Optimal. Leaf size=73 \[ -\frac {2 d \sqrt {a+b x^3}}{3 b^2 c^2}-\frac {2 \left (a c^2-d^2\right ) \log \left (c \sqrt {a+b x^3}+d\right )}{3 b^2 c^3}+\frac {x^3}{3 b c} \]
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Rubi [A] time = 0.20, antiderivative size = 73, 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} \[ -\frac {2 \left (a c^2-d^2\right ) \log \left (c \sqrt {a+b x^3}+d\right )}{3 b^2 c^3}-\frac {2 d \sqrt {a+b x^3}}{3 b^2 c^2}+\frac {x^3}{3 b c} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2155
Rubi steps
\begin {align*} \int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{a c+b c x+d \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {-a+x^2}{d+c x} \, dx,x,\sqrt {a+b x^3}\right )}{3 b^2}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {d}{c^2}+\frac {x}{c}+\frac {-a c^2+d^2}{c^2 (d+c x)}\right ) \, dx,x,\sqrt {a+b x^3}\right )}{3 b^2}\\ &=\frac {x^3}{3 b c}-\frac {2 d \sqrt {a+b x^3}}{3 b^2 c^2}-\frac {2 \left (a c^2-d^2\right ) \log \left (d+c \sqrt {a+b x^3}\right )}{3 b^2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 63, normalized size = 0.86 \[ \frac {\left (2 d^2-2 a c^2\right ) \log \left (c \sqrt {a+b x^3}+d\right )+c \left (b c x^3-2 d \sqrt {a+b x^3}\right )}{3 b^2 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 118, normalized size = 1.62 \[ \frac {b c^{2} x^{3} - 2 \, \sqrt {b x^{3} + a} c d - {\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right ) + {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 72, normalized size = 0.99 \[ -\frac {\frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac {{\left (b x^{3} + a\right )} b c - 2 \, \sqrt {b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 943, normalized size = 12.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 62, normalized size = 0.85 \[ \frac {\frac {{\left (b x^{3} + a\right )} c - 2 \, \sqrt {b x^{3} + a} d}{c^{2}} - \frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 119, normalized size = 1.63 \[ \frac {x^3}{3\,b\,c}-\frac {2\,d\,\sqrt {b\,x^3+a}}{3\,b^2\,c^2}+\frac {\ln \left (\frac {d-c\,\sqrt {b\,x^3+a}}{d+c\,\sqrt {b\,x^3+a}}\right )\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3}-\frac {\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3} \]
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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