Optimal. Leaf size=253 \[ \frac {(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac {9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac {27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac {81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}} \]
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Rubi [A]
time = 0.14, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {424, 393, 198,
197} \begin {gather*} \frac {x (b c-a d) (4 a d+15 b c)}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {81 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac {27 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac {9 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac {x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac {x \left (c+d x^3\right ) (b c-a d)}{16 a b \left (a+b x^3\right )^{16/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 393
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx &=\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac {\int \frac {c (15 b c+a d)+4 d (3 b c+a d) x^3}{\left (a+b x^3\right )^{16/3}} \, dx}{16 a b}\\ &=\frac {(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac {\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) \int \frac {1}{\left (a+b x^3\right )^{13/3}} \, dx}{52 a^2 b^2}\\ &=\frac {(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac {\left (9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\left (a+b x^3\right )^{10/3}} \, dx}{520 a^3 b^2}\\ &=\frac {(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac {9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac {\left (27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{1820 a^4 b^2}\\ &=\frac {(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac {9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac {27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac {\left (81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{7280 a^5 b^2}\\ &=\frac {(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac {\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac {9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac {27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac {81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}\\ \end {align*}
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Mathematica [A]
time = 1.26, size = 169, normalized size = 0.67 \begin {gather*} \frac {x \left (3645 b^5 c^2 x^{15}+486 a b^4 c x^{12} \left (40 c+d x^3\right )+81 a^2 b^3 x^9 \left (520 c^2+32 c d x^3+d^2 x^6\right )+520 a^5 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+144 a^3 b^2 x^6 \left (325 c^2+39 c d x^3+3 d^2 x^6\right )+156 a^4 b x^3 \left (175 c^2+40 c d x^3+6 d^2 x^6\right )\right )}{7280 a^6 \left (a+b x^3\right )^{16/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 197, normalized size = 0.78
method | result | size |
gosper | \(\frac {x \left (81 a^{2} b^{3} d^{2} x^{15}+486 a \,b^{4} c d \,x^{15}+3645 b^{5} c^{2} x^{15}+432 a^{3} b^{2} d^{2} x^{12}+2592 a^{2} b^{3} c d \,x^{12}+19440 a \,b^{4} c^{2} x^{12}+936 a^{4} b \,d^{2} x^{9}+5616 a^{3} b^{2} c d \,x^{9}+42120 a^{2} b^{3} c^{2} x^{9}+1040 a^{5} d^{2} x^{6}+6240 a^{4} b c d \,x^{6}+46800 a^{3} b^{2} c^{2} x^{6}+3640 a^{5} c d \,x^{3}+27300 a^{4} b \,c^{2} x^{3}+7280 c^{2} a^{5}\right )}{7280 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{6}}\) | \(197\) |
trager | \(\frac {x \left (81 a^{2} b^{3} d^{2} x^{15}+486 a \,b^{4} c d \,x^{15}+3645 b^{5} c^{2} x^{15}+432 a^{3} b^{2} d^{2} x^{12}+2592 a^{2} b^{3} c d \,x^{12}+19440 a \,b^{4} c^{2} x^{12}+936 a^{4} b \,d^{2} x^{9}+5616 a^{3} b^{2} c d \,x^{9}+42120 a^{2} b^{3} c^{2} x^{9}+1040 a^{5} d^{2} x^{6}+6240 a^{4} b c d \,x^{6}+46800 a^{3} b^{2} c^{2} x^{6}+3640 a^{5} c d \,x^{3}+27300 a^{4} b \,c^{2} x^{3}+7280 c^{2} a^{5}\right )}{7280 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{6}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 261, normalized size = 1.03 \begin {gather*} -\frac {{\left (455 \, b^{3} - \frac {1680 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {2184 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {1040 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} d^{2} x^{16}}{7280 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} + \frac {{\left (455 \, b^{4} - \frac {2240 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {4368 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {4160 \, {\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac {1820 \, {\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} c d x^{16}}{3640 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{5}} - \frac {{\left (91 \, b^{5} - \frac {560 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {1456 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {2080 \, {\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac {1820 \, {\left (b x^{3} + a\right )}^{4} b}{x^{12}} - \frac {1456 \, {\left (b x^{3} + a\right )}^{5}}{x^{15}}\right )} c^{2} x^{16}}{1456 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.91, size = 246, normalized size = 0.97 \begin {gather*} \frac {{\left (81 \, {\left (45 \, b^{5} c^{2} + 6 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{16} + 432 \, {\left (45 \, a b^{4} c^{2} + 6 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{13} + 936 \, {\left (45 \, a^{2} b^{3} c^{2} + 6 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{10} + 7280 \, a^{5} c^{2} x + 1040 \, {\left (45 \, a^{3} b^{2} c^{2} + 6 \, a^{4} b c d + a^{5} d^{2}\right )} x^{7} + 1820 \, {\left (15 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{4}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{7280 \, {\left (a^{6} b^{6} x^{18} + 6 \, a^{7} b^{5} x^{15} + 15 \, a^{8} b^{4} x^{12} + 20 \, a^{9} b^{3} x^{9} + 15 \, a^{10} b^{2} x^{6} + 6 \, a^{11} b x^{3} + a^{12}\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 = 1.48, size = 257, normalized size = 1.02 \begin {gather*} \frac {x\,\left (\frac {c^2}{16\,a}+\frac {a\,\left (\frac {d^2}{16\,b}-\frac {c\,d}{8\,a}\right )}{b}\right )}{{\left (b\,x^3+a\right )}^{16/3}}-\frac {x\,\left (\frac {d^2}{13\,b^2}-\frac {-a^2\,d^2+2\,a\,b\,c\,d+15\,b^2\,c^2}{208\,a^2\,b^2}\right )}{{\left (b\,x^3+a\right )}^{13/3}}+\frac {x\,\left (a^2\,d^2+6\,a\,b\,c\,d+45\,b^2\,c^2\right )}{520\,a^3\,b^2\,{\left (b\,x^3+a\right )}^{10/3}}+\frac {x\,\left (9\,a^2\,d^2+54\,a\,b\,c\,d+405\,b^2\,c^2\right )}{3640\,a^4\,b^2\,{\left (b\,x^3+a\right )}^{7/3}}+\frac {x\,\left (27\,a^2\,d^2+162\,a\,b\,c\,d+1215\,b^2\,c^2\right )}{7280\,a^5\,b^2\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {x\,\left (81\,a^2\,d^2+486\,a\,b\,c\,d+3645\,b^2\,c^2\right )}{7280\,a^6\,b^2\,{\left (b\,x^3+a\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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