Optimal. Leaf size=190 \[ \frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1823, 829,
858, 223, 209, 272, 65, 214} \begin {gather*} \frac {125}{128} d^8 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-d^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 829
Rule 858
Rule 1823
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx &=-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-8 d^3 e^2-25 d^2 e^3 x-24 d e^4 x^2\right )}{x} \, dx}{8 e^2}\\ &=-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int \frac {\left (56 d^3 e^4+175 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx}{56 e^4}\\ &=\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int \frac {\left (-336 d^5 e^6-875 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{336 e^6}\\ &=\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int \frac {\left (1344 d^7 e^8+2625 d^6 e^9 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{1344 e^8}\\ &=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int \frac {-2688 d^9 e^{10}-2625 d^8 e^{11} x}{x \sqrt {d^2-e^2 x^2}} \, dx}{2688 e^{10}}\\ &=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+d^9 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{128} \left (125 d^8 e\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {1}{2} d^9 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{128} \left (125 d^8 e\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^9 \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2}\\ &=\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 184, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (14848 d^7+27195 d^6 e x+7424 d^5 e^2 x^2-17710 d^4 e^3 x^3-14592 d^3 e^4 x^4+1960 d^2 e^5 x^5+5760 d e^6 x^6+1680 e^7 x^7\right )}{13440}+2 d^8 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {125 d^8 e \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {-e^2}} \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. \(352\) vs.
\(2(164)=328\).
time = 0.08, size = 353, normalized size = 1.86
method | result | size |
default | \(e^{3} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )-\frac {3 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7}+3 d^{2} e \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )+d^{3} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )\) | \(353\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 200, normalized size = 1.05 \begin {gather*} \frac {125}{128} \, d^{8} \arcsin \left (\frac {x e}{d}\right ) - d^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {125}{128} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6} x e + \sqrt {-x^{2} e^{2} + d^{2}} d^{7} + \frac {125}{192} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e + \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} + \frac {25}{48} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x e + \frac {1}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} - \frac {1}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x e - \frac {3}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 142, normalized size = 0.75 \begin {gather*} -\frac {125}{64} \, d^{8} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + d^{8} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + \frac {1}{13440} \, {\left (1680 \, x^{7} e^{7} + 5760 \, d x^{6} e^{6} + 1960 \, d^{2} x^{5} e^{5} - 14592 \, d^{3} x^{4} e^{4} - 17710 \, d^{4} x^{3} e^{3} + 7424 \, d^{5} x^{2} e^{2} + 27195 \, d^{6} x e + 14848 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 39.16, size = 1263, normalized size = 6.65 \begin {gather*} d^{7} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + 3 d^{6} e \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) + d^{5} e^{2} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) - 5 d^{4} e^{3} \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - 5 d^{3} e^{4} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) + d^{2} e^{5} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + 3 d e^{6} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) + e^{7} \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 143, normalized size = 0.75 \begin {gather*} \frac {125}{128} \, d^{8} \arcsin \left (\frac {x e}{d}\right ) \mathrm {sgn}\left (d\right ) - d^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {1}{13440} \, {\left (14848 \, d^{7} + {\left (27195 \, d^{6} e + 2 \, {\left (3712 \, d^{5} e^{2} - {\left (8855 \, d^{4} e^{3} + 4 \, {\left (1824 \, d^{3} e^{4} - 5 \, {\left (49 \, d^{2} e^{5} + 6 \, {\left (7 \, x e^{7} + 24 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \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 {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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