Optimal. Leaf size=80 \[ -\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {4 \sqrt {c-a c x}}{a^3} \]
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Rubi [A] time = 0.22, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6167, 6130, 21, 77} \[ -\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {4 \sqrt {c-a c x}}{a^3} \]
Antiderivative was successfully verified.
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Rule 21
Rule 77
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\\ &=-\int \frac {x^2 (1+a x) \sqrt {c-a c x}}{1-a x} \, dx\\ &=-\left (c \int \frac {x^2 (1+a x)}{\sqrt {c-a c x}} \, dx\right )\\ &=-\left (c \int \left (\frac {2}{a^2 \sqrt {c-a c x}}-\frac {5 \sqrt {c-a c x}}{a^2 c}+\frac {4 (c-a c x)^{3/2}}{a^2 c^2}-\frac {(c-a c x)^{5/2}}{a^2 c^3}\right ) \, dx\right )\\ &=\frac {4 \sqrt {c-a c x}}{a^3}-\frac {10 (c-a c x)^{3/2}}{3 a^3 c}+\frac {8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac {2 (c-a c x)^{7/2}}{7 a^3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 0.50 \[ \frac {2 \left (15 a^3 x^3+39 a^2 x^2+52 a x+104\right ) \sqrt {c-a c x}}{105 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 36, normalized size = 0.45 \[ \frac {2 \, {\left (15 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 52 \, a x + 104\right )} \sqrt {-a c x + c}}{105 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 142, normalized size = 1.78 \[ \frac {2 \, {\left (\frac {7 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-a c x + c} c^{2}\right )}}{a^{2} c^{2}} + \frac {3 \, {\left (5 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 35 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-a c x + c} c^{3}\right )}}{a^{2} c^{3}}\right )}}{105 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.46 \[ \frac {2 \sqrt {-a c x +c}\, \left (15 x^{3} a^{3}+39 a^{2} x^{2}+52 a x +104\right )}{105 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 60, normalized size = 0.75 \[ -\frac {2 \, {\left (15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 84 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 175 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 210 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 66, normalized size = 0.82 \[ \frac {4\,\sqrt {c-a\,c\,x}}{a^3}-\frac {10\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^3\,c}+\frac {8\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^3\,c^2}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^3\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.31, size = 68, normalized size = 0.85 \[ - \frac {2 \left (- 2 c^{3} \sqrt {- a c x + c} + \frac {5 c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {4 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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