Optimal. Leaf size=151 \[ \frac {(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac {(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac {9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac {27 (12 b c+a d) x}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac {81 (12 b c+a d) x}{1820 a^5 b \sqrt [3]{a+b x^3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {393, 198, 197}
\begin {gather*} \frac {81 x (a d+12 b c)}{1820 a^5 b \sqrt [3]{a+b x^3}}+\frac {27 x (a d+12 b c)}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac {9 x (a d+12 b c)}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac {x (a d+12 b c)}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac {x (b c-a d)}{13 a b \left (a+b x^3\right )^{13/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 393
Rubi steps
\begin {align*} \int \frac {c+d x^3}{\left (a+b x^3\right )^{16/3}} \, dx &=\frac {(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac {(12 b c+a d) \int \frac {1}{\left (a+b x^3\right )^{13/3}} \, dx}{13 a b}\\ &=\frac {(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac {(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac {(9 (12 b c+a d)) \int \frac {1}{\left (a+b x^3\right )^{10/3}} \, dx}{130 a^2 b}\\ &=\frac {(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac {(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac {9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac {(27 (12 b c+a d)) \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{455 a^3 b}\\ &=\frac {(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac {(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac {9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac {27 (12 b c+a d) x}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac {(81 (12 b c+a d)) \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{1820 a^4 b}\\ &=\frac {(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac {(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac {9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac {27 (12 b c+a d) x}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac {81 (12 b c+a d) x}{1820 a^5 b \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 100, normalized size = 0.66 \begin {gather*} \frac {x \left (972 b^4 c x^{12}+455 a^4 \left (4 c+d x^3\right )+351 a^2 b^2 x^6 \left (20 c+d x^3\right )+81 a b^3 x^9 \left (52 c+d x^3\right )+195 a^3 b x^3 \left (28 c+3 d x^3\right )\right )}{1820 a^5 \left (a+b x^3\right )^{13/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 105, normalized size = 0.70
method | result | size |
gosper | \(\frac {x \left (81 a \,b^{3} d \,x^{12}+972 b^{4} c \,x^{12}+351 a^{2} b^{2} d \,x^{9}+4212 a \,b^{3} c \,x^{9}+585 a^{3} b d \,x^{6}+7020 a^{2} b^{2} c \,x^{6}+455 a^{4} d \,x^{3}+5460 a^{3} b c \,x^{3}+1820 c \,a^{4}\right )}{1820 \left (b \,x^{3}+a \right )^{\frac {13}{3}} a^{5}}\) | \(105\) |
trager | \(\frac {x \left (81 a \,b^{3} d \,x^{12}+972 b^{4} c \,x^{12}+351 a^{2} b^{2} d \,x^{9}+4212 a \,b^{3} c \,x^{9}+585 a^{3} b d \,x^{6}+7020 a^{2} b^{2} c \,x^{6}+455 a^{4} d \,x^{3}+5460 a^{3} b c \,x^{3}+1820 c \,a^{4}\right )}{1820 \left (b \,x^{3}+a \right )^{\frac {13}{3}} a^{5}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 154, normalized size = 1.02 \begin {gather*} -\frac {{\left (140 \, b^{3} - \frac {546 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {780 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {455 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} d x^{13}}{1820 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{4}} + \frac {{\left (35 \, b^{4} - \frac {182 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {390 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {455 \, {\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac {455 \, {\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} c x^{13}}{455 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.31, size = 155, normalized size = 1.03 \begin {gather*} \frac {{\left (81 \, {\left (12 \, b^{4} c + a b^{3} d\right )} x^{13} + 351 \, {\left (12 \, a b^{3} c + a^{2} b^{2} d\right )} x^{10} + 585 \, {\left (12 \, a^{2} b^{2} c + a^{3} b d\right )} x^{7} + 1820 \, a^{4} c x + 455 \, {\left (12 \, a^{3} b c + a^{4} d\right )} x^{4}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{1820 \, {\left (a^{5} b^{5} x^{15} + 5 \, a^{6} b^{4} x^{12} + 10 \, a^{7} b^{3} x^{9} + 10 \, a^{8} b^{2} x^{6} + 5 \, a^{9} b x^{3} + a^{10}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 1.45, size = 132, normalized size = 0.87 \begin {gather*} \frac {x\,\left (\frac {c}{13\,a}-\frac {d}{13\,b}\right )}{{\left (b\,x^3+a\right )}^{13/3}}+\frac {x\,\left (a\,d+12\,b\,c\right )}{130\,a^2\,b\,{\left (b\,x^3+a\right )}^{10/3}}+\frac {x\,\left (9\,a\,d+108\,b\,c\right )}{910\,a^3\,b\,{\left (b\,x^3+a\right )}^{7/3}}+\frac {x\,\left (27\,a\,d+324\,b\,c\right )}{1820\,a^4\,b\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {x\,\left (81\,a\,d+972\,b\,c\right )}{1820\,a^5\,b\,{\left (b\,x^3+a\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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