Optimal. Leaf size=190 \[ \frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, 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 = {1809, 815, 844, 217, 203, 266, 63, 208} \[ \frac {1}{128} d^6 (128 d+125 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{192} d^4 (64 d+125 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{240} d^2 (48 d+125 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {3}{7} d \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{8} e x \left (d^2-e^2 x^2\right )^{7/2}+\frac {125}{128} d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 815
Rule 844
Rule 1809
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 \operatorname {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 ) \operatorname {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 \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.36, size = 168, normalized size = 0.88 \[ d^8 \left (-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+\frac {125 d^7 \sqrt {d^2-e^2 x^2} \sin ^{-1}\left (\frac {e x}{d}\right )}{128 \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {\sqrt {d^2-e^2 x^2} \left (14848 d^7+27195 d^6 e x+7424 d^5 e^2 x^2-17710 d^4 e^3 x^3-14592 d^3 e^4 x^4+1960 d^2 e^5 x^5+5760 d e^6 x^6+1680 e^7 x^7\right )}{13440} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 151, normalized size = 0.79 \[ -\frac {125}{64} \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \frac {1}{13440} \, {\left (1680 \, e^{7} x^{7} + 5760 \, d e^{6} x^{6} + 1960 \, d^{2} e^{5} x^{5} - 14592 \, d^{3} e^{4} x^{4} - 17710 \, d^{4} e^{3} x^{3} + 7424 \, d^{5} e^{2} x^{2} + 27195 \, d^{6} e x + 14848 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 143, normalized size = 0.75 \[ \frac {125}{128} \, d^{8} \arcsin \left (\frac {x e}{d}\right ) \mathrm {sgn}\relax (d) - d^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {1}{13440} \, {\left (14848 \, d^{7} + {\left (27195 \, d^{6} e + 2 \, {\left (3712 \, d^{5} e^{2} - {\left (8855 \, d^{4} e^{3} + 4 \, {\left (1824 \, d^{3} e^{4} - 5 \, {\left (49 \, d^{2} e^{5} + 6 \, {\left (7 \, x e^{7} + 24 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 231, normalized size = 1.22 \[ -\frac {d^{9} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+\frac {125 d^{8} e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}+\frac {125 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e x}{128}+\sqrt {-e^{2} x^{2}+d^{2}}\, d^{7}+\frac {125 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{4} e x}{192}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5}}{3}+\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2} e x}{48}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3}}{5}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e x}{8}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.00, size = 204, normalized size = 1.07 \[ \frac {125}{128} \, d^{8} \arcsin \left (\frac {e x}{d}\right ) - d^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {125}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{7} + \frac {125}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e x + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} + \frac {25}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e x + \frac {1}{5} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} - \frac {1}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x - \frac {3}{7} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 47.72, size = 1263, normalized size = 6.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________