Optimal. Leaf size=253 \[ -\frac {21 a^{9/2} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{1024 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.25, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6720, 279, 321, 217, 206} \[ -\frac {21 a^6 c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {21 a^5 c x \sqrt {c \left (a+b x^2\right )^3}}{1024 b \left (a+b x^2\right )}+\frac {21 a^4 c x^3 \sqrt {c \left (a+b x^2\right )^3}}{512 \left (a+b x^2\right )}+\frac {7}{128} a^3 c x^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{320} a^2 c x^3 \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {3}{40} a c x^3 \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{12} c x^3 \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 6720
Rubi steps
\begin {align*} \int x^2 \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{9/2} \, dx}{\left (a+b x^2\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 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{7/2} \, dx}{4 \left (a+b x^2\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^2 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \left (a+b x^2\right )^{5/2} \, dx}{40 \left (a+b x^2\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 (a+b x^2\right )^{3/2} \, dx}{64 \left (a+b x^2\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^4 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int x^2 \sqrt {a+b x^2} \, dx}{128 \left (a+b x^2\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^5 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{512 \left (a+b x^2\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^6 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{1024 b \left (a+b x^2\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^6 c \sqrt {c \left (a+b x^2\right )^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{1024 b \left (a+b x^2\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^6 c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{3/2} \left (a+b x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 143, normalized size = 0.57 \[ \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt {b} x \sqrt {\frac {b x^2}{a}+1} \left (315 a^5+4910 a^4 b x^2+11432 a^3 b^2 x^4+12144 a^2 b^3 x^6+6272 a b^4 x^8+1280 b^5 x^{10}\right )-315 a^{11/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{15360 b^{3/2} \left (a+b x^2\right )^4 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 433, normalized size = 1.71 \[ \left [\frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {\frac {c}{b}} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c - 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {\frac {c}{b}}}{b x^{2} + a}\right ) + 2 \, {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{30720 \, {\left (b^{2} x^{2} + a b\right )}}, \frac {315 \, {\left (a^{6} b c x^{2} + a^{7} c\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {-\frac {c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) + {\left (1280 \, b^{5} c x^{11} + 6272 \, a b^{4} c x^{9} + 12144 \, a^{2} b^{3} c x^{7} + 11432 \, a^{3} b^{2} c x^{5} + 4910 \, a^{4} b c x^{3} + 315 \, a^{5} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{15360 \, {\left (b^{2} x^{2} + a b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 177, normalized size = 0.70 \[ \frac {1}{15360} \, {\left (\frac {315 \, a^{6} c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b} + {\left (\frac {315 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{b} + 2 \, {\left (2455 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (1429 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (759 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 8 \, {\left (10 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 49 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 236, normalized size = 0.93 \[ -\frac {\left (\left (b \,x^{2}+a \right )^{3} c \right )^{\frac {3}{2}} \left (315 a^{6} c^{3} \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right )-1280 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, b^{3} x^{7}+315 \sqrt {b c}\, \sqrt {b c \,x^{2}+a c}\, a^{5} c^{2} x -3712 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a \,b^{2} x^{5}+210 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} a^{4} c x -3440 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{2} b \,x^{3}-840 \sqrt {b c}\, \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} a^{3} x \right )}{15360 \left (b \,x^{2}+a \right )^{3} \left (\left (b \,x^{2}+a \right ) c \right )^{\frac {3}{2}} \sqrt {b c}\, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2} \,d x \]
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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