Optimal. Leaf size=137 \[ \frac {\left (-8 a c+3 b^2-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1357, 738, 779, 621, 206} \[ \frac {\left (-8 a c+3 b^2-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 738
Rule 779
Rule 1357
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {2 \operatorname {Subst}\left (\int \frac {x (4 a+2 b x)}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{3 \left (b^2-4 a c\right )}\\ &=\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{2 c^2}\\ &=\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^3}{\sqrt {a+b x^3+c x^6}}\right )}{c^2}\\ &=\frac {2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (3 b^2-8 a c-2 b c x^3\right ) \sqrt {a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 137, normalized size = 1.00 \[ \frac {\frac {2 \sqrt {c} \left (8 a^2 c+a \left (-3 b^2+10 b c x^3+4 c^2 x^6\right )-b^2 x^3 \left (3 b+c x^3\right )\right )}{\sqrt {a+b x^3+c x^6}}+3 b \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{6 c^{5/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 459, normalized size = 3.35 \[ \left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{12 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{6 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{11}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{11}}{\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^{11}}{{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{11}}{\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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