Optimal. Leaf size=101 \[ -\frac {8 c \sqrt {1-a^2 x^2}}{5 a^2 \sqrt {c-a c x}}-\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{5 a^2}-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c} \]
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Rubi [A]
time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6263, 809, 671,
663} \begin {gather*} -\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a^2 c}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{5 a^2}-\frac {8 c \sqrt {1-a^2 x^2}}{5 a^2 \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rule 809
Rule 6263
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=\frac {\int \frac {x (c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c}-\frac {3 \int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx}{5 a c}\\ &=-\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{5 a^2}-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c}-\frac {4 \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac {8 c \sqrt {1-a^2 x^2}}{5 a^2 \sqrt {c-a c x}}-\frac {2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{5 a^2}-\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a^2 c}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 46, normalized size = 0.46 \begin {gather*} -\frac {2 c \sqrt {1-a^2 x^2} \left (6-3 a x+a^2 x^2\right )}{5 a^2 \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.31, size = 48, normalized size = 0.48
method | result | size |
gosper | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c x a +c}\, \left (a^{2} x^{2}-3 a x +6\right )}{5 \left (a x -1\right ) a^{2}}\) | \(47\) |
default | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (a^{2} x^{2}-3 a x +6\right )}{5 \left (a x -1\right ) a^{2}}\) | \(48\) |
risch | \(\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c \left (a^{2} x^{2}-3 a x +6\right ) \left (a x +1\right )}{5 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a^{2} \sqrt {\left (a x +1\right ) c}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 50, normalized size = 0.50 \begin {gather*} -\frac {2 \, {\left (a^{2} \sqrt {c} x^{2} - 3 \, a \sqrt {c} x + 6 \, \sqrt {c}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x - a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 49, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left (a^{2} x^{2} - 3 \, a x + 6\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{5 \, {\left (a^{3} x - a^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {x \sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 65, normalized size = 0.64 \begin {gather*} \frac {8 \, \sqrt {2} {\left | c \right |}}{5 \, a^{2} \sqrt {c}} - \frac {4 \, \sqrt {a c x + c} {\left | c \right |}}{a^{2} c} - \frac {2 \, {\left ({\left (a c x + c\right )}^{\frac {5}{2}} {\left | c \right |} - 5 \, {\left (a c x + c\right )}^{\frac {3}{2}} c {\left | c \right |}\right )}}{5 \, a^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 75, normalized size = 0.74 \begin {gather*} \frac {\sqrt {c-a\,c\,x}\,\left (\frac {12\,\sqrt {1-a^2\,x^2}}{5\,a^3}-\frac {6\,x\,\sqrt {1-a^2\,x^2}}{5\,a^2}+\frac {2\,x^2\,\sqrt {1-a^2\,x^2}}{5\,a}\right )}{x-\frac {1}{a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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