Optimal. Leaf size=136 \[ \frac {256 c^3 \sqrt {1-a^2 x^2}}{35 a \sqrt {c-a c x}}+\frac {64 c^2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{35 a}+\frac {24 c (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{35 a}+\frac {2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{7 a} \]
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Rubi [A]
time = 0.08, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6262, 671, 663}
\begin {gather*} \frac {256 c^3 \sqrt {1-a^2 x^2}}{35 a \sqrt {c-a c x}}+\frac {64 c^2 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{35 a}+\frac {24 c \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{35 a}+\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{7 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rule 6262
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac {\int \frac {(c-a c x)^{7/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{7 a}+\frac {12}{7} \int \frac {(c-a c x)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {24 c (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{35 a}+\frac {2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{7 a}+\frac {1}{35} (96 c) \int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {64 c^2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{35 a}+\frac {24 c (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{35 a}+\frac {2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{7 a}+\frac {1}{35} \left (128 c^2\right ) \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {256 c^3 \sqrt {1-a^2 x^2}}{35 a \sqrt {c-a c x}}+\frac {64 c^2 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{35 a}+\frac {24 c (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{35 a}+\frac {2 (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{7 a}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 57, normalized size = 0.42 \begin {gather*} -\frac {2 c^3 \sqrt {1-a^2 x^2} \left (-177+71 a x-27 a^2 x^2+5 a^3 x^3\right )}{35 a \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.24, size = 60, normalized size = 0.44
method | result | size |
gosper | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \left (-c x a +c \right )^{\frac {5}{2}} \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right )}{35 \left (a x -1\right )^{3} a}\) | \(56\) |
default | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, c^{2} \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right )}{35 \left (a x -1\right ) a}\) | \(60\) |
risch | \(\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{3} \left (5 a^{3} x^{3}-27 a^{2} x^{2}+71 a x -177\right ) \left (a x +1\right )}{35 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {\left (a x +1\right ) c}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 60, normalized size = 0.44 \begin {gather*} -\frac {2 \, {\left (5 \, a^{3} c^{\frac {5}{2}} x^{3} - 27 \, a^{2} c^{\frac {5}{2}} x^{2} + 71 \, a c^{\frac {5}{2}} x - 177 \, c^{\frac {5}{2}}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{35 \, {\left (a^{2} x - a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 69, normalized size = 0.51 \begin {gather*} \frac {2 \, {\left (5 \, a^{3} c^{2} x^{3} - 27 \, a^{2} c^{2} x^{2} + 71 \, a c^{2} x - 177 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{35 \, {\left (a^{2} x - a\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 )\right )^{\frac {5}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \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 [B]
time = 0.99, size = 80, normalized size = 0.59 \begin {gather*} \frac {2\,c^2\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}\,\left (5\,a^2\,x^2-22\,a\,x+49\right )}{35\,a}-\frac {256\,c^2\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{35\,a\,\left (a\,x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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