Optimal. Leaf size=101 \[ -\frac {4 \sqrt {c-a c x}}{a^4}+\frac {14 (c-a c x)^{3/2}}{3 a^4 c}-\frac {18 (c-a c x)^{5/2}}{5 a^4 c^2}+\frac {10 (c-a c x)^{7/2}}{7 a^4 c^3}-\frac {2 (c-a c x)^{9/2}}{9 a^4 c^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6265, 21, 78}
\begin {gather*} -\frac {2 (c-a c x)^{9/2}}{9 a^4 c^4}+\frac {10 (c-a c x)^{7/2}}{7 a^4 c^3}-\frac {18 (c-a c x)^{5/2}}{5 a^4 c^2}+\frac {14 (c-a c x)^{3/2}}{3 a^4 c}-\frac {4 \sqrt {c-a c x}}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 78
Rule 6265
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx &=\int \frac {x^3 (1+a x) \sqrt {c-a c x}}{1-a x} \, dx\\ &=c \int \frac {x^3 (1+a x)}{\sqrt {c-a c x}} \, dx\\ &=c \int \left (\frac {2}{a^3 \sqrt {c-a c x}}-\frac {7 \sqrt {c-a c x}}{a^3 c}+\frac {9 (c-a c x)^{3/2}}{a^3 c^2}-\frac {5 (c-a c x)^{5/2}}{a^3 c^3}+\frac {(c-a c x)^{7/2}}{a^3 c^4}\right ) \, dx\\ &=-\frac {4 \sqrt {c-a c x}}{a^4}+\frac {14 (c-a c x)^{3/2}}{3 a^4 c}-\frac {18 (c-a c x)^{5/2}}{5 a^4 c^2}+\frac {10 (c-a c x)^{7/2}}{7 a^4 c^3}-\frac {2 (c-a c x)^{9/2}}{9 a^4 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 48, normalized size = 0.48 \begin {gather*} -\frac {2 \sqrt {c-a c x} \left (272+136 a x+102 a^2 x^2+85 a^3 x^3+35 a^4 x^4\right )}{315 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.72, size = 75, normalized size = 0.74
method | result | size |
gosper | \(-\frac {2 \sqrt {-c x a +c}\, \left (35 a^{4} x^{4}+85 a^{3} x^{3}+102 a^{2} x^{2}+136 a x +272\right )}{315 a^{4}}\) | \(45\) |
trager | \(-\frac {2 \sqrt {-c x a +c}\, \left (35 a^{4} x^{4}+85 a^{3} x^{3}+102 a^{2} x^{2}+136 a x +272\right )}{315 a^{4}}\) | \(45\) |
risch | \(\frac {2 c \left (35 a^{4} x^{4}+85 a^{3} x^{3}+102 a^{2} x^{2}+136 a x +272\right ) \left (a x -1\right )}{315 a^{4} \sqrt {-c \left (a x -1\right )}}\) | \(52\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-c x a +c \right )^{\frac {9}{2}}}{9}-\frac {5 c \left (-c x a +c \right )^{\frac {7}{2}}}{7}+\frac {9 c^{2} \left (-c x a +c \right )^{\frac {5}{2}}}{5}-\frac {7 c^{3} \left (-c x a +c \right )^{\frac {3}{2}}}{3}+2 c^{4} \sqrt {-c x a +c}\right )}{c^{4} a^{4}}\) | \(75\) |
default | \(-\frac {2 \left (\frac {\left (-c x a +c \right )^{\frac {9}{2}}}{9}-\frac {5 c \left (-c x a +c \right )^{\frac {7}{2}}}{7}+\frac {9 c^{2} \left (-c x a +c \right )^{\frac {5}{2}}}{5}-\frac {7 c^{3} \left (-c x a +c \right )^{\frac {3}{2}}}{3}+2 c^{4} \sqrt {-c x a +c}\right )}{c^{4} a^{4}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 74, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (35 \, {\left (-a c x + c\right )}^{\frac {9}{2}} - 225 \, {\left (-a c x + c\right )}^{\frac {7}{2}} c + 567 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c^{2} - 735 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{3} + 630 \, \sqrt {-a c x + c} c^{4}\right )}}{315 \, a^{4} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 44, normalized size = 0.44 \begin {gather*} -\frac {2 \, {\left (35 \, a^{4} x^{4} + 85 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 136 \, a x + 272\right )} \sqrt {-a c x + c}}{315 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.34, size = 83, normalized size = 0.82 \begin {gather*} \frac {2 \left (- 2 c^{4} \sqrt {- a c x + c} + \frac {7 c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {9 c^{2} \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {5 c \left (- a c x + c\right )^{\frac {7}{2}}}{7} - \frac {\left (- a c x + c\right )^{\frac {9}{2}}}{9}\right )}{a^{4} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (83) = 166\).
time = 0.40, size = 189, normalized size = 1.87 \begin {gather*} -\frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 35 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-a c x + c} c^{3}\right )}}{a^{3} c^{3}} + \frac {35 \, {\left (a c x - c\right )}^{4} \sqrt {-a c x + c} + 180 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} c + 378 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c^{2} - 420 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {-a c x + c} c^{4}}{a^{3} c^{4}}\right )}}{315 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 83, normalized size = 0.82 \begin {gather*} \frac {14\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^4\,c}-\frac {4\,\sqrt {c-a\,c\,x}}{a^4}-\frac {18\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^4\,c^2}+\frac {10\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^4\,c^3}-\frac {2\,{\left (c-a\,c\,x\right )}^{9/2}}{9\,a^4\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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