Optimal. Leaf size=105 \[ -\frac {11 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {11 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6268, 25, 528,
457, 79, 44, 65, 214} \begin {gather*} -\frac {11 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}-\frac {11 x \sqrt {c-\frac {c}{a x}}}{8 a^2}-\frac {1}{3} x^3 \sqrt {c-\frac {c}{a x}}-\frac {11 x^2 \sqrt {c-\frac {c}{a x}}}{12 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 44
Rule 65
Rule 79
Rule 214
Rule 457
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} x^2 (1+a x)}{1-a x} \, dx\\ &=-\frac {c \int \frac {x (1+a x)}{\sqrt {c-\frac {c}{a x}}} \, dx}{a}\\ &=-\frac {c \int \frac {\left (a+\frac {1}{x}\right ) x^2}{\sqrt {c-\frac {c}{a x}}} \, dx}{a}\\ &=\frac {c \text {Subst}\left (\int \frac {a+x}{x^4 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {(11 c) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 a}\\ &=-\frac {11 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {(11 c) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=-\frac {11 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {(11 c) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^3}\\ &=-\frac {11 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {11 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{8 a^2}\\ &=-\frac {11 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {11 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 50, normalized size = 0.48 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a x}} \left (a^3 x^3+11 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-\frac {1}{a x}\right )\right )}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.30, size = 155, normalized size = 1.48
method | result | size |
risch | \(-\frac {\left (8 a^{2} x^{2}+22 a x +33\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2}}-\frac {11 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{16 a^{2} \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(119\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {5}{2}}+60 a^{\frac {5}{2}} \sqrt {a \,x^{2}-x}\, x +96 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}}-30 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}}+48 a \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-15 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \right )}{48 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}}\) | \(155\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 163, normalized size = 1.55 \begin {gather*} \left [-\frac {2 \, {\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 33 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{48 \, a^{3}}, -\frac {{\left (8 \, a^{3} x^{3} + 22 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 33 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\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^{2} \sqrt {c - \frac {c}{a x}}}{a x - 1}\, dx - \int \frac {a x^{3} \sqrt {c - \frac {c}{a x}}}{a x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 127, normalized size = 1.21 \begin {gather*} -\frac {1}{24} \, \sqrt {a^{2} c x^{2} - a c x} {\left (2 \, x {\left (\frac {4 \, x {\left | a \right |}}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {11 \, {\left | a \right |}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} + \frac {33 \, {\left | a \right |}}{a^{4} \mathrm {sgn}\left (x\right )}\right )} - \frac {11 \, \sqrt {c} \log \left ({\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, a^{3}} + \frac {11 \, \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} \sqrt {c} {\left | a \right |} + a c \right |}\right )}{16 \, a^{3} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 90, normalized size = 0.86 \begin {gather*} \frac {11\,x^3\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-\frac {21\,x^3\,\sqrt {c-\frac {c}{a\,x}}}{8}-\frac {11\,x^3\,{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{8\,c^2}+\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-\frac {c}{a\,x}}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,11{}\mathrm {i}}{8\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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