Optimal. Leaf size=132 \[ \frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {7 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}+\frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {655, 671, 641, 195, 217, 203} \begin {gather*} \frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {7 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 203
Rule 217
Rule 641
Rule 655
Rule 671
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx &=\int (d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{6} (7 d) \int (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{6} \left (7 d^2\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{8} \left (7 d^4\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{16} \left (7 d^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{16} \left (7 d^6\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 {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {7 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 102, normalized size = 0.77 \begin {gather*} \frac {105 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\sqrt {d^2-e^2 x^2} \left (96 d^5+135 d^4 e x-192 d^3 e^2 x^2+10 d^2 e^3 x^3+96 d e^4 x^4-40 e^5 x^5\right )}{240 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.40, size = 125, normalized size = 0.95 \begin {gather*} \frac {7 d^6 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{16 e^2}+\frac {\sqrt {d^2-e^2 x^2} \left (96 d^5+135 d^4 e x-192 d^3 e^2 x^2+10 d^2 e^3 x^3+96 d e^4 x^4-40 e^5 x^5\right )}{240 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 105, normalized size = 0.80 \begin {gather*} -\frac {210 \, d^{6} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (40 \, e^{5} x^{5} - 96 \, d e^{4} x^{4} - 10 \, d^{2} e^{3} x^{3} + 192 \, d^{3} e^{2} x^{2} - 135 \, d^{4} e x - 96 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 228, normalized size = 1.73 \begin {gather*} \frac {7 d^{6} \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 )}{16 \sqrt {e^{2}}}+\frac {7 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4} x}{16}+\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{2} x}{24}+\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} x}{30}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{5 d e}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {9}{2}}}{5 \left (x +\frac {d}{e}\right )^{2} d \,e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 3.10, size = 139, normalized size = 1.05 \begin {gather*} -\frac {7 i \, d^{6} \arcsin \left (\frac {e x}{d} + 2\right )}{16 \, e} + \frac {7}{16} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x + \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{8 \, e} + \frac {7}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{6 \, {\left (e^{2} x + d e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{30 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 17.75, size = 495, normalized size = 3.75 \begin {gather*} d^{4} \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 ) - 2 d^{3} e \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 ) + 2 d e^{3} \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 ) - e^{4} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________