Integrand size = 28, antiderivative size = 313 \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {a^5 (d x)^{1+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac {5 a^4 b (d x)^{4+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )}+\frac {10 a^3 b^2 (d x)^{7+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^7 (7+m) \left (a+b x^3\right )}+\frac {10 a^2 b^3 (d x)^{10+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^{10} (10+m) \left (a+b x^3\right )}+\frac {5 a b^4 (d x)^{13+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^{13} (13+m) \left (a+b x^3\right )}+\frac {b^5 (d x)^{16+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^{16} (16+m) \left (a+b x^3\right )} \]
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Time = 0.10 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1369, 276} \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+16}}{d^{16} (m+16) \left (a+b x^3\right )}+\frac {5 a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+13}}{d^{13} (m+13) \left (a+b x^3\right )}+\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )}+\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )} \]
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Rule 276
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int (d x)^m \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 (d x)^m+\frac {5 a^4 b^6 (d x)^{3+m}}{d^3}+\frac {10 a^3 b^7 (d x)^{6+m}}{d^6}+\frac {10 a^2 b^8 (d x)^{9+m}}{d^9}+\frac {5 a b^9 (d x)^{12+m}}{d^{12}}+\frac {b^{10} (d x)^{15+m}}{d^{15}}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {a^5 (d x)^{1+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac {5 a^4 b (d x)^{4+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )}+\frac {10 a^3 b^2 (d x)^{7+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^7 (7+m) \left (a+b x^3\right )}+\frac {10 a^2 b^3 (d x)^{10+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^{10} (10+m) \left (a+b x^3\right )}+\frac {5 a b^4 (d x)^{13+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^{13} (13+m) \left (a+b x^3\right )}+\frac {b^5 (d x)^{16+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^{16} (16+m) \left (a+b x^3\right )} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.35 \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {x (d x)^m \left (\left (a+b x^3\right )^2\right )^{5/2} \left (\frac {a^5}{1+m}+\frac {5 a^4 b x^3}{4+m}+\frac {10 a^3 b^2 x^6}{7+m}+\frac {10 a^2 b^3 x^9}{10+m}+\frac {5 a b^4 x^{12}}{13+m}+\frac {b^5 x^{15}}{16+m}\right )}{\left (a+b x^3\right )^5} \]
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Time = 0.04 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.45
method | result | size |
gosper | \(\frac {x \left (b^{5} m^{5} x^{15}+35 b^{5} m^{4} x^{15}+445 b^{5} m^{3} x^{15}+5 a \,b^{4} m^{5} x^{12}+2485 b^{5} m^{2} x^{15}+190 a \,b^{4} m^{4} x^{12}+5714 m \,x^{15} b^{5}+2555 a \,b^{4} m^{3} x^{12}+3640 b^{5} x^{15}+10 a^{2} b^{3} m^{5} x^{9}+14810 a \,b^{4} m^{2} x^{12}+410 a^{2} b^{3} m^{4} x^{9}+34840 m \,x^{12} b^{4} a +5950 a^{2} b^{3} m^{3} x^{9}+22400 a \,b^{4} x^{12}+10 a^{3} b^{2} m^{5} x^{6}+36550 a^{2} b^{3} m^{2} x^{9}+440 a^{3} b^{2} m^{4} x^{6}+89240 m \,x^{9} a^{2} b^{3}+6970 a^{3} b^{2} m^{3} x^{6}+58240 a^{2} b^{3} x^{9}+5 a^{4} b \,m^{5} x^{3}+47260 a^{3} b^{2} m^{2} x^{6}+235 a^{4} b \,m^{4} x^{3}+123920 m \,x^{6} a^{3} b^{2}+4085 a^{4} b \,m^{3} x^{3}+83200 a^{3} b^{2} x^{6}+a^{5} m^{5}+31685 a^{4} b \,m^{2} x^{3}+50 a^{5} m^{4}+100630 m \,x^{3} b \,a^{4}+955 a^{5} m^{3}+72800 a^{4} b \,x^{3}+8650 a^{5} m^{2}+36824 m \,a^{5}+58240 a^{5}\right ) \left (d x \right )^{m} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{\left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right ) \left (b \,x^{3}+a \right )^{5}}\) | \(453\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (b^{5} m^{5} x^{15}+35 b^{5} m^{4} x^{15}+445 b^{5} m^{3} x^{15}+5 a \,b^{4} m^{5} x^{12}+2485 b^{5} m^{2} x^{15}+190 a \,b^{4} m^{4} x^{12}+5714 m \,x^{15} b^{5}+2555 a \,b^{4} m^{3} x^{12}+3640 b^{5} x^{15}+10 a^{2} b^{3} m^{5} x^{9}+14810 a \,b^{4} m^{2} x^{12}+410 a^{2} b^{3} m^{4} x^{9}+34840 m \,x^{12} b^{4} a +5950 a^{2} b^{3} m^{3} x^{9}+22400 a \,b^{4} x^{12}+10 a^{3} b^{2} m^{5} x^{6}+36550 a^{2} b^{3} m^{2} x^{9}+440 a^{3} b^{2} m^{4} x^{6}+89240 m \,x^{9} a^{2} b^{3}+6970 a^{3} b^{2} m^{3} x^{6}+58240 a^{2} b^{3} x^{9}+5 a^{4} b \,m^{5} x^{3}+47260 a^{3} b^{2} m^{2} x^{6}+235 a^{4} b \,m^{4} x^{3}+123920 m \,x^{6} a^{3} b^{2}+4085 a^{4} b \,m^{3} x^{3}+83200 a^{3} b^{2} x^{6}+a^{5} m^{5}+31685 a^{4} b \,m^{2} x^{3}+50 a^{5} m^{4}+100630 m \,x^{3} b \,a^{4}+955 a^{5} m^{3}+72800 a^{4} b \,x^{3}+8650 a^{5} m^{2}+36824 m \,a^{5}+58240 a^{5}\right ) x \left (d x \right )^{m}}{\left (b \,x^{3}+a \right ) \left (1+m \right ) \left (4+m \right ) \left (7+m \right ) \left (10+m \right ) \left (13+m \right ) \left (16+m \right )}\) | \(453\) |
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Time = 0.27 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.18 \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {{\left ({\left (b^{5} m^{5} + 35 \, b^{5} m^{4} + 445 \, b^{5} m^{3} + 2485 \, b^{5} m^{2} + 5714 \, b^{5} m + 3640 \, b^{5}\right )} x^{16} + 5 \, {\left (a b^{4} m^{5} + 38 \, a b^{4} m^{4} + 511 \, a b^{4} m^{3} + 2962 \, a b^{4} m^{2} + 6968 \, a b^{4} m + 4480 \, a b^{4}\right )} x^{13} + 10 \, {\left (a^{2} b^{3} m^{5} + 41 \, a^{2} b^{3} m^{4} + 595 \, a^{2} b^{3} m^{3} + 3655 \, a^{2} b^{3} m^{2} + 8924 \, a^{2} b^{3} m + 5824 \, a^{2} b^{3}\right )} x^{10} + 10 \, {\left (a^{3} b^{2} m^{5} + 44 \, a^{3} b^{2} m^{4} + 697 \, a^{3} b^{2} m^{3} + 4726 \, a^{3} b^{2} m^{2} + 12392 \, a^{3} b^{2} m + 8320 \, a^{3} b^{2}\right )} x^{7} + 5 \, {\left (a^{4} b m^{5} + 47 \, a^{4} b m^{4} + 817 \, a^{4} b m^{3} + 6337 \, a^{4} b m^{2} + 20126 \, a^{4} b m + 14560 \, a^{4} b\right )} x^{4} + {\left (a^{5} m^{5} + 50 \, a^{5} m^{4} + 955 \, a^{5} m^{3} + 8650 \, a^{5} m^{2} + 36824 \, a^{5} m + 58240 \, a^{5}\right )} x\right )} \left (d x\right )^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \]
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\[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int \left (d x\right )^{m} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.78 \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {{\left ({\left (m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640\right )} b^{5} d^{m} x^{16} + 5 \, {\left (m^{5} + 38 \, m^{4} + 511 \, m^{3} + 2962 \, m^{2} + 6968 \, m + 4480\right )} a b^{4} d^{m} x^{13} + 10 \, {\left (m^{5} + 41 \, m^{4} + 595 \, m^{3} + 3655 \, m^{2} + 8924 \, m + 5824\right )} a^{2} b^{3} d^{m} x^{10} + 10 \, {\left (m^{5} + 44 \, m^{4} + 697 \, m^{3} + 4726 \, m^{2} + 12392 \, m + 8320\right )} a^{3} b^{2} d^{m} x^{7} + 5 \, {\left (m^{5} + 47 \, m^{4} + 817 \, m^{3} + 6337 \, m^{2} + 20126 \, m + 14560\right )} a^{4} b d^{m} x^{4} + {\left (m^{5} + 50 \, m^{4} + 955 \, m^{3} + 8650 \, m^{2} + 36824 \, m + 58240\right )} a^{5} d^{m} x\right )} x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \]
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Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (247) = 494\).
Time = 0.36 (sec) , antiderivative size = 900, normalized size of antiderivative = 2.88 \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \]
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