Optimal. Leaf size=201 \[ \frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {847, 794, 201,
223, 209} \begin {gather*} \frac {3 d^9 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rubi steps
\begin {align*} \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x^3 \left (-4 d^2 e-9 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{9 e^2}\\ &=-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {\int x^2 \left (27 d^3 e^2+32 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{72 e^4}\\ &=-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x \left (-64 d^4 e^3-189 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{504 e^6}\\ &=-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {d^5 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^4}\\ &=\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.22, size = 158, normalized size = 0.79 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-1024 d^8-945 d^7 e x-512 d^6 e^2 x^2-630 d^5 e^3 x^3-384 d^4 e^4 x^4+7560 d^3 e^5 x^5+6400 d^2 e^6 x^6-5040 d e^7 x^7-4480 e^8 x^8\right )}{40320 e^5}+\frac {3 d^9 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 215, normalized size = 1.07
method | result | size |
risch | \(-\frac {\left (4480 e^{8} x^{8}+5040 d \,e^{7} x^{7}-6400 d^{2} e^{6} x^{6}-7560 d^{3} e^{5} x^{5}+384 d^{4} x^{4} e^{4}+630 d^{5} e^{3} x^{3}+512 d^{6} e^{2} x^{2}+945 d^{7} e x +1024 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{40320 e^{5}}+\frac {3 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\) | \(141\) |
default | \(e \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{9 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )}{9 e^{2}}\right )+d \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )}{8 e^{2}}\right )\) | \(215\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 163, normalized size = 0.81 \begin {gather*} \frac {3}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} + \frac {3}{128} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7} x e^{\left (-4\right )} + \frac {1}{64} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x e^{\left (-4\right )} - \frac {1}{9} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{4} e^{\left (-1\right )} - \frac {1}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3} e^{\left (-2\right )} - \frac {4}{63} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2} e^{\left (-3\right )} - \frac {1}{16} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x e^{\left (-4\right )} - \frac {8}{315} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.58, size = 128, normalized size = 0.64 \begin {gather*} -\frac {1}{40320} \, {\left (1890 \, d^{9} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (4480 \, x^{8} e^{8} + 5040 \, d x^{7} e^{7} - 6400 \, d^{2} x^{6} e^{6} - 7560 \, d^{3} x^{5} e^{5} + 384 \, d^{4} x^{4} e^{4} + 630 \, d^{5} x^{3} e^{3} + 512 \, d^{6} x^{2} e^{2} + 945 \, d^{7} x e + 1024 \, d^{8}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 25.94, size = 830, normalized size = 4.13 \begin {gather*} d^{3} \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 ) + d^{2} e \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 ) - d e^{2} \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 ) - e^{3} \left (\begin {cases} - \frac {16 d^{8} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac {8 d^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac {2 d^{4} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac {x^{8} \sqrt {d^{2} - e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.41, size = 117, normalized size = 0.58 \begin {gather*} \frac {3}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{40320} \, {\left (1024 \, d^{8} e^{\left (-5\right )} + {\left (945 \, d^{7} e^{\left (-4\right )} + 2 \, {\left (256 \, d^{6} e^{\left (-3\right )} + {\left (315 \, d^{5} e^{\left (-2\right )} + 4 \, {\left (48 \, d^{4} e^{\left (-1\right )} - 5 \, {\left (189 \, d^{3} + 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, x e^{3} + 9 \, d e^{2}\right )} x\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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________