Optimal. Leaf size=141 \[ \frac {a x^3 \sqrt {c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}-\frac {\sqrt {a} (3 b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {d}} \]
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Rubi [A]
time = 0.11, antiderivative size = 141, 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, 481, 537,
223, 212, 385, 211} \begin {gather*} -\frac {\sqrt {a} (3 b c-2 a d) \text {ArcTan}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac {a x^3 \sqrt {c+d x^6}}{6 b \left (a+b x^6\right ) (b c-a d)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 223
Rule 385
Rule 476
Rule 481
Rule 537
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^3\right )\\ &=\frac {a x^3 \sqrt {c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}-\frac {\text {Subst}\left (\int \frac {a c-2 (b c-a d) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 b (b c-a d)}\\ &=\frac {a x^3 \sqrt {c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,x^3\right )}{3 b^2}-\frac {(a (3 b c-2 a d)) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 b^2 (b c-a d)}\\ &=\frac {a x^3 \sqrt {c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{3 b^2}-\frac {(a (3 b c-2 a d)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{6 b^2 (b c-a d)}\\ &=\frac {a x^3 \sqrt {c+d x^6}}{6 b (b c-a d) \left (a+b x^6\right )}-\frac {\sqrt {a} (3 b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 b^2 (b c-a d)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {d} x^3}{\sqrt {c+d x^6}}\right )}{3 b^2 \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 1.95, size = 152, normalized size = 1.08 \begin {gather*} \frac {\frac {a b x^3 \sqrt {c+d x^6}}{(b c-a d) \left (a+b x^6\right )}+\frac {\sqrt {a} (-3 b c+2 a 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 )}{(b c-a d)^{3/2}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^6}}{\sqrt {d} x^3}\right )}{\sqrt {d}}}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {x^{14}}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 3.16, size = 1077, normalized size = 7.64 \begin {gather*} \left [\frac {4 \, \sqrt {d x^{6} + c} a b d x^{3} + 4 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 )}{24 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {4 \, \sqrt {d x^{6} + c} a b d x^{3} - 8 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 )}{24 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{6} + c} a b d x^{3} + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {d} \log \left (-2 \, d x^{6} - 2 \, \sqrt {d x^{6} + c} \sqrt {d} x^{3} - c\right )}{12 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}, \frac {2 \, \sqrt {d x^{6} + c} a b d x^{3} - 4 \, {\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x^{3}}{\sqrt {d x^{6} + c}}\right ) + {\left ({\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{6} + 3 \, a b c d - 2 \, a^{2} d^{2}\right )} \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 )}{12 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + a b^{3} c d - a^{2} b^{2} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{14}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 343 vs.
\(2 (117) = 234\).
time = 1.90, size = 343, normalized size = 2.43 \begin {gather*} -\frac {{\left (3 \, a b c \sqrt {-d} \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - 2 \, a^{2} \sqrt {-d} d \arctan \left (\frac {a \sqrt {d}}{\sqrt {a b c - a^{2} d}}\right ) - 2 \, \sqrt {a b c - a^{2} d} b c \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right ) + 2 \, \sqrt {a b c - a^{2} d} a d \arctan \left (\frac {\sqrt {d}}{\sqrt {-d}}\right ) + \sqrt {a b c - a^{2} d} a \sqrt {-d} \sqrt {d}\right )} \mathrm {sgn}\left (x\right )}{6 \, {\left (\sqrt {a b c - a^{2} d} b^{3} c \sqrt {-d} - \sqrt {a b c - a^{2} d} a b^{2} \sqrt {-d} d\right )}} + \frac {a c \sqrt {d + \frac {c}{x^{6}}}}{6 \, {\left (b^{2} c \mathrm {sgn}\left (x\right ) - a b d \mathrm {sgn}\left (x\right )\right )} {\left (b c + a {\left (d + \frac {c}{x^{6}}\right )} - a d\right )}} + \frac {{\left (3 \, a b c - 2 \, a^{2} d\right )} \arctan \left (\frac {a \sqrt {d + \frac {c}{x^{6}}}}{\sqrt {a b c - a^{2} d}}\right )}{6 \, {\left (b^{3} c \mathrm {sgn}\left (x\right ) - a b^{2} d \mathrm {sgn}\left (x\right )\right )} \sqrt {a b c - a^{2} d}} - \frac {\arctan \left (\frac {\sqrt {d + \frac {c}{x^{6}}}}{\sqrt {-d}}\right )}{3 \, b^{2} \sqrt {-d} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{14}}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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