Optimal. Leaf size=169 \[ -\frac {2 (a x+1)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}-\frac {2 (a x+1)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac {4 \sqrt {a x+1} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{a^3 c (1-a x)^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6130, 23, 88, 50, 63, 208} \[ -\frac {2 (a x+1)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}-\frac {2 (a x+1)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac {4 \sqrt {a x+1} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{a^3 c (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 50
Rule 63
Rule 88
Rule 208
Rule 6130
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx &=\int \frac {x^2 (1+a x)^{3/2} \sqrt {c-a c x}}{(1-a x)^{3/2}} \, dx\\ &=\frac {(c-a c x)^{3/2} \int \frac {x^2 (1+a x)^{3/2}}{c-a c x} \, dx}{(1-a x)^{3/2}}\\ &=\frac {(c-a c x)^{3/2} \int \left (-\frac {(1+a x)^{5/2}}{a^2 c}+\frac {(1+a x)^{3/2}}{a^2 (c-a c x)}\right ) \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac {(c-a c x)^{3/2} \int \frac {(1+a x)^{3/2}}{c-a c x} \, dx}{a^2 (1-a x)^{3/2}}\\ &=-\frac {2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac {\left (2 (c-a c x)^{3/2}\right ) \int \frac {\sqrt {1+a x}}{c-a c x} \, dx}{a^2 (1-a x)^{3/2}}\\ &=-\frac {4 \sqrt {1+a x} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac {\left (4 (c-a c x)^{3/2}\right ) \int \frac {1}{\sqrt {1+a x} (c-a c x)} \, dx}{a^2 (1-a x)^{3/2}}\\ &=-\frac {4 \sqrt {1+a x} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac {\left (8 (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 c-c x^2} \, dx,x,\sqrt {1+a x}\right )}{a^3 (1-a x)^{3/2}}\\ &=-\frac {4 \sqrt {1+a x} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac {2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{a^3 c (1-a x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.50 \[ -\frac {2 \sqrt {c-a c x} \left (\sqrt {a x+1} \left (3 a^3 x^3+9 a^2 x^2+16 a x+52\right )-42 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )\right )}{21 a^3 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 258, normalized size = 1.53 \[ \left [\frac {2 \, {\left (21 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (3 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 16 \, a x + 52\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{21 \, {\left (a^{4} x - a^{3}\right )}}, \frac {2 \, {\left (42 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (3 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 16 \, a x + 52\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{21 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 130, normalized size = 0.77 \[ \frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} {\left (21 \, c \arctan \left (\frac {\sqrt {c}}{\sqrt {-c}}\right ) + 40 \, \sqrt {-c} \sqrt {c}\right )}}{a^{2} \sqrt {-c} c} - \frac {\frac {42 \, \sqrt {2} c^{4} \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + 3 \, {\left (a c x + c\right )}^{\frac {7}{2}} + 7 \, {\left (a c x + c\right )}^{\frac {3}{2}} c^{2} + 42 \, \sqrt {a c x + c} c^{3}}{a^{2} c^{4}}\right )}}{21 \, a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 129, normalized size = 0.76 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (-3 x^{3} a^{3} \sqrt {c \left (a x +1\right )}-9 x^{2} a^{2} \sqrt {c \left (a x +1\right )}+42 \sqrt {c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-16 x a \sqrt {c \left (a x +1\right )}-52 \sqrt {c \left (a x +1\right )}\right )}{21 \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, a^{3}} \]
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 \[ \int \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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