Optimal. Leaf size=163 \[ \frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{3/2}}-\frac {\left (x^3 \left (8 a c+b^2\right )+2 a b\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9} \]
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Rubi [A] time = 0.18, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1357, 732, 810, 843, 621, 206, 724} \[ \frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{3/2}}-\frac {\left (x^3 \left (8 a c+b^2\right )+2 a b\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 732
Rule 810
Rule 843
Rule 1357
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^{10}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^3\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} b \left (b^2-12 a c\right )-8 a c^2 x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^3\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9}+\frac {1}{3} c^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )-\frac {\left (b \left (b^2-12 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{48 a}\\ &=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^3\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9}+\frac {1}{3} \left (2 c^2\right ) \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 )+\frac {\left (b \left (b^2-12 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 )}{24 a}\\ &=-\frac {\left (2 a b+\left (b^2+8 a c\right ) x^3\right ) \sqrt {a+b x^3+c x^6}}{24 a x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9}+\frac {b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{3/2}}+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 149, normalized size = 0.91 \[ \frac {1}{144} \left (\frac {3 b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{a^{3/2}}-\frac {2 \sqrt {a+b x^3+c x^6} \left (8 a^2+14 a b x^3+32 a c x^6+3 b^2 x^6\right )}{a x^9}+48 c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 771, normalized size = 4.73 \[ \left [\frac {48 \, a^{2} c^{\frac {3}{2}} x^{9} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{9} \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^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, a^{2} x^{9}}, -\frac {96 \, a^{2} \sqrt {-c} c x^{9} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {a} x^{9} \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^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{288 \, a^{2} x^{9}}, \frac {24 \, a^{2} c^{\frac {3}{2}} x^{9} \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 ) - 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{9} \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^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, a^{2} x^{9}}, -\frac {48 \, a^{2} \sqrt {-c} c x^{9} \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 ) + 3 \, {\left (b^{3} - 12 \, a b c\right )} \sqrt {-a} x^{9} \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^{2} + 32 \, a^{2} c\right )} x^{6} + 14 \, a^{2} b x^{3} + 8 \, a^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{144 \, a^{2} x^{9}}\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 \[ \int \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{10}}\,{d x} \]
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 \[ \int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{10}}\, dx \]
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 \[ \int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^{10}} \,d x \]
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 \[ \int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{10}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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