Optimal. Leaf size=140 \[ -\frac {2 d \left (a+b x^3\right )^{3/2}}{9 b^3 c^2}+\frac {2 \left (a c^2-d^2\right )^2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b^3 c^5}+\frac {2 d \sqrt {a+b x^3} \left (2 a c^2-d^2\right )}{3 b^3 c^4}+\frac {\left (a+b x^3\right )^2}{6 b^3 c}-\frac {x^3 \left (2 a c^2-d^2\right )}{3 b^2 c^3} \]
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Rubi [A] time = 0.30, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2155, 697} \[ -\frac {x^3 \left (2 a c^2-d^2\right )}{3 b^2 c^3}+\frac {2 d \sqrt {a+b x^3} \left (2 a c^2-d^2\right )}{3 b^3 c^4}+\frac {2 \left (a c^2-d^2\right )^2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b^3 c^5}-\frac {2 d \left (a+b x^3\right )^{3/2}}{9 b^3 c^2}+\frac {\left (a+b x^3\right )^2}{6 b^3 c} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2155
Rubi steps
\begin {align*} \int \frac {x^8}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{a c+b c x+d \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (a-x^2\right )^2}{d+c x} \, dx,x,\sqrt {a+b x^3}\right )}{3 b^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {2 a c^2 d-d^3}{c^4}-\frac {\left (2 a c^2-d^2\right ) x}{c^3}-\frac {d x^2}{c^2}+\frac {x^3}{c}+\frac {\left (a c^2-d^2\right )^2}{c^4 (d+c x)}\right ) \, dx,x,\sqrt {a+b x^3}\right )}{3 b^3}\\ &=-\frac {\left (2 a c^2-d^2\right ) x^3}{3 b^2 c^3}+\frac {2 d \left (2 a c^2-d^2\right ) \sqrt {a+b x^3}}{3 b^3 c^4}-\frac {2 d \left (a+b x^3\right )^{3/2}}{9 b^3 c^2}+\frac {\left (a+b x^3\right )^2}{6 b^3 c}+\frac {2 \left (a c^2-d^2\right )^2 \log \left (d+c \sqrt {a+b x^3}\right )}{3 b^3 c^5}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 126, normalized size = 0.90 \[ \frac {c \left (a \left (20 c^2 d \sqrt {a+b x^3}-6 b c^3 x^3\right )+2 b c d x^3 \left (3 d-2 c \sqrt {a+b x^3}\right )-12 d^3 \sqrt {a+b x^3}+3 b^2 c^3 x^6\right )+12 \left (d^2-a c^2\right )^2 \log \left (c \sqrt {a+b x^3}+d\right )}{18 b^3 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 191, normalized size = 1.36 \[ \frac {3 \, b^{2} c^{4} x^{6} - 6 \, {\left (a b c^{4} - b c^{2} d^{2}\right )} x^{3} + 6 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + 6 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right ) - 6 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt {b x^{3} + a} c - d\right ) - 4 \, {\left (b c^{3} d x^{3} - 5 \, a c^{3} d + 3 \, c d^{3}\right )} \sqrt {b x^{3} + a}}{18 \, b^{3} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 156, normalized size = 1.11 \[ \frac {2 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{3 \, b^{3} c^{5}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{9} c^{3} - 12 \, {\left (b x^{3} + a\right )} a b^{9} c^{3} - 4 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{9} c^{2} d + 24 \, \sqrt {b x^{3} + a} a b^{9} c^{2} d + 6 \, {\left (b x^{3} + a\right )} b^{9} c d^{2} - 12 \, \sqrt {b x^{3} + a} b^{9} d^{3}}{18 \, b^{12} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 1473, normalized size = 10.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 125, normalized size = 0.89 \[ \frac {\frac {3 \, {\left (b x^{3} + a\right )}^{2} c^{3} - 4 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} c^{2} d - 6 \, {\left (2 \, a c^{3} - c d^{2}\right )} {\left (b x^{3} + a\right )} + 12 \, {\left (2 \, a c^{2} d - d^{3}\right )} \sqrt {b x^{3} + a}}{c^{4}} + \frac {12 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right )}{c^{5}}}{18 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 200, normalized size = 1.43 \[ \frac {\left (\frac {2\,d\,\left (a\,c^2-d^2\right )}{b^2\,c^4}+\frac {4\,a\,d}{3\,b^2\,c^2}\right )\,\sqrt {b\,x^3+a}}{3\,b}+\frac {x^6}{6\,b\,c}-\frac {x^3\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3}+\frac {\ln \left (\frac {d+c\,\sqrt {b\,x^3+a}}{d-c\,\sqrt {b\,x^3+a}}\right )\,{\left (a\,c^2-d^2\right )}^2}{3\,b^3\,c^5}+\frac {\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )\,\left (a^2\,c^4-2\,a\,c^2\,d^2+d^4\right )}{3\,b^3\,c^5}-\frac {2\,d\,x^3\,\sqrt {b\,x^3+a}}{9\,b^2\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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