Optimal. Leaf size=91 \[ \frac {3 x (a d+6 b c)}{28 a^3 b \sqrt [3]{a+b x^3}}+\frac {x (a d+6 b c)}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {x (b c-a d)}{7 a b \left (a+b x^3\right )^{7/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {385, 192, 191} \[ \frac {3 x (a d+6 b c)}{28 a^3 b \sqrt [3]{a+b x^3}}+\frac {x (a d+6 b c)}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {x (b c-a d)}{7 a b \left (a+b x^3\right )^{7/3}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 385
Rubi steps
\begin {align*} \int \frac {c+d x^3}{\left (a+b x^3\right )^{10/3}} \, dx &=\frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{7 a b}\\ &=\frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) x}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {(3 (6 b c+a d)) \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{28 a^2 b}\\ &=\frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) x}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {3 (6 b c+a d) x}{28 a^3 b \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 0.65 \[ \frac {7 a^2 \left (4 c x+d x^4\right )+3 a b x^4 \left (14 c+d x^3\right )+18 b^2 c x^7}{28 a^3 \left (a+b x^3\right )^{7/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 87, normalized size = 0.96 \[ \frac {{\left (3 \, {\left (6 \, b^{2} c + a b d\right )} x^{7} + 7 \, {\left (6 \, a b c + a^{2} d\right )} x^{4} + 28 \, a^{2} c x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{28 \, {\left (a^{3} b^{3} x^{9} + 3 \, a^{4} b^{2} x^{6} + 3 \, a^{5} b x^{3} + a^{6}\right )}} \]
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 \[ \int \frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {10}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.63 \[ \frac {\left (3 a b d \,x^{6}+18 b^{2} c \,x^{6}+7 a^{2} d \,x^{3}+42 a b c \,x^{3}+28 a^{2} c \right ) x}{28 \left (b \,x^{3}+a \right )^{\frac {7}{3}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 86, normalized size = 0.95 \[ -\frac {{\left (4 \, b - \frac {7 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} d x^{7}}{28 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a^{2}} + \frac {{\left (2 \, b^{2} - \frac {7 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {14 \, {\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} c x^{7}}{14 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 87, normalized size = 0.96 \[ \frac {3\,a\,d\,x\,{\left (b\,x^3+a\right )}^2-4\,a^3\,d\,x+18\,b\,c\,x\,{\left (b\,x^3+a\right )}^2+a^2\,d\,x\,\left (b\,x^3+a\right )+4\,a^2\,b\,c\,x+6\,a\,b\,c\,x\,\left (b\,x^3+a\right )}{28\,a^3\,b\,{\left (b\,x^3+a\right )}^{7/3}} \]
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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