Optimal. Leaf size=80 \[ -\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1821, 821, 272,
65, 214} \begin {gather*} -\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {\int \frac {-4 d^3 e-3 d^2 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{2} \left (3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{4} \left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3}{2} \text {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 )\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 116, normalized size = 1.45 \begin {gather*} -\frac {(d+4 e x) \sqrt {d^2-e^2 x^2}-3 e^2 x^2 \log \left (d \left (-d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )\right )+3 e^2 x^2 \log \left (d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 d x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 139, normalized size = 1.74
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e x +d \right )}{2 d \,x^{2}}-\frac {3 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}\) | \(72\) |
default | \(d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )-\frac {2 e \sqrt {-e^{2} x^{2}+d^{2}}}{d x}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 80, normalized size = 1.00 \begin {gather*} -\frac {3 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d} - \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} e}{d x} - \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.55, size = 61, normalized size = 0.76 \begin {gather*} \frac {3 \, x^{2} e^{2} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - \sqrt {-x^{2} e^{2} + d^{2}} {\left (4 \, x e + d\right )}}{2 \, d x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.84, size = 214, normalized size = 2.68 \begin {gather*} d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {d}{e x} \right )}}{d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i \operatorname {asin}{\left (\frac {d}{e x} \right )}}{d} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 166 vs.
\(2 (67) = 134\).
time = 0.80, size = 166, normalized size = 2.08 \begin {gather*} \frac {x^{2} {\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}}{x} + e^{2}\right )} e^{4}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d} - \frac {3 \, e^{2} \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 )}{2 \, d} - \frac {\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-2\right )}}{x^{2}} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d}{x}}{8 \, d^{2}} \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+e\,x\right )}^2}{x^3\,\sqrt {d^2-e^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________