Optimal. Leaf size=120 \[ -\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
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Rubi [A]
time = 0.12, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6268, 25, 528,
382, 79, 52, 65, 214} \begin {gather*} \frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-x \left (c-\frac {c}{a x}\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 52
Rule 65
Rule 79
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 )^{7/2} \, dx &=\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)}{1-a x} \, dx\\ &=-\frac {c \int \frac {\left (c-\frac {c}{a x}\right )^{5/2} (1+a x)}{x} \, dx}{a}\\ &=-\frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{5/2} \, dx}{a}\\ &=\frac {c \text {Subst}\left (\int \frac {(a+x) \left (c-\frac {c x}{a}\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {(3 c) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {\left (3 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}-\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}-\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}-\left (c-\frac {c}{a x}\right )^{7/2} x+\frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 83, normalized size = 0.69 \begin {gather*} -\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (-2+4 a x+8 a^2 x^2+5 a^3 x^3\right )}{5 a^3 x^2}+\frac {3 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 144, normalized size = 1.20
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (30 a^{\frac {7}{2}} \sqrt {a \,x^{2}-x}\, x^{4}-20 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-15 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}-4 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x +4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{10 x^{3} \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}}}\) | \(144\) |
risch | \(-\frac {\left (5 a^{4} x^{4}+3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +2\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{5 x^{2} a^{3} \left (a x -1\right )}+\frac {3 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c x a}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {a c x \left (a x -1\right )}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(147\) |
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 = 0.35, size = 212, normalized size = 1.77 \begin {gather*} \left [\frac {15 \, a^{2} c^{\frac {7}{2}} x^{2} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, a^{3} x^{2}}, -\frac {15 \, a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, a^{3} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 14.86, size = 731, normalized size = 6.09 \begin {gather*} - c^{3} \left (\begin {cases} - \frac {\sqrt {c} \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {\sqrt {c} \sqrt {x} \sqrt {a x - 1}}{\sqrt {a}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {i \sqrt {a} \sqrt {c} x^{\frac {3}{2}}}{\sqrt {- a x + 1}} + \frac {i \sqrt {c} \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} + \frac {i \sqrt {c} \sqrt {x}}{\sqrt {a} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) - \frac {2 c^{4} \operatorname {atan}{\left (\frac {\sqrt {c - \frac {c}{a x}}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {2 c^{3} \sqrt {c - \frac {c}{a x}}}{a} + \frac {c^{3} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {2 a \left (c - \frac {c}{a x}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {c^{3} \left (\begin {cases} - \frac {4 a^{\frac {11}{2}} \sqrt {c} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 a^{\frac {9}{2}} \sqrt {c} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 a^{5} \sqrt {c} x^{3} \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {2 a^{4} \sqrt {c} x^{2} \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {8 a^{3} \sqrt {c} x \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {6 a^{2} \sqrt {c} \sqrt {a x - 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {4 a^{\frac {11}{2}} \sqrt {c} x^{\frac {7}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 a^{\frac {9}{2}} \sqrt {c} x^{\frac {5}{2}}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {4 i a^{5} \sqrt {c} x^{3} \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {2 i a^{4} \sqrt {c} x^{2} \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} - \frac {8 i a^{3} \sqrt {c} x \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} + \frac {6 i a^{2} \sqrt {c} \sqrt {- a x + 1}}{15 a^{\frac {7}{2}} x^{\frac {7}{2}} - 15 a^{\frac {5}{2}} x^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right )}{a^{3}} \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 )}^{7/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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