Optimal. Leaf size=139 \[ \frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {9 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.15, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6268, 25, 528,
382, 100, 159, 162, 65, 214} \begin {gather*} \frac {9 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}+\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-x \left (c-\frac {c}{a x}\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 65
Rule 100
Rule 159
Rule 162
Rule 214
Rule 382
Rule 528
Rule 6268
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^{5/2} (1-a x)}{1+a x} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{7/2} x}{1+a x} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{7/2}}{a+\frac {1}{x}} \, dx}{c}\\ &=\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2} \left (\frac {9 c^2}{2}+\frac {c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {2 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}} \left (\frac {27 c^3}{4}-\frac {21 c^3 x}{4 a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {4 \text {Subst}\left (\int \frac {\frac {27 c^4}{8}-\frac {69 c^4 x}{8 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {\left (9 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}+\frac {\left (16 c^3\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\left (c-\frac {c}{a x}\right )^{5/2} x+\left (9 c^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )-\left (32 c^2\right ) \text {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\frac {7 c^2 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}-\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {9 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {16 \sqrt {2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 116, normalized size = 0.83 \begin {gather*} \frac {c^2 \sqrt {c-\frac {c}{a x}} \left (-2+26 a x-3 a^2 x^2\right )+27 a c^{5/2} x \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-48 \sqrt {2} a c^{5/2} x \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{3 a^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs.
\(2(116)=232\).
time = 0.41, size = 257, normalized size = 1.85
method | result | size |
risch | \(-\frac {\left (3 a^{3} x^{3}-29 a^{2} x^{2}+28 a x -2\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \,a^{2} \left (a x -1\right )}-\frac {\left (-\frac {9 a^{2} \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right )}{2 \sqrt {a^{2} c}}-\frac {8 a \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{\sqrt {c}}\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{a^{2} \left (a x -1\right )}\) | \(214\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (48 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}\, x^{3}-90 a^{\frac {5}{2}} \sqrt {a \,x^{2}-x}\, \sqrt {\frac {1}{a}}\, x^{3}-48 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{3}+48 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+45 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-72 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x^{3}-4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{6 x^{2} a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, \sqrt {\frac {1}{a}}}\) | \(257\) |
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 = 0.37, size = 282, normalized size = 2.03 \begin {gather*} \left [\frac {48 \, \sqrt {2} a c^{\frac {5}{2}} x \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 27 \, a c^{\frac {5}{2}} x \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (3 \, a^{2} c^{2} x^{2} - 26 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, a^{2} x}, \frac {48 \, \sqrt {2} a \sqrt {-c} c^{2} x \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - 27 \, a \sqrt {-c} c^{2} x \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{2} c^{2} x^{2} - 26 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x}\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 \left (- \frac {3 c^{2} \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {3 c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int \left (- \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\right )\, dx - \int \frac {a c^{2} x \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} -\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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