Optimal. Leaf size=207 \[ \frac {63 a^{7/2} c \sqrt {c \left (a+b x^2\right )^3} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/2}}+\frac {63 a^4 c x \sqrt {c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.07, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6720, 195, 217, 206} \[ \frac {63 a^4 c x \sqrt {c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^5 c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 \sqrt {b} \left (a+b x^2\right )^{3/2}}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 6720
Rubi steps
\begin {align*} \int \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 \left (a+b x^2\right )^{9/2} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (9 a c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{7/2} \, dx}{10 \left (a+b x^2\right )^{3/2}}\\ &=\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (63 a^2 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{80 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \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 \left (a+b x^2\right )^{3/2} \, dx}{32 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (63 a^4 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \sqrt {a+b x^2} \, dx}{128 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^4 c x \sqrt {c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (63 a^5 c \sqrt {c \left (a+b x^2\right )^3}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^4 c x \sqrt {c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {\left (63 a^5 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 )}{256 \left (a+b x^2\right )^{3/2}}\\ &=\frac {21}{128} a^3 c x \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^4 c x \sqrt {c \left (a+b x^2\right )^3}}{256 \left (a+b x^2\right )}+\frac {21}{160} a^2 c x \left (a+b x^2\right ) \sqrt {c \left (a+b x^2\right )^3}+\frac {9}{80} a c x \left (a+b x^2\right )^2 \sqrt {c \left (a+b x^2\right )^3}+\frac {1}{10} c x \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^3}+\frac {63 a^5 c \sqrt {c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 \sqrt {b} \left (a+b x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 132, normalized size = 0.64 \[ \frac {\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (315 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\sqrt {b} x \sqrt {\frac {b x^2}{a}+1} \left (965 a^4+1490 a^3 b x^2+1368 a^2 b^2 x^4+656 a b^3 x^6+128 b^4 x^8\right )\right )}{1280 \sqrt {b} \left (a+b x^2\right )^4 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 402, normalized size = 1.94 \[ \left [\frac {315 \, {\left (a^{5} b c x^{2} + a^{6} c\right )} \sqrt {\frac {c}{b}} \log \left (-\frac {2 \, b^{2} c x^{4} + 3 \, a b c x^{2} + a^{2} c + 2 \, \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {\frac {c}{b}}}{b x^{2} + a}\right ) + 2 \, {\left (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{2560 \, {\left (b x^{2} + a\right )}}, -\frac {315 \, {\left (a^{5} b c x^{2} + a^{6} c\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {\sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} b x \sqrt {-\frac {c}{b}}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) - {\left (128 \, b^{4} c x^{9} + 656 \, a b^{3} c x^{7} + 1368 \, a^{2} b^{2} c x^{5} + 1490 \, a^{3} b c x^{3} + 965 \, a^{4} c x\right )} \sqrt {b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{1280 \, {\left (b x^{2} + a\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 153, normalized size = 0.74 \[ -\frac {1}{1280} \, {\left (\frac {315 \, a^{5} c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c}} - {\left (965 \, a^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (745 \, a^{3} b \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (171 \, a^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, {\left (8 \, b^{4} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 41 \, a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b c x^{2} + a c} x\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 205, normalized size = 0.99 \[ \frac {\left (\left (b \,x^{2}+a \right )^{3} c \right )^{\frac {3}{2}} \left (315 a^{5} c^{3} \ln \left (\frac {b c x +\sqrt {b c \,x^{2}+a c}\, \sqrt {b c}}{\sqrt {b c}}\right )+315 \sqrt {b c \,x^{2}+a c}\, \sqrt {b c}\, a^{4} c^{2} x +128 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, b^{2} x^{5}+210 \left (b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {b c}\, a^{3} c x +400 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a b \,x^{3}+440 \left (b c \,x^{2}+a c \right )^{\frac {5}{2}} \sqrt {b c}\, a^{2} x \right )}{1280 \left (b \,x^{2}+a \right )^{3} \left (\left (b \,x^{2}+a \right ) c \right )^{\frac {3}{2}} \sqrt {b c}\, c} \]
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 \[ \int \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2} \,d x \]
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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