Optimal. Leaf size=139 \[ \frac {2 c^3 (1+a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {4 c^3 (1+a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {c^3 (1+a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6278, 6275, 45}
\begin {gather*} \frac {c^3 (a x+1)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}-\frac {4 c^3 (a x+1)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {2 c^3 (a x+1)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^2 (1+a x)^5 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int \left (4 (1+a x)^5-4 (1+a x)^6+(1+a x)^7\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 c^3 (1+a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {4 c^3 (1+a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {c^3 (1+a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 60, normalized size = 0.43 \begin {gather*} \frac {c^3 (1+a x)^6 \left (37-54 a x+21 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{168 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 98, normalized size = 0.71
method | result | size |
gosper | \(\frac {x \left (21 a^{7} x^{7}+72 x^{6} a^{6}+28 x^{5} a^{5}-168 a^{4} x^{4}-210 a^{3} x^{3}+56 a^{2} x^{2}+252 a x +168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{168 \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\) | \(97\) |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c^{3} x \left (21 a^{7} x^{7}+72 x^{6} a^{6}+28 x^{5} a^{5}-168 a^{4} x^{4}-210 a^{3} x^{3}+56 a^{2} x^{2}+252 a x +168\right )}{168 \left (a^{2} x^{2}-1\right )}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs.
\(2 (121) = 242\).
time = 0.30, size = 323, normalized size = 2.32 \begin {gather*} \frac {1}{5} \, a^{4} c^{\frac {7}{2}} x^{5} - \frac {2}{3} \, a^{2} c^{\frac {7}{2}} x^{3} + c^{\frac {7}{2}} x - \frac {1}{24} \, {\left (\frac {3 \, a^{6} c^{4} x^{10}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {11 \, a^{4} c^{4} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {14 \, a^{2} c^{4} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {6 \, c^{4} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} + \frac {1}{35} \, {\left (15 \, a^{4} c^{\frac {7}{2}} x^{7} - 42 \, a^{2} c^{\frac {7}{2}} x^{5} + 35 \, c^{\frac {7}{2}} x^{3}\right )} a^{2} - \frac {1}{2} \, {\left (\frac {a^{6} c^{4} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {4 \, a^{4} c^{4} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {6 \, a^{2} c^{4} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {3 \, c^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 120, normalized size = 0.86 \begin {gather*} -\frac {{\left (21 \, a^{7} c^{3} x^{8} + 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} + 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} + 168 \, c^{3} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{168 \, {\left (a^{2} x^{2} - 1\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 {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, 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 [B]
time = 1.09, size = 107, normalized size = 0.77 \begin {gather*} \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {a^7\,c^3\,x^8}{8}+\frac {3\,a^6\,c^3\,x^7}{7}+\frac {a^5\,c^3\,x^6}{6}-a^4\,c^3\,x^5-\frac {5\,a^3\,c^3\,x^4}{4}+\frac {a^2\,c^3\,x^3}{3}+\frac {3\,a\,c^3\,x^2}{2}+c^3\,x\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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