Optimal. Leaf size=187 \[ \frac {3 c^2 x \sqrt {c-a^2 c x^2}}{64 a^3}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {3 c^{5/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{64 a^4} \]
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Rubi [A]
time = 0.25, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6286, 1823,
847, 794, 201, 223, 209} \begin {gather*} -\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}+\frac {3 c^{5/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{64 a^4}-\frac {(315 a x+208) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {3 c^2 x \sqrt {c-a^2 c x^2}}{64 a^3}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rule 6286
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x^3 (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {\int x^3 \left (-13 a^2 c-18 a^3 c x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{9 a^2}\\ &=-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}+\frac {\int x^2 \left (54 a^3 c^2+104 a^4 c^2 x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{72 a^4 c}\\ &=-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {\int x \left (-208 a^4 c^3-378 a^5 c^3 x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{504 a^6 c^2}\\ &=-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {c \int \left (c-a^2 c x^2\right )^{3/2} \, dx}{8 a^3}\\ &=\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {\left (3 c^2\right ) \int \sqrt {c-a^2 c x^2} \, dx}{32 a^3}\\ &=\frac {3 c^2 x \sqrt {c-a^2 c x^2}}{64 a^3}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{64 a^3}\\ &=\frac {3 c^2 x \sqrt {c-a^2 c x^2}}{64 a^3}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{64 a^3}\\ &=\frac {3 c^2 x \sqrt {c-a^2 c x^2}}{64 a^3}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {3 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{64 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 131, normalized size = 0.70 \begin {gather*} -\frac {c^2 \left (\sqrt {c-a^2 c x^2} \left (1664+945 a x+832 a^2 x^2+630 a^3 x^3-4416 a^4 x^4-7560 a^5 x^5-320 a^6 x^6+5040 a^7 x^7+2240 a^8 x^8\right )+945 \sqrt {c} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )\right )}{20160 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(469\) vs.
\(2(155)=310\).
time = 0.06, size = 470, normalized size = 2.51
method | result | size |
risch | \(\frac {\left (2240 a^{8} x^{8}+5040 a^{7} x^{7}-320 x^{6} a^{6}-7560 x^{5} a^{5}-4416 a^{4} x^{4}+630 a^{3} x^{3}+832 a^{2} x^{2}+945 a x +1664\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{20160 a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{3}}{64 a^{3} \sqrt {c \,a^{2}}}\) | \(133\) |
default | \(\frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{9 c \,a^{2}}+\frac {20 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{63 c \,a^{4}}-\frac {2 \left (-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8 c \,a^{2}}+\frac {\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )}{6}}{8 a^{2}}\right )}{a}-\frac {2 \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )}{6}\right )}{a^{3}}-\frac {2 \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{5}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )}{a^{4}}\) | \(470\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.66, size = 239, normalized size = 1.28 \begin {gather*} \frac {1}{20160} \, {\left (\frac {2240 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x^{2}}{a^{3} c} - \frac {7560 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{4}} + \frac {5040 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x}{a^{4} c} + \frac {630 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x}{a^{4}} + \frac {15120 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x}{a^{4}} - \frac {14175 \, \sqrt {-a^{2} c x^{2} + c} c^{2} x}{a^{4}} - \frac {14175 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{a^{5}} - \frac {8064 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{5}} + \frac {6400 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{a^{5} c} - \frac {30240 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{a^{5}} + \frac {15120 \, c^{4} \arcsin \left (a x - 2\right )}{a^{8} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 307, normalized size = 1.64 \begin {gather*} \left [\frac {945 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{40320 \, a^{4}}, -\frac {945 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{20160 \, a^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 34.68, size = 763, normalized size = 4.08 \begin {gather*} - a^{4} c^{2} \left (\begin {cases} \frac {x^{8} \sqrt {- a^{2} c x^{2} + c}}{9} - \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{63 a^{2}} - \frac {2 x^{4} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 x^{2} \sqrt {- a^{2} c x^{2} + c}}{315 a^{6}} - \frac {16 \sqrt {- a^{2} c x^{2} + c}}{315 a^{8}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{8}}{8} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{9}}{8 \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} x^{7}}{48 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{5}}{192 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{3}}{384 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i \sqrt {c} x}{128 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{128 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{9}}{8 \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} x^{7}}{48 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{5}}{192 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{3}}{384 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 \sqrt {c} x}{128 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} \operatorname {asin}{\left (a x \right )}}{128 a^{7}} & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \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^{2} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 155, normalized size = 0.83 \begin {gather*} \frac {1}{20160} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (552 \, c^{2} + 5 \, {\left (189 \, a c^{2} + 2 \, {\left (4 \, a^{2} c^{2} - 7 \, {\left (4 \, a^{4} c^{2} x + 9 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {315 \, c^{2}}{a}\right )} x - \frac {416 \, c^{2}}{a^{2}}\right )} x - \frac {945 \, c^{2}}{a^{3}}\right )} x - \frac {1664 \, c^{2}}{a^{4}}\right )} - \frac {3 \, c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{64 \, a^{3} \sqrt {-c} {\left | a \right |}} \end {gather*}
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 \begin {gather*} \int -\frac {x^3\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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