Optimal. Leaf size=74 \[ \frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 (a x+1)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6141, 653, 192, 191} \[ \frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 (a x+1)}{5 a \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 653
Rule 6141
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {3}{5} \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac {2 (1+a x)}{5 a \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x}{5 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.72 \[ \frac {2 a^3 x^3-4 a^2 x^2+a x+2}{5 a c^2 (a x-1)^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 75, normalized size = 1.01 \[ -\frac {{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right )} \sqrt {-a^{2} c x^{2} + c}}{5 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 47, normalized size = 0.64 \[ \frac {\left (2 x^{3} a^{3}-4 a^{2} x^{2}+a x +2\right ) \left (a x +1\right )^{2}}{5 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 218, normalized size = 2.95 \[ \frac {1}{5} \, a {\left (\frac {a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c} - \frac {a}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{4} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c} - \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c} - \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{3} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{2} c} + \frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} a c^{2}} + \frac {x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 56, normalized size = 0.76 \[ -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a^3\,x^3-4\,a^2\,x^2+a\,x+2\right )}{5\,a\,c^3\,{\left (a\,x-1\right )}^3\,\left (a\,x+1\right )} \]
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 \[ - \int \frac {a x}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} - c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a^{5} c^{2} x^{5} \sqrt {- a^{2} c x^{2} + c} - a^{4} c^{2} x^{4} \sqrt {- a^{2} c x^{2} + c} - 2 a^{3} c^{2} x^{3} \sqrt {- a^{2} c x^{2} + c} + 2 a^{2} c^{2} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{2} x \sqrt {- a^{2} c x^{2} + c} - c^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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