Optimal. Leaf size=223 \[ \frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac {b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{2048 c^{11/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 223, 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, 756, 793,
626, 635, 212} \begin {gather*} -\frac {b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 c^{11/2}}+\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac {\left (-16 a c+21 b^2-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}+\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 756
Rule 793
Rule 1371
Rubi steps
\begin {align*} \int x^{11} \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^3 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\text {Subst}\left (\int x \left (-2 a-\frac {9 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{21 c}\\ &=\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac {\left (b \left (3 b^2-4 a c\right )\right ) \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{48 c^3}\\ &=-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}+\frac {\left (b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{256 c^4}\\ &=\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac {\left (b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^3\right )}{2048 c^5}\\ &=\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac {\left (b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right )\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 )}{1024 c^5}\\ &=\frac {b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1024 c^5}-\frac {b \left (3 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 c^4}+\frac {x^6 \left (a+b x^3+c x^6\right )^{5/2}}{21 c}+\frac {\left (21 b^2-16 a c-30 b c x^3\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 c^3}-\frac {b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 c^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 220, normalized size = 0.99 \begin {gather*} \frac {\sqrt {a+b x^3+c x^6} \left (315 b^6-210 b^5 c x^3+16 b^3 c^2 x^3 \left (91 a-9 c x^6\right )+168 b^4 c \left (-15 a+c x^6\right )+1024 c^3 \left (a+c x^6\right )^2 \left (-2 a+5 c x^6\right )+16 b^2 c^2 \left (343 a^2-62 a c x^6+8 c^2 x^{12}\right )+32 b c^3 x^3 \left (-73 a^2+22 a c x^6+200 c^2 x^{12}\right )\right )}{107520 c^5}+\frac {b \left (b^2-4 a c\right )^2 \left (3 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 )}{2048 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{11} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}\, dx\]
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 = 0.43, size = 535, normalized size = 2.40 \begin {gather*} \left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\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 (5120 \, c^{7} x^{18} + 6400 \, b c^{6} x^{15} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{12} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{9} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{6} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{430080 \, c^{6}}, \frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\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 (5120 \, c^{7} x^{18} + 6400 \, b c^{6} x^{15} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{12} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{9} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{6} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{215040 \, c^{6}}\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 x^{11} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}\, 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 = -0.00 \begin {gather*} \int x^{11}\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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