Optimal. Leaf size=150 \[ -\frac {b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{256 c^{7/2}}+\frac {b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{128 c^3}-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]
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Rubi [A] time = 0.12, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1357, 640, 612, 621, 206} \[ \frac {b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{128 c^3}-\frac {b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{256 c^{7/2}}-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1357
Rubi steps
\begin {align*} \int x^5 \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac {b \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{6 c}\\ &=-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c}+\frac {\left (b \left (b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{32 c^2}\\ &=\frac {b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{128 c^3}-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac {\left (b \left (b^2-4 a c\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{256 c^3}\\ &=\frac {b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{128 c^3}-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac {\left (b \left (b^2-4 a c\right )^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 )}{128 c^3}\\ &=\frac {b \left (b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{128 c^3}-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c}-\frac {b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{256 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 149, normalized size = 0.99 \[ -\frac {b \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )-2 \sqrt {c} \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}\right )}{256 c^{7/2}}-\frac {b \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 c^2}+\frac {\left (a+b x^3+c x^6\right )^{5/2}}{15 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 361, normalized size = 2.41 \[ \left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{6} - 8 \, b c x^{3} - b^{2} + 4 \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (128 \, c^{5} x^{12} + 176 \, b c^{4} x^{9} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{7680 \, c^{4}}, \frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{6} + b x^{3} + a} {\left (2 \, c x^{3} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{6} + b c x^{3} + a c\right )}}\right ) + 2 \, {\left (128 \, c^{5} x^{12} + 176 \, b c^{4} x^{9} + 8 \, {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{6} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} - 2 \, {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{3840 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 172, normalized size = 1.15 \[ \frac {1}{1920} \, \sqrt {c x^{6} + b x^{3} + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c x^{3} + 11 \, b\right )} x^{3} + \frac {b^{2} c^{3} + 32 \, a c^{4}}{c^{4}}\right )} x^{3} - \frac {5 \, b^{3} c^{2} - 28 \, a b c^{3}}{c^{4}}\right )} x^{3} + \frac {15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}}{c^{4}}\right )} + \frac {{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{3} - \sqrt {c x^{6} + b x^{3} + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}} x^{5}\, dx \]
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 [B] time = 1.58, size = 223, normalized size = 1.49 \[ \frac {{\left (c\,x^6+b\,x^3+a\right )}^{5/2}}{15\,c}-\frac {b\,\left (\frac {3\,a\,\left (\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (2\,c\,x^3+b\right )\,\sqrt {c\,x^6+b\,x^3+a}}{4\,c}\right )}{4}+\frac {x^3\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{4}-\frac {3\,b^2\,\left (\ln \left (\sqrt {c\,x^6+b\,x^3+a}+\frac {c\,x^3+\frac {b}{2}}{\sqrt {c}}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (2\,c\,x^3+b\right )\,\sqrt {c\,x^6+b\,x^3+a}}{4\,c}\right )}{16\,c}+\frac {b\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{8\,c}\right )}{6\,c} \]
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 \[ \int x^{5} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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