Optimal. Leaf size=267 \[ \frac {5 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt {1+a x}}+\frac {70 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt {1+a x}}-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {15 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6269, 6264,
100, 155, 148, 56, 221} \begin {gather*} -\frac {15 a^{7/2} x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}+\frac {5 a^4 x^5 (587-109 a x) \left (c-\frac {c}{a x}\right )^{9/2}}{7 (1-a x)^{9/2} \sqrt {a x+1}}+\frac {70 a^3 x^4 \left (c-\frac {c}{a x}\right )^{9/2}}{(1-a x)^{5/2} \sqrt {a x+1}}-\frac {50 a^2 x^3 \left (c-\frac {c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2} \sqrt {a x+1}}+\frac {10 a x^2 \left (c-\frac {c}{a x}\right )^{9/2}}{7 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 x \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{9/2}}{7 \sqrt {a x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 148
Rule 155
Rule 221
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^6}{x^{9/2} (1+a x)^{3/2}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {\left (2 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^4 \left (\frac {25 a}{2}-\frac {5 a^2 x}{2}\right )}{x^{7/2} (1+a x)^{3/2}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {\left (4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^3 \left (-\frac {375 a^2}{4}-\frac {25 a^3 x}{4}\right )}{x^{5/2} (1+a x)^{3/2}} \, dx}{35 (1-a x)^{9/2}}\\ &=-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {\left (8 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^2 \left (\frac {3675 a^3}{8}+\frac {825 a^4 x}{8}\right )}{x^{3/2} (1+a x)^{3/2}} \, dx}{105 (1-a x)^{9/2}}\\ &=\frac {70 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt {1+a x}}-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {\left (16 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x) \left (-\frac {21225 a^4}{16}-\frac {8175 a^5 x}{16}\right )}{\sqrt {x} (1+a x)^{3/2}} \, dx}{105 (1-a x)^{9/2}}\\ &=\frac {5 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt {1+a x}}+\frac {70 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt {1+a x}}-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {\left (15 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=\frac {5 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt {1+a x}}+\frac {70 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt {1+a x}}-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {\left (15 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{9/2}}\\ &=\frac {5 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5 (587-109 a x)}{7 (1-a x)^{9/2} \sqrt {1+a x}}+\frac {70 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt {1+a x}}-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {15 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 5.24, size = 234, normalized size = 0.88 \begin {gather*} -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {-a x (1+a x)} \left (3091-12955 a x+34100 a^2 x^2-59750 a^3 x^3-375805 a^4 x^4+84329 a^5 x^5+165830 a^6 x^6-214760 a^7 x^7+70000 a^8 x^8\right )+105 \left (-27+81 a x-54 a^2 x^2-54 a^3 x^3+209 a^4 x^4+101 a^5 x^5\right ) \text {ArcSin}\left (\sqrt {-a x}\right )-\frac {7168 (-1+a x)^4 (-a x (1+a x))^{5/2} \, _5F_4\left (-\frac {3}{2},2,2,2,\frac {5}{2};1,1,1,\frac {7}{2};-a x\right )}{(1+a x)^{3/2}}\right )}{896 a^4 x^3 \sqrt {-a x (1+a x)} \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.75, size = 227, normalized size = 0.85
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (14 a^{\frac {11}{2}} \sqrt {-\left (a x +1\right ) x}\, x^{5}+105 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a^{5} x^{5}+3510 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, x^{4}+105 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a^{4} x^{4}+1440 a^{\frac {7}{2}} x^{3} \sqrt {-\left (a x +1\right ) x}-220 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}+40 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}-4 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {-a^{2} x^{2}+1}}{14 x^{3} a^{\frac {9}{2}} \left (a x +1\right ) \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )}\) | \(227\) |
risch | \(\frac {\left (7 a^{5} x^{5}+859 a^{4} x^{4}+720 a^{3} x^{3}-110 a^{2} x^{2}+20 a x -2\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{7 x^{3} \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (-\frac {15 a^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right )}{2 \sqrt {a^{2} c}}-\frac {128 a^{2} \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+\left (x +\frac {1}{a}\right ) a c}}{c \left (x +\frac {1}{a}\right )}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}\, a^{4}}\) | \(255\) |
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 = 408, normalized size = 1.53 \begin {gather*} \left [\frac {105 \, {\left (a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\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 ) - 4 \, {\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{28 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, \frac {105 \, {\left (a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\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 ) - 2 \, {\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{14 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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