Integrand size = 19, antiderivative size = 253 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {21 a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1024 b^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1973, 285, 327, 221} \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=-\frac {21 a^{9/2} c \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \sqrt {c \left (a+b x^2\right )^3}}{1024 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
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Rule 221
Rule 285
Rule 327
Rule 1973
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^{9/2} \, dx}{\left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (3 a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^{7/2} \, dx}{4 \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^{5/2} \, dx}{40 \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^{3/2} \, dx}{64 \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \sqrt {1+\frac {b x^2}{a}} \, dx}{128 \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (21 a^3 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {x^2}{\sqrt {1+\frac {b x^2}{a}}} \, dx}{512 \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {\left (21 a^4 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {1}{\sqrt {1+\frac {b x^2}{a}}} \, dx}{1024 b \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ & = \frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}-\frac {21 a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1024 b^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.56 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {b} x \sqrt {a+b x^2} \left (315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right )+630 a^6 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )\right )}{15360 b^{3/2} \left (a+b x^2\right )^{9/2}} \]
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Time = 2.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.61
method | result | size |
risch | \(\frac {x \left (1280 x^{10} b^{5}+6272 a \,x^{8} b^{4}+12144 a^{2} x^{6} b^{3}+11432 a^{3} x^{4} b^{2}+4910 x^{2} a^{4} b +315 a^{5}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}}{15360 \left (b \,x^{2}+a \right ) b}-\frac {21 a^{6} \ln \left (\frac {b c x}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a c}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{3}}\, \sqrt {c \left (b \,x^{2}+a \right )}}{1024 b \sqrt {b c}\, \left (b \,x^{2}+a \right )^{2}}\) | \(155\) |
default | \(-\frac {{\left (c \left (b \,x^{2}+a \right )^{3}\right )}^{\frac {3}{2}} \left (-1280 x^{7} \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} b^{3} \sqrt {b c}-3712 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a \,b^{2} x^{5}-3440 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{2} b \,x^{3}+315 \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right ) a^{6} c^{3}-840 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{3} x +210 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{4} c x +315 \sqrt {b c}\, \sqrt {b c \,x^{2}+a c}\, a^{5} c^{2} x \right )}{15360 b \left (b \,x^{2}+a \right )^{3} {\left (c \left (b \,x^{2}+a \right )\right )}^{\frac {3}{2}} c \sqrt {b c}}\) | \(236\) |
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Time = 0.31 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.71 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\left [\frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {\frac {c}{b}} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c - 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {\frac {c}{b}}}{b x^{2} + a}\right ) + 2 \, {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{30720 \, {\left (b^{2} x^{2} + a b\right )}}, \frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {-\frac {c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) + {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{15360 \, {\left (b^{2} x^{2} + a b\right )}}\right ] \]
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\[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int x^{2} \left (c \left (a + b x^{2}\right )^{3}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int { \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}} x^{2} \,d x } \]
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Time = 0.46 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.70 \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {1}{15360} \, {\left (\frac {315 \, a^{6} c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b} + {\left (\frac {315 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{b} + 2 \, {\left (2455 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (1429 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (759 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 8 \, {\left (10 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 49 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \]
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Timed out. \[ \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int x^2\,{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2} \,d x \]
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