Optimal. Leaf size=249 \[ \frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {23 (1-a x)^{5/2} \sqrt {1+a x}}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}-\frac {7 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}+\frac {79 (1-a x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{8 \sqrt {2} a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6269, 6264,
99, 154, 159, 163, 56, 221, 95, 212} \begin {gather*} -\frac {7 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {79 (1-a x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{8 \sqrt {2} a^{7/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {23 \sqrt {a x+1} (1-a x)^{5/2}}{8 a^3 x^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 \sqrt {a x+1} (1-a x)^{3/2}}{8 a^2 x \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\sqrt {a x+1} \sqrt {1-a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 212
Rule 221
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx &=\frac {(1-a x)^{5/2} \int \frac {e^{\tanh ^{-1}(a x)} x^{5/2}}{(1-a x)^{5/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {(1-a x)^{5/2} \int \frac {x^{5/2} \sqrt {1+a x}}{(1-a x)^3} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {(1-a x)^{5/2} \int \frac {x^{3/2} \left (\frac {5}{2}+3 a x\right )}{(1-a x)^2 \sqrt {1+a x}} \, dx}{2 a \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {(1-a x)^{5/2} \int \frac {\sqrt {x} \left (-\frac {33 a}{4}-\frac {23 a^2 x}{2}\right )}{(1-a x) \sqrt {1+a x}} \, dx}{4 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {23 (1-a x)^{5/2} \sqrt {1+a x}}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1-a x)^{5/2} \int \frac {\frac {23 a^2}{4}+14 a^3 x}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{4 a^5 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {23 (1-a x)^{5/2} \sqrt {1+a x}}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}-\frac {\left (7 (1-a x)^{5/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}+\frac {\left (79 (1-a x)^{5/2}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{16 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {23 (1-a x)^{5/2} \sqrt {1+a x}}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}-\frac {\left (7 (1-a x)^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}+\frac {\left (79 (1-a x)^{5/2}\right ) \text {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {23 (1-a x)^{5/2} \sqrt {1+a x}}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}-\frac {7 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}+\frac {79 (1-a x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{8 \sqrt {2} a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 139, normalized size = 0.56 \begin {gather*} \frac {-2 \sqrt {a} \sqrt {x} \sqrt {1+a x} \left (23-35 a x+8 a^2 x^2\right )-112 (-1+a x)^2 \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )+79 \sqrt {2} (-1+a x)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{16 a^{3/2} c^2 \sqrt {c-\frac {c}{a x}} \sqrt {x} (1-a x)^{3/2}} \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. \(389\) vs.
\(2(193)=386\).
time = 0.61, size = 390, normalized size = 1.57
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-a c x \left (a x +1\right )}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right )}{2 a^{3} \sqrt {a^{2} c}}+\frac {\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{2 a^{6} c \left (x -\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{8 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {79 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{16 a^{4} \sqrt {-2 c}}\right ) a^{2} \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, c^{2}}\) | \(365\) |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (16 a^{\frac {7}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}-70 a^{\frac {5}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +79 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}-56 a^{3} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+46 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}+112 a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x -158 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x -56 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+79 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{32 a^{\frac {3}{2}} c^{3} \left (a x -1\right )^{3} \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\) | \(390\) |
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 = 0.46, size = 600, normalized size = 2.41 \begin {gather*} \left [-\frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (8 \, a^{3} x^{3} - 35 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (8 \, a^{3} x^{3} - 35 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\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 {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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