Optimal. Leaf size=225 \[ \frac {13 a^{5/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{7/2}}-\frac {a^3 x^4 (2525-427 a x) \left (c-\frac {c}{a x}\right )^{7/2}}{15 (1-a x)^{7/2} \sqrt {a x+1}}-\frac {398 a^2 x^3 \left (c-\frac {c}{a x}\right )^{7/2}}{15 (1-a x)^{3/2} \sqrt {a x+1}}+\frac {38 a x^2 \left (c-\frac {c}{a x}\right )^{7/2}}{15 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 x \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{7/2}}{5 \sqrt {a x+1}} \]
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Rubi [A] time = 0.21, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6134, 6129, 98, 150, 143, 54, 215} \[ -\frac {a^3 x^4 (2525-427 a x) \left (c-\frac {c}{a x}\right )^{7/2}}{15 (1-a x)^{7/2} \sqrt {a x+1}}-\frac {398 a^2 x^3 \left (c-\frac {c}{a x}\right )^{7/2}}{15 (1-a x)^{3/2} \sqrt {a x+1}}+\frac {13 a^{5/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{7/2}}+\frac {38 a x^2 \left (c-\frac {c}{a x}\right )^{7/2}}{15 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 x \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{7/2}}{5 \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 143
Rule 150
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} (1-a x)^{7/2}}{x^{7/2}} \, dx}{(1-a x)^{7/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {(1-a x)^5}{x^{7/2} (1+a x)^{3/2}} \, dx}{(1-a x)^{7/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}-\frac {\left (2 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {(1-a x)^3 \left (\frac {19 a}{2}-\frac {3 a^2 x}{2}\right )}{x^{5/2} (1+a x)^{3/2}} \, dx}{5 (1-a x)^{7/2}}\\ &=\frac {38 a \left (c-\frac {c}{a x}\right )^{7/2} x^2}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}-\frac {\left (4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {(1-a x)^2 \left (-\frac {199 a^2}{4}-\frac {29 a^3 x}{4}\right )}{x^{3/2} (1+a x)^{3/2}} \, dx}{15 (1-a x)^{7/2}}\\ &=-\frac {398 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {38 a \left (c-\frac {c}{a x}\right )^{7/2} x^2}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}-\frac {\left (8 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {(1-a x) \left (\frac {1165 a^3}{8}+\frac {427 a^4 x}{8}\right )}{\sqrt {x} (1+a x)^{3/2}} \, dx}{15 (1-a x)^{7/2}}\\ &=-\frac {a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^4 (2525-427 a x)}{15 (1-a x)^{7/2} \sqrt {1+a x}}-\frac {398 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {38 a \left (c-\frac {c}{a x}\right )^{7/2} x^2}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}+\frac {\left (13 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{7/2}}\\ &=-\frac {a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^4 (2525-427 a x)}{15 (1-a x)^{7/2} \sqrt {1+a x}}-\frac {398 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {38 a \left (c-\frac {c}{a x}\right )^{7/2} x^2}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}+\frac {\left (13 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{7/2}}\\ &=-\frac {a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^4 (2525-427 a x)}{15 (1-a x)^{7/2} \sqrt {1+a x}}-\frac {398 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^3}{15 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {38 a \left (c-\frac {c}{a x}\right )^{7/2} x^2}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1-a x}}{5 \sqrt {1+a x}}+\frac {13 a^{5/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 3.76, size = 200, normalized size = 0.89 \[ -\frac {c^3 x \sqrt {c-\frac {c}{a x}} \left (1040 a^4 x^4 (a x-1)^3 \sqrt {a x+1} \sqrt {-a x (a x+1)} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-a x\right )+585 \left (19 a^4 x^4-86 a^3 x^3+70 a x-35\right ) \sin ^{-1}\left (\sqrt {-a x}\right )-3 \sqrt {-a x (a x+1)} \left (520 a^6 x^6-2470 a^5 x^5+6325 a^4 x^4-28706 a^3 x^3-21508 a^2 x^2+19192 a x-6921\right )\right )}{720 (-a x)^{7/2} \sqrt {a x+1} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 386, normalized size = 1.72 \[ \left [\frac {195 \, {\left (a^{4} c^{3} x^{4} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (15 \, a^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{5} x^{4} - a^{3} x^{2}\right )}}, \frac {195 \, {\left (a^{4} c^{3} x^{4} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (15 \, a^{4} c^{3} x^{4} + 1591 \, a^{3} c^{3} x^{3} + 548 \, a^{2} c^{3} x^{2} - 62 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{5} x^{4} - a^{3} x^{2}\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 209, normalized size = 0.93 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (30 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, x^{4}+195 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x^{4} a^{4}+3182 a^{\frac {7}{2}} x^{3} \sqrt {-\left (a x +1\right ) x}+195 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x^{3} a^{3}+1096 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}-124 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+12 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {-a^{2} x^{2}+1}}{30 x^{2} a^{\frac {7}{2}} \left (a x +1\right ) \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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