Optimal. Leaf size=204 \[ -\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{2 \sqrt {c} \sqrt {a+b x^3+c x^6}}\right )}{3072 c^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 204, 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, 654,
626, 635, 212} \begin {gather*} \frac {\left (b^2-4 a c\right )^2 \left (7 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 )}{3072 c^{9/2}}-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 626
Rule 635
Rule 654
Rule 756
Rule 1371
Rubi steps
\begin {align*} \int x^8 \left (a+b x^3+c x^6\right )^{3/2} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )\\ &=\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\text {Subst}\left (\int \left (-a-\frac {7 b x}{2}\right ) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{18 c}\\ &=-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (7 b^2-4 a c\right ) \text {Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^3\right )}{72 c^2}\\ &=\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}-\frac {\left (\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^3\right )}{384 c^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 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 )}{3072 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (7 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 )}{1536 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \sqrt {a+b x^3+c x^6}}{1536 c^4}+\frac {\left (7 b^2-4 a c\right ) \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{576 c^3}-\frac {7 b \left (a+b x^3+c x^6\right )^{5/2}}{180 c^2}+\frac {x^3 \left (a+b x^3+c x^6\right )^{5/2}}{18 c}+\frac {\left (b^2-4 a c\right )^2 \left (7 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 )}{3072 c^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.53, size = 194, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+b x^3+c x^6} \left (-105 b^5+70 b^4 c x^3+8 b^3 c \left (95 a-7 c x^6\right )+48 b^2 c^2 x^3 \left (-9 a+c x^6\right )+160 c^3 x^3 \left (3 a^2+14 a c x^6+8 c^2 x^{12}\right )+16 b c^2 \left (-81 a^2+18 a c x^6+104 c^2 x^{12}\right )\right )-15 \left (b^2-4 a c\right )^2 \left (7 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 )}{46080 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{8} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 451, normalized size = 2.21 \begin {gather*} \left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} 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 (1280 \, c^{6} x^{15} + 1664 \, b c^{5} x^{12} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{9} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{92160 \, c^{5}}, -\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} 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 (1280 \, c^{6} x^{15} + 1664 \, b c^{5} x^{12} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{9} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{6} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x^{3}\right )} \sqrt {c x^{6} + b x^{3} + a}}{46080 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{8} \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^8\,{\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.
[In]
[Out]
________________________________________________________________________________________