Optimal. Leaf size=141 \[ \frac {256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt {c-a c x}}+\frac {8 c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{21 a}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a} \]
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Rubi [A] time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 657, 649} \[ \frac {256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt {c-a c x}}+\frac {8 c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{21 a}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rule 6127
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=c \int (c-a c x)^{5/2} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac {1}{3} \left (4 c^2\right ) \int (c-a c x)^{3/2} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {8 c^3 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac {2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac {1}{21} \left (32 c^3\right ) \int \sqrt {c-a c x} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt {c-a c x}}+\frac {8 c^3 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac {2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}+\frac {1}{105} \left (128 c^4\right ) \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {256 c^5 \left (1-a^2 x^2\right )^{3/2}}{315 a (c-a c x)^{3/2}}+\frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a \sqrt {c-a c x}}+\frac {8 c^3 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{21 a}+\frac {2 c^2 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{9 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.44 \[ -\frac {2 c^3 (a x+1)^{3/2} \left (35 a^3 x^3-165 a^2 x^2+321 a x-319\right ) \sqrt {c-a c x}}{315 a \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 80, normalized size = 0.57 \[ \frac {2 \, {\left (35 \, a^{4} c^{3} x^{4} - 130 \, a^{3} c^{3} x^{3} + 156 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{315 \, {\left (a^{2} x - a\right )}} \]
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 = 0.03, size = 63, normalized size = 0.45 \[ \frac {2 \left (a x +1\right )^{2} \left (35 x^{3} a^{3}-165 a^{2} x^{2}+321 a x -319\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{3} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 128, normalized size = 0.91 \[ -\frac {2 \, {\left (5 \, a^{5} c^{\frac {7}{2}} x^{5} - 20 \, a^{4} c^{\frac {7}{2}} x^{4} + 32 \, a^{3} c^{\frac {7}{2}} x^{3} - 34 \, a^{2} c^{\frac {7}{2}} x^{2} + 91 \, a c^{\frac {7}{2}} x + 182 \, c^{\frac {7}{2}}\right )}}{45 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (5 \, a^{4} c^{\frac {7}{2}} x^{4} - 22 \, a^{3} c^{\frac {7}{2}} x^{3} + 44 \, a^{2} c^{\frac {7}{2}} x^{2} - 106 \, a c^{\frac {7}{2}} x - 177 \, c^{\frac {7}{2}}\right )}}{35 \, \sqrt {a x + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 79, normalized size = 0.56 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {634\,c^3\,x}{315}+\frac {638\,c^3}{315\,a}-\frac {316\,a\,c^3\,x^2}{315}-\frac {52\,a^2\,c^3\,x^3}{315}+\frac {38\,a^3\,c^3\,x^4}{63}-\frac {2\,a^4\,c^3\,x^5}{9}\right )}{\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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