Optimal. Leaf size=294 \[ -\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{6 (1-a x)^2}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 (1-a x)}+\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2}-\frac {25 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}+\frac {2 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}+\frac {9 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{8 (1-a x)^{5/2} (a x+1)^{5/2}}+\frac {17 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{12 (1-a x)^2 (a x+1)} \]
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Rubi [A] time = 0.45, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac {25 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2 (a x+1)^2}+\frac {17 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{12 (1-a x)^2 (a x+1)}+\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{8 (1-a x)^2}-\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{6 (1-a x)^2}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{5/2}}{4 (1-a x)}+\frac {2 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}+\frac {9 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{8 (1-a x)^{5/2} (a x+1)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 97
Rule 149
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6129
Rule 6159
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{5/2} (1+a x)^{5/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{7/2}}{x^5} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1-a x} (1+a x)^{5/2} \left (2 a-5 a^2 x\right )}{x^4} \, dx}{4 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1+a x)^{5/2} \left (-15 a^2+13 a^3 x\right )}{x^3 \sqrt {1-a x}} \, dx}{12 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {(1+a x)^{3/2} \left (-34 a^3+41 a^4 x\right )}{x^2 \sqrt {1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}+\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {\sqrt {1+a x} \left (-27 a^4+75 a^5 x\right )}{x \sqrt {1-a x}} \, dx}{24 (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}+\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {27 a^5-48 a^6 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 a (1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}+\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}-\frac {\left (9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 (1-a x)^{5/2} (1+a x)^{5/2}}+\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}+\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {\left (9 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}+\frac {\left (2 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{5/2} (1+a x)^{5/2}}\\ &=\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^3}{8 (1-a x)^2}-\frac {25 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5}{8 (1-a x)^2 (1+a x)^2}+\frac {17 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^4}{12 (1-a x)^2 (1+a x)}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^2 (1+a x)}{6 (1-a x)^2}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} x (1+a x)}{4 (1-a x)}+\frac {2 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \sin ^{-1}(a x)}{(1-a x)^{5/2} (1+a x)^{5/2}}+\frac {9 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{5/2} x^5 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 (1-a x)^{5/2} (1+a x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 134, normalized size = 0.46 \[ -\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (48 a^4 x^4 \log \left (\sqrt {a^2 x^2-1}+a x\right )+27 a^4 x^4 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )+\sqrt {a^2 x^2-1} \left (24 a^4 x^4-64 a^3 x^3-3 a^2 x^2+16 a x+6\right )\right )}{24 a^4 x^3 \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.50, size = 394, normalized size = 1.34 \[ \left [\frac {96 \, a^{3} \sqrt {-c} c^{2} x^{3} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 27 \, a^{3} \sqrt {-c} c^{2} x^{3} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, a^{4} x^{3}}, -\frac {27 \, a^{3} c^{\frac {5}{2}} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 24 \, a^{3} c^{\frac {5}{2}} x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (24 \, a^{4} c^{2} x^{4} - 64 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} + 16 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, a^{4} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 6.54, size = 416, normalized size = 1.41 \[ \frac {1}{12} \, {\left (\frac {27 \, c^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} + \frac {24 \, c^{\frac {5}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {12 \, \sqrt {a^{2} c x^{2} - c} c^{2} \mathrm {sgn}\relax (x)}{a^{2}} - \frac {3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{3} {\left | a \right |} \mathrm {sgn}\relax (x) - 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\relax (x) - 192 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{5} {\left | a \right |} \mathrm {sgn}\relax (x) - 160 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{6} {\left | a \right |} \mathrm {sgn}\relax (x) - 64 \, a c^{\frac {13}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 625, normalized size = 2.13 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x \left (-80 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{5} a^{7} c +80 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} x^{3} a^{7}+48 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{4} a^{6} c +27 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x^{4} a^{6} c +60 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{5} a^{5} c^{2}-75 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} x^{2} a^{6}+100 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{5} a^{5} c^{2}-80 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} x \,a^{5}-45 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{4} a^{4} c^{2}-90 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{5} a^{3} c^{3}-150 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{5} a^{3} c^{3}-30 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}+150 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{4} a +90 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {7}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{4} a +135 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{4} a^{2} c^{3}+135 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{4} c^{4}\right )}{120 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} c} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{a^{2} x^{2} - 1}\,{d x} \]
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 \[ \int -\frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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