Optimal. Leaf size=192 \[ -\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}+\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{576 a^4 x^3}-\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{384 a^{9/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1371, 758, 848,
820, 738, 212} \begin {gather*} \frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{576 a^4 x^3}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (48 a^2 c^2-120 a b^2 c+35 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^{9/2}}-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 758
Rule 820
Rule 848
Rule 1371
Rubi steps
\begin {align*} \int \frac {1}{x^{13} \sqrt {a+b x^3+c x^6}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}-\frac {\text {Subst}\left (\int \frac {\frac {7 b}{2}+3 c x}{x^4 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{12 a}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} \left (35 b^2-36 a c\right )+7 b c x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{36 a^2}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}-\frac {\text {Subst}\left (\int \frac {\frac {5}{8} b \left (21 b^2-44 a c\right )+\frac {1}{4} c \left (35 b^2-36 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{72 a^3}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}+\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{576 a^4 x^3}+\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{384 a^4}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}+\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{576 a^4 x^3}-\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \text {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^4}\\ &=-\frac {\sqrt {a+b x^3+c x^6}}{12 a x^{12}}+\frac {7 b \sqrt {a+b x^3+c x^6}}{72 a^2 x^9}-\frac {\left (35 b^2-36 a c\right ) \sqrt {a+b x^3+c x^6}}{288 a^3 x^6}+\frac {5 b \left (21 b^2-44 a c\right ) \sqrt {a+b x^3+c x^6}}{576 a^4 x^3}-\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{384 a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 141, normalized size = 0.73 \begin {gather*} \frac {\sqrt {a+b x^3+c x^6} \left (-48 a^3+56 a^2 b x^3-70 a b^2 x^6+72 a^2 c x^6+105 b^3 x^9-220 a b c x^9\right )}{576 a^4 x^{12}}+\frac {\left (35 b^4-120 a b^2 c+48 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{192 a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{13} \sqrt {c \,x^{6}+b \,x^{3}+a}}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 327, normalized size = 1.70 \begin {gather*} \left [\frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, 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 (5 \, {\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} x^{9} + 56 \, a^{3} b x^{3} - 2 \, {\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} x^{6} - 48 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{2304 \, a^{5} x^{12}}, \frac {3 \, {\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, 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 (5 \, {\left (21 \, a b^{3} - 44 \, a^{2} b c\right )} x^{9} + 56 \, a^{3} b x^{3} - 2 \, {\left (35 \, a^{2} b^{2} - 36 \, a^{3} c\right )} x^{6} - 48 \, a^{4}\right )} \sqrt {c x^{6} + b x^{3} + a}}{1152 \, a^{5} x^{12}}\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 {1}{x^{13} \sqrt {a + b x^{3} + c x^{6}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {1}{x^{13}\,\sqrt {c\,x^6+b\,x^3+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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