Optimal. Leaf size=93 \[ \frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^3 c^{3/2}}+\frac {(a x+1)^2}{3 a^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {5 (a x+1)}{3 a^3 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6151, 1635, 778, 217, 203} \[ \frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^3 c^{3/2}}+\frac {(a x+1)^2}{3 a^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {5 (a x+1)}{3 a^3 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 778
Rule 1635
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {x^2 (1+a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {(1+a x)^2}{3 a^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{3} \int \frac {\left (\frac {2}{a^2}+\frac {3 x}{a}\right ) (1+a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {(1+a x)^2}{3 a^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {5 (1+a x)}{3 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^2}{3 a^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {5 (1+a x)}{3 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{a^2 c}\\ &=\frac {(1+a x)^2}{3 a^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {5 (1+a x)}{3 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^3 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 0.88 \[ \frac {\frac {(5 a x-4) \sqrt {c-a^2 c x^2}}{(a x-1)^2}-3 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )}{3 a^3 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 215, normalized size = 2.31 \[ \left [-\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a x - 4\right )}}{6 \, {\left (a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right )}}, -\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (5 \, a x - 4\right )}}{3 \, {\left (a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 44, normalized size = 0.47 \[ -\frac {\log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{a^{2} \sqrt {-c} c {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 166, normalized size = 1.78 \[ -\frac {3 x}{a^{2} c \sqrt {-a^{2} c \,x^{2}+c}}+\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a^{2} c \sqrt {a^{2} c}}-\frac {2}{a^{3} c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2}{3 a^{4} c \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}+\frac {4 x}{3 a^{2} c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 230, normalized size = 2.47 \[ \frac {1}{3} \, a {\left (\frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{6} c x + \sqrt {-a^{2} c x^{2} + c} a^{5} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{6} c x - \sqrt {-a^{2} c x^{2} + c} a^{5} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{5} c x + \sqrt {-a^{2} c x^{2} + c} a^{4} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{5} c x - \sqrt {-a^{2} c x^{2} + c} a^{4} c} - \frac {5 \, x}{\sqrt {-a^{2} c x^{2} + c} a^{3} c} + \frac {3 \, \arcsin \left (a x\right )}{a^{4} c^{\frac {3}{2}}} - \frac {6}{\sqrt {-a^{2} c x^{2} + c} a^{4} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {x^2\,{\left (a\,x+1\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {a x^{3}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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