Optimal. Leaf size=111 \[ \frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^2}-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}+\frac {c x \sqrt {c-a^2 c x^2}}{4 a}-\frac {(15 a x+14) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2} \]
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Rubi [A] time = 0.19, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6151, 1809, 780, 195, 217, 203} \[ \frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^2}-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}+\frac {c x \sqrt {c-a^2 c x^2}}{4 a}-\frac {(15 a x+14) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 780
Rule 1809
Rule 6151
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x (1+a x)^2 \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac {\int x \left (-7 a^2 c-10 a^3 c x\right ) \sqrt {c-a^2 c x^2} \, dx}{5 a^2}\\ &=-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac {(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac {c \int \sqrt {c-a^2 c x^2} \, dx}{2 a}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{4 a}-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac {(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac {c^2 \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{4 a}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{4 a}-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac {(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\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 )}{4 a}\\ &=\frac {c x \sqrt {c-a^2 c x^2}}{4 a}-\frac {1}{5} x^2 \left (c-a^2 c x^2\right )^{3/2}-\frac {(14+15 a x) \left (c-a^2 c x^2\right )^{3/2}}{30 a^2}+\frac {c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 97, normalized size = 0.87 \[ \frac {c \left (12 a^4 x^4+30 a^3 x^3+16 a^2 x^2-15 a x-28\right ) \sqrt {c-a^2 c x^2}-15 c^{3/2} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )}{60 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 198, normalized size = 1.78 \[ \left [\frac {15 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (12 \, a^{4} c x^{4} + 30 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} - 15 \, a c x - 28 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{120 \, a^{2}}, -\frac {15 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (12 \, a^{4} c x^{4} + 30 \, a^{3} c x^{3} + 16 \, a^{2} c x^{2} - 15 \, a c x - 28 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{60 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 98, normalized size = 0.88 \[ \frac {1}{60} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, {\left (2 \, a^{2} c x + 5 \, a c\right )} x + 8 \, c\right )} x - \frac {15 \, c}{a}\right )} x - \frac {28 \, c}{a^{2}}\right )} - \frac {c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{4 \, a \sqrt {-c} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 222, normalized size = 2.00 \[ \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 a^{2} c}-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 a}-\frac {3 c x \sqrt {-a^{2} c \,x^{2}+c}}{4 a}-\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{4 a \sqrt {a^{2} c}}-\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a^{2}}+\frac {c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{a}+\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{a \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 167, normalized size = 1.50 \[ -\frac {1}{60} \, a {\left (\frac {30 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {60 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x}{a^{2}} + \frac {45 \, \sqrt {-a^{2} c x^{2} + c} c x}{a^{2}} + \frac {45 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{a^{3}} + \frac {40 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{3}} - \frac {12 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{3} c} + \frac {120 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a^{3}} - \frac {60 \, c^{3} \arcsin \left (a x - 2\right )}{a^{6} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \]
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 \[ \int -\frac {x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.12, size = 306, normalized size = 2.76 \[ a^{2} c \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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