Optimal. Leaf size=161 \[ \frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{384 a^{7/2}}-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{192 a^3 x^6}+\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 a^2 x^9}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}} \]
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Rubi [A] time = 0.15, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1357, 744, 806, 720, 724, 206} \[ -\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{192 a^3 x^6}+\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{384 a^{7/2}}+\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 a^2 x^9}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 744
Rule 806
Rule 1357
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3+c x^6}}{x^{13}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^5} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}}-\frac {\operatorname {Subst}\left (\int \frac {\left (\frac {5 b}{2}+c x\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx,x,x^3\right )}{12 a}\\ &=-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}}+\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 a^2 x^9}+\frac {\left (5 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 )}{48 a^2}\\ &=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{192 a^3 x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}}+\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 a^2 x^9}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{384 a^3}\\ &=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{192 a^3 x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}}+\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 a^2 x^9}+\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \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 )}{192 a^3}\\ &=-\frac {\left (5 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt {a+b x^3+c x^6}}{192 a^3 x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{12 a x^{12}}+\frac {5 b \left (a+b x^3+c x^6\right )^{3/2}}{72 a^2 x^9}+\frac {\left (b^2-4 a c\right ) \left (5 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 )}{384 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 139, normalized size = 0.86 \[ \frac {\left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{384 a^{7/2}}-\frac {\sqrt {a+b x^3+c x^6} \left (48 a^3+8 a^2 x^3 \left (b+3 c x^3\right )-2 a b x^6 \left (5 b+26 c x^3\right )+15 b^3 x^9\right )}{576 a^3 x^{12}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 325, normalized size = 2.02 \[ \left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {a} x^{12} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 4 \, {\left ({\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{9} + 8 \, a^{3} b x^{3} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{6} + 48 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{2304 \, a^{4} x^{12}}, -\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-a} x^{12} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (b x^{3} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{6} + a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{9} + 8 \, a^{3} b x^{3} - 2 \, {\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{6} + 48 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{1152 \, a^{4} x^{12}}\right ] \]
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 \[ \int \frac {\sqrt {c x^{6} + b x^{3} + a}}{x^{13}}\,{d x} \]
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 \[ \int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{x^{13}}\, dx \]
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 \[ \int \frac {\sqrt {c\,x^6+b\,x^3+a}}{x^{13}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x^{3} + c x^{6}}}{x^{13}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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