Optimal. Leaf size=161 \[ -\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}+\frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4}-\frac {(105 a x+88) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3} \]
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Rubi [A] time = 0.34, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1809, 833, 780, 195, 217, 203} \[ \frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}+\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {(105 a x+88) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 780
Rule 833
Rule 1809
Rule 6151
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x^3 (1+a x)^2 \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {\int x^3 \left (-11 a^2 c-14 a^3 c x\right ) \sqrt {c-a^2 c x^2} \, dx}{7 a^2}\\ &=-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}+\frac {\int x^2 \left (42 a^3 c^2+66 a^4 c^2 x\right ) \sqrt {c-a^2 c x^2} \, dx}{42 a^4 c}\\ &=-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {\int x \left (-132 a^4 c^3-210 a^5 c^3 x\right ) \sqrt {c-a^2 c x^2} \, dx}{210 a^6 c^2}\\ &=-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c \int \sqrt {c-a^2 c x^2} \, dx}{4 a^3}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c^2 \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{8 a^3}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^3}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{35 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}-\frac {(88+105 a x) \left (c-a^2 c x^2\right )^{3/2}}{420 a^4}+\frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 113, normalized size = 0.70 \[ \frac {c \left (120 a^6 x^6+280 a^5 x^5+144 a^4 x^4-70 a^3 x^3-88 a^2 x^2-105 a x-176\right ) \sqrt {c-a^2 c x^2}-105 c^{3/2} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )}{840 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 234, normalized size = 1.45 \[ \left [\frac {105 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{1680 \, a^{4}}, -\frac {105 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{840 \, a^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 117, normalized size = 0.73 \[ \frac {1}{840} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (3 \, a^{2} c x + 7 \, a c\right )} x + 18 \, c\right )} x - \frac {35 \, c}{a}\right )} x - \frac {44 \, c}{a^{2}}\right )} x - \frac {105 \, c}{a^{3}}\right )} x - \frac {176 \, c}{a^{4}}\right )} - \frac {c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{3} \sqrt {-c} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 268, normalized size = 1.66 \[ \frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{7 a^{2} c}+\frac {16 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{35 c \,a^{4}}+\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 a^{3} c}-\frac {7 x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{12 a^{3}}-\frac {7 c x \sqrt {-a^{2} c \,x^{2}+c}}{8 a^{3}}-\frac {7 c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 a^{3} \sqrt {a^{2} c}}-\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{4}}+\frac {c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{a^{3}}+\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a^{3} \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 213, normalized size = 1.32 \[ \frac {1}{840} \, a {\left (\frac {120 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}}{a^{3} c} - \frac {490 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{4}} + \frac {280 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{4} c} + \frac {840 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x}{a^{4}} - \frac {735 \, \sqrt {-a^{2} c x^{2} + c} c x}{a^{4}} - \frac {735 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{a^{5}} - \frac {560 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{5}} + \frac {384 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{5} c} - \frac {1680 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a^{5}} + \frac {840 \, c^{3} \arcsin \left (a x - 2\right )}{a^{8} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {x^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.31, size = 420, normalized size = 2.61 \[ a^{2} c \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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