Optimal. Leaf size=169 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {852, 1807, 835, 807, 266, 47, 63, 208} \[ -\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 852
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^7} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}-\frac {\int \frac {\left (12 d^3 e-9 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^6} \, dx}{6 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}+\frac {\int \frac {\left (45 d^4 e^2-24 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{30 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}-\frac {\int \frac {\left (96 d^5 e^3-45 d^4 e^4 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{120 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {\left (3 e^4\right ) \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{8 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {\left (3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d^2}\\ &=-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}-\frac {\left (3 e^6\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{32 d^2}\\ &=-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {\left (3 e^4\right ) \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 )}{16 d^2}\\ &=-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 117, normalized size = 0.69 \[ -\frac {-45 e^6 x^6 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (40 d^5-96 d^4 e x+50 d^3 e^2 x^2+32 d^2 e^3 x^3-45 d e^4 x^4+64 e^5 x^5\right )+45 e^6 x^6 \log (x)}{240 d^3 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 108, normalized size = 0.64 \[ -\frac {45 \, e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (64 \, e^{5} x^{5} - 45 \, d e^{4} x^{4} + 32 \, d^{2} e^{3} x^{3} + 50 \, d^{3} e^{2} x^{2} - 96 \, d^{4} e x + 40 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 566, normalized size = 3.35 \[ \frac {3 e^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}\, d^{2}}-\frac {13 e^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{4 \sqrt {e^{2}}\, d^{3}}+\frac {13 e^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 \sqrt {e^{2}}\, d^{3}}-\frac {13 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{7} x}{4 d^{5}}+\frac {13 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7} x}{4 d^{5}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6}}{16 d^{4}}-\frac {13 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{7} x}{6 d^{7}}+\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7} x}{6 d^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6}}{16 d^{6}}+\frac {26 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7} x}{15 d^{9}}-\frac {26 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{6}}{15 d^{8}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}{80 d^{8}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{4}}{3 \left (x +\frac {d}{e}\right )^{2} d^{8}}+\frac {26 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{15 d^{9} x}-\frac {23 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{16 d^{8} x^{2}}+\frac {16 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{15 d^{7} x^{3}}-\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{24 d^{6} x^{4}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{5 d^{5} x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{4} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 180, normalized size = 1.07 \[ \frac {3 \, e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{3}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}}{16 \, d^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{16 \, d^{4} x^{2}} + \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{15 \, d^{3} x^{3}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{8 \, d^{2} x^{4}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{5 \, d x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^7\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 19.72, size = 808, normalized size = 4.78 \[ d^{2} \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________