Optimal. Leaf size=113 \[ -\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac {5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac {15 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {15 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {663, 665, 217, 203} \[ -\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac {5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac {15 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {15 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 217
Rule 663
Rule 665
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-5 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx\\ &=-\frac {5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac {1}{2} (15 d) \int \frac {\sqrt {d^2-e^2 x^2}}{d+e x} \, dx\\ &=-\frac {15 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac {1}{2} \left (15 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {15 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac {1}{2} \left (15 d^2\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 {15 d \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 \left (d^2-e^2 x^2\right )^{3/2}}{2 e (d+e x)}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^3}-\frac {15 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 75, normalized size = 0.66 \[ \sqrt {d^2-e^2 x^2} \left (-\frac {8 d^2}{e (d+e x)}-\frac {4 d}{e}+\frac {x}{2}\right )-\frac {15 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 99, normalized size = 0.88 \[ -\frac {24 \, d^{2} e x + 24 \, d^{3} - 30 \, {\left (d^{2} e x + d^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (e^{2} x^{2} - 7 \, d e x - 24 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, {\left (e^{2} x + d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 284, normalized size = 2.51 \[ -\frac {15 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {15 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, x}{2}-\frac {5 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} x}{d^{2}}-\frac {4 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}}}{d^{3} e}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{4} d \,e^{5}}-\frac {3 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{3} d^{2} e^{4}}-\frac {4 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{2} d^{3} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 134, normalized size = 1.19 \[ -\frac {15 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{2 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{2} x + d e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________