Optimal. Leaf size=133 \[ -\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{128 a^{5/2}}+\frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}} \]
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Rubi [A] time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1357, 720, 724, 206} \[ \frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{128 a^{5/2}}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 1357
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^3\right )}{16 a}\\ &=\frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}+\frac {\left (b^2-4 a c\right )^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{128 a^2}\\ &=\frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right )^2 \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^3}{\sqrt {a+b x^3+c x^6}}\right )}{64 a^2}\\ &=\frac {\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{64 a^2 x^6}-\frac {\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}}-\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{128 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 138, normalized size = 1.04 \[ -\frac {\frac {3 \left (b^2-4 a c\right ) \left (x^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )-2 \sqrt {a} \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}\right )}{8 a^{3/2} x^6}+\frac {2 \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{x^{12}}}{48 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 319, normalized size = 2.40 \[ \left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{12} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} - 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left ({\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{9} - 24 \, a^{3} b x^{3} - 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{6} - 16 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{768 \, a^{3} x^{12}}, \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{12} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{3} - 20 \, a^{2} b c\right )} x^{9} - 24 \, a^{3} b x^{3} - 2 \, {\left (a^{2} b^{2} + 20 \, a^{3} c\right )} x^{6} - 16 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{384 \, a^{3} x^{12}}\right ] \]
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 \[ \int \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{13}}\, dx \]
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 \[ \int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{13}} \,d x \]
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 \[ \int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{13}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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