________________________________________________________________