Optimal. Leaf size=150 \[ \frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-\frac {1}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {\left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 \sqrt {c}} \]
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Rubi [A]
time = 0.12, antiderivative size = 150, 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 = {1371, 746, 828,
857, 635, 212, 738} \begin {gather*} \frac {\left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{8 \sqrt {c}}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {1}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 828
Rule 857
Rule 1371
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{2} \text {Subst}\left (\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-\frac {\text {Subst}\left (\int \frac {-4 a b c-c \left (b^2+4 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{8 c}\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}+\frac {1}{2} (a b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^3\right )-\frac {1}{8} \left (-b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )\\ &=\frac {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-(a b) \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 )-\frac {1}{4} \left (-b^2-4 a c\right ) \text {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 {1}{4} \left (3 b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}-\frac {\left (a+b x^3+c x^6\right )^{3/2}}{3 x^3}-\frac {1}{2} \sqrt {a} b \tanh ^{-1}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )+\frac {\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 )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 131, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x^3+c x^6} \left (-4 a+5 b x^3+2 c x^6\right )}{12 x^3}+\sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )-\frac {\left (b^2+4 a c\right ) \log \left (b+2 c x^3-2 \sqrt {c} \sqrt {a+b x^3+c x^6}\right )}{8 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{4}}\, dx\]
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 = 0.45, size = 713, normalized size = 4.75 \begin {gather*} \left [\frac {12 \, \sqrt {a} b c x^{3} \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 ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{3} \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 (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{48 \, c x^{3}}, \frac {6 \, \sqrt {a} b c x^{3} \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 ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{3} \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 (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{24 \, c x^{3}}, \frac {24 \, \sqrt {-a} b c x^{3} \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 ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{3} \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 (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{48 \, c x^{3}}, \frac {12 \, \sqrt {-a} b c x^{3} \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 ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{3} \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 (2 \, c^{2} x^{6} + 5 \, b c x^{3} - 4 \, a c\right )} \sqrt {c x^{6} + b x^{3} + a}}{24 \, c x^{3}}\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 {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{4}}\, 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 {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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