Optimal. Leaf size=99 \[ -\frac {\left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{2 d^3}-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{2 d^2 x^2}-\frac {c \sqrt {d-e x} \sqrt {d+e x}}{e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, 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 = {520, 1251, 897, 1157, 388, 208} \[ -\frac {\sqrt {d^2-e^2 x^2} \left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c \left (d^2-e^2 x^2\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 388
Rule 520
Rule 897
Rule 1157
Rule 1251
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} \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 [B] time = 0.21, size = 233, normalized size = 2.35 \[ \frac {-e^2 x^2 \sqrt {d^2-e^2 x^2} \left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-a d^3 e^2+a d e^4 x^2-4 c d^{9/2} x^2 \sqrt {d-e x} \sqrt {\frac {e x}{d}+1} \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right )-2 c d^5 x^2+4 c d^4 x^2 \sqrt {d-e x} \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )+2 c d^3 e^2 x^4}{2 d^3 e^2 x^2 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 98, normalized size = 0.99 \[ -\frac {2 \, c d^{4} x^{2} - {\left (2 \, b d^{2} e^{2} + a e^{4}\right )} x^{2} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) + {\left (2 \, c d^{3} x^{2} + a d e^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{2 \, d^{3} e^{2} x^{2}} \]
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 [C] time = 0.02, size = 163, normalized size = 1.65 \[ -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (a \,e^{4} x^{2} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+2 b \,d^{2} e^{2} x^{2} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+2 \sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{3} x^{2} \mathrm {csgn}\relax (d )+\sqrt {-e^{2} x^{2}+d^{2}}\, a d \,e^{2} \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.02, size = 123, normalized size = 1.24 \[ -\frac {b \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {a e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} c}{e^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{2 \, d^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.15, size = 422, normalized size = 4.26 \[ \frac {b\,\left (\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )-\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )\right )}{d}-\frac {\left (\frac {c\,d}{e^2}+\frac {c\,x}{e}\right )\,\sqrt {d-e\,x}}{\sqrt {d+e\,x}}-\frac {\frac {a\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {a\,e^2}{2}+\frac {15\,a\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}}{\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {32\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}+\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}}-\frac {a\,e^2\,\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{2\,d^3}+\frac {a\,e^2\,\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )}{2\,d^3}+\frac {a\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{32\,d^3\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 133.79, size = 270, normalized size = 2.73 \[ \frac {i a e^{2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & 2, 2, \frac {5}{2} \\\frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} - \frac {a e^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, 1 & \\\frac {5}{4}, \frac {7}{4} & 1, \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} + \frac {i b {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {b {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i c d {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {c d {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________