Optimal. Leaf size=116 \[ -\frac {5 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 d (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {5 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 116, 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 = {671, 641, 217, 203} \[ -\frac {5 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 d (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {5 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 217
Rule 641
Rule 671
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx &=-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{3} (5 d) \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {5 d (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{2} \left (5 d^2\right ) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {5 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 d (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{2} \left (5 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {5 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 d (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{2} \left (5 d^3\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 {5 d^2 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {5 d (d+e x) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {5 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 70, normalized size = 0.60 \[ \frac {15 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\sqrt {d^2-e^2 x^2} \left (22 d^2+9 d e x+2 e^2 x^2\right )}{6 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.02, size = 72, normalized size = 0.62 \[ -\frac {30 \, d^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (2 \, e^{2} x^{2} + 9 \, d e x + 22 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 52, normalized size = 0.45 \[ \frac {5}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (22 \, d^{2} e^{\left (-1\right )} + {\left (2 \, x e + 9 \, d\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 94, normalized size = 0.81 \[ \frac {5 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e \,x^{2}}{3}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d x}{2}-\frac {11 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.94, size = 76, normalized size = 0.66 \[ -\frac {1}{3} \, \sqrt {-e^{2} x^{2} + d^{2}} e x^{2} + \frac {5 \, d^{3} \arcsin \left (\frac {e x}{d}\right )}{2 \, e} - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d x - \frac {11 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^3}{\sqrt {d^2-e^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.58, size = 337, normalized size = 2.91 \[ d^{3} \left (\begin {cases} \frac {\sqrt {\frac {d^{2}}{e^{2}}} \operatorname {asin}{\left (x \sqrt {\frac {e^{2}}{d^{2}}} \right )}}{\sqrt {d^{2}}} & \text {for}\: d^{2} > 0 \wedge e^{2} > 0 \\\frac {\sqrt {- \frac {d^{2}}{e^{2}}} \operatorname {asinh}{\left (x \sqrt {- \frac {e^{2}}{d^{2}}} \right )}}{\sqrt {d^{2}}} & \text {for}\: d^{2} > 0 \wedge e^{2} < 0 \\\frac {\sqrt {\frac {d^{2}}{e^{2}}} \operatorname {acosh}{\left (x \sqrt {\frac {e^{2}}{d^{2}}} \right )}}{\sqrt {- d^{2}}} & \text {for}\: d^{2} < 0 \wedge e^{2} < 0 \end {cases}\right ) + 3 d^{2} e \left (\begin {cases} \frac {x^{2}}{2 \sqrt {d^{2}}} & \text {for}\: e^{2} = 0 \\- \frac {\sqrt {d^{2} - e^{2} x^{2}}}{e^{2}} & \text {otherwise} \end {cases}\right ) + 3 d e^{2} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e^{3}} - \frac {i d x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e^{3}} - \frac {d x}{2 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {x^{3}}{2 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} - \frac {2 d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \sqrt {d^{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________