Optimal. Leaf size=99 \[ -\frac {c \sqrt {d-e x} \sqrt {d+e x}}{e^2}-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{2 d^2 x^2}-\frac {\left (2 b d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{2 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 155, normalized size of antiderivative = 1.57, number of steps
used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {534, 1265, 911,
1171, 396, 214} \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 396
Rule 534
Rule 911
Rule 1171
Rule 1265
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^3 \sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {a+b x+c x^2}{x^2 \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {\frac {c d^4+b d^2 e^2+a e^4}{e^4}-\frac {\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac {c x^4}{e^4}}{\left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^2} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {-a-\frac {2 \left (c d^4+b d^2 e^2\right )}{e^4}+\frac {2 c d^2 x^2}{e^4}}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {c \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (e^2 \left (\frac {2 c d^4}{e^6}+\frac {-a-\frac {2 \left (c d^4+b d^2 e^2\right )}{e^4}}{e^2}\right ) \sqrt {d^2-e^2 x^2}\right ) \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 )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {c \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (2 b d^2+a e^2\right ) \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 85, normalized size = 0.86 \begin {gather*} -\frac {\frac {\sqrt {d-e x} \sqrt {d+e x} \left (a d e^2+2 c d^3 x^2\right )}{e^2 x^2}+2 \left (2 b d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.16, size = 163, normalized size = 1.65
method | result | size |
default | \(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (2 \,\mathrm {csgn}\left (d \right ) c \,d^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+\ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (d \right )+d \right )}{x}\right ) a \,e^{4} x^{2}+2 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (d \right )+d \right )}{x}\right ) b \,d^{2} e^{2} x^{2}+\mathrm {csgn}\left (d \right ) a d \,e^{2} \sqrt {-e^{2} x^{2}+d^{2}}\right ) \mathrm {csgn}\left (d \right )}{2 d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}}\) | \(163\) |
risch | \(-\frac {a \sqrt {-e x +d}\, \sqrt {e x +d}}{2 d^{2} x^{2}}+\frac {\left (-\frac {c \sqrt {-\left (e x -d \right ) \left (e x +d \right )}}{e^{2}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) a \,e^{2}}{2 d^{2} \sqrt {d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) b}{\sqrt {d^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{\sqrt {e x +d}\, \sqrt {-e x +d}}\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 117, normalized size = 1.18 \begin {gather*} -\sqrt {-x^{2} e^{2} + d^{2}} c e^{\left (-2\right )} - \frac {b \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {a 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^{3}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} a}{2 \, d^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 98, normalized size = 0.99 \begin {gather*} -\frac {2 \, c d^{4} x^{2} - {\left (2 \, b d^{2} e^{2} + a e^{4}\right )} x^{2} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) + {\left (2 \, c d^{3} x^{2} + a d e^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{2 \, d^{3} e^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 5.15, size = 422, normalized size = 4.26 \begin {gather*} \frac {b\,\left (\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )-\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )\right )}{d}-\frac {\left (\frac {c\,d}{e^2}+\frac {c\,x}{e}\right )\,\sqrt {d-e\,x}}{\sqrt {d+e\,x}}-\frac {\frac {a\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {a\,e^2}{2}+\frac {15\,a\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}}{\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {32\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}+\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}}-\frac {a\,e^2\,\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{2\,d^3}+\frac {a\,e^2\,\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )}{2\,d^3}+\frac {a\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{32\,d^3\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________