Optimal. Leaf size=116 \[ \frac {\sqrt {d x-1} \sqrt {d x+1} \left (2 a d^2+3 c\right )}{3 x}+\frac {a \sqrt {d x-1} \sqrt {d x+1}}{3 x^3}+\frac {1}{2} b d^2 \tan ^{-1}\left (\sqrt {d x-1} \sqrt {d x+1}\right )+\frac {b \sqrt {d x-1} \sqrt {d x+1}}{2 x^2} \]
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Rubi [A] time = 0.22, antiderivative size = 171, normalized size of antiderivative = 1.47, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1610, 1807, 835, 807, 266, 63, 205} \[ -\frac {\left (1-d^2 x^2\right ) \left (2 a d^2+3 c\right )}{3 x \sqrt {d x-1} \sqrt {d x+1}}-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {d x-1} \sqrt {d x+1}}-\frac {b \left (1-d^2 x^2\right )}{2 x^2 \sqrt {d x-1} \sqrt {d x+1}}+\frac {b d^2 \sqrt {d^2 x^2-1} \tan ^{-1}\left (\sqrt {d^2 x^2-1}\right )}{2 \sqrt {d x-1} \sqrt {d x+1}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 266
Rule 807
Rule 835
Rule 1610
Rule 1807
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{x^4 \sqrt {-1+d x} \sqrt {1+d x}} \, dx &=\frac {\sqrt {-1+d^2 x^2} \int \frac {a+b x+c x^2}{x^4 \sqrt {-1+d^2 x^2}} \, dx}{\sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\sqrt {-1+d^2 x^2} \int \frac {3 b+\left (3 c+2 a d^2\right ) x}{x^3 \sqrt {-1+d^2 x^2}} \, dx}{3 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\sqrt {-1+d^2 x^2} \int \frac {2 \left (3 c+2 a d^2\right )+3 b d^2 x}{x^2 \sqrt {-1+d^2 x^2}} \, dx}{6 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (3 c+2 a d^2\right ) \left (1-d^2 x^2\right )}{3 x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (b d^2 \sqrt {-1+d^2 x^2}\right ) \int \frac {1}{x \sqrt {-1+d^2 x^2}} \, dx}{2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (3 c+2 a d^2\right ) \left (1-d^2 x^2\right )}{3 x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (b d^2 \sqrt {-1+d^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+d^2 x}} \, dx,x,x^2\right )}{4 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (3 c+2 a d^2\right ) \left (1-d^2 x^2\right )}{3 x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (b \sqrt {-1+d^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{d^2}+\frac {x^2}{d^2}} \, dx,x,\sqrt {-1+d^2 x^2}\right )}{2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{3 x^3 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (3 c+2 a d^2\right ) \left (1-d^2 x^2\right )}{3 x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {b d^2 \sqrt {-1+d^2 x^2} \tan ^{-1}\left (\sqrt {-1+d^2 x^2}\right )}{2 \sqrt {-1+d x} \sqrt {1+d x}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 94, normalized size = 0.81 \[ \frac {\left (d^2 x^2-1\right ) \left (a \left (4 d^2 x^2+2\right )+3 x (b+2 c x)\right )+3 b d^2 x^3 \sqrt {d^2 x^2-1} \tan ^{-1}\left (\sqrt {d^2 x^2-1}\right )}{6 x^3 \sqrt {d x-1} \sqrt {d x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 90, normalized size = 0.78 \[ \frac {6 \, b d^{2} x^{3} \arctan \left (-d x + \sqrt {d x + 1} \sqrt {d x - 1}\right ) + 2 \, {\left (2 \, a d^{3} + 3 \, c d\right )} x^{3} + {\left (2 \, {\left (2 \, a d^{2} + 3 \, c\right )} x^{2} + 3 \, b x + 2 \, a\right )} \sqrt {d x + 1} \sqrt {d x - 1}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.35, size = 197, normalized size = 1.70 \[ -\frac {3 \, b d^{3} \arctan \left (\frac {1}{2} \, {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, b d^{3} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{10} - 12 \, c d^{2} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{8} - 96 \, a d^{4} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{4} - 96 \, c d^{2} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{4} - 48 \, b d^{3} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{2} - 128 \, a d^{4} - 192 \, c d^{2}\right )}}{{\left ({\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{4} + 4\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 123, normalized size = 1.06 \[ -\frac {\sqrt {d x -1}\, \sqrt {d x +1}\, \left (3 b \,d^{2} x^{3} \arctan \left (\frac {1}{\sqrt {d^{2} x^{2}-1}}\right )-4 \sqrt {d^{2} x^{2}-1}\, a \,d^{2} x^{2}-6 \sqrt {d^{2} x^{2}-1}\, c \,x^{2}-3 \sqrt {d^{2} x^{2}-1}\, b x -2 \sqrt {d^{2} x^{2}-1}\, a \right ) \mathrm {csgn}\relax (d )^{2}}{6 \sqrt {d^{2} x^{2}-1}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 86, normalized size = 0.74 \[ -\frac {1}{2} \, b d^{2} \arcsin \left (\frac {1}{d {\left | x \right |}}\right ) + \frac {2 \, \sqrt {d^{2} x^{2} - 1} a d^{2}}{3 \, x} + \frac {\sqrt {d^{2} x^{2} - 1} c}{x} + \frac {\sqrt {d^{2} x^{2} - 1} b}{2 \, x^{2}} + \frac {\sqrt {d^{2} x^{2} - 1} a}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.44, size = 304, normalized size = 2.62 \[ \frac {\frac {b\,d^2\,1{}\mathrm {i}}{32}+\frac {b\,d^2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,{\left (\sqrt {d\,x+1}-1\right )}^2}-\frac {b\,d^2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,{\left (\sqrt {d\,x+1}-1\right )}^4}}{\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {d\,x+1}-1\right )}^4}+\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {d\,x+1}-1\right )}^6}}-\frac {b\,d^2\,\ln \left (\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2}+\frac {b\,d^2\,\ln \left (\frac {\sqrt {d\,x-1}-\mathrm {i}}{\sqrt {d\,x+1}-1}\right )\,1{}\mathrm {i}}{2}+\frac {c\,\sqrt {d\,x-1}\,\sqrt {d\,x+1}}{x}+\frac {\sqrt {d\,x-1}\,\left (\frac {2\,a\,d^3\,x^3}{3}+\frac {2\,a\,d^2\,x^2}{3}+\frac {a\,d\,x}{3}+\frac {a}{3}\right )}{x^3\,\sqrt {d\,x+1}}+\frac {b\,d^2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,{\left (\sqrt {d\,x+1}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 129.78, size = 219, normalized size = 1.89 \[ - \frac {a d^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {5}{2}, \frac {5}{2}, 3 \\2, \frac {9}{4}, \frac {5}{2}, \frac {11}{4}, 3 & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i a d^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 1 & \\\frac {7}{4}, \frac {9}{4} & \frac {3}{2}, 2, 2, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {b d^{2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & 2, 2, \frac {5}{2} \\\frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i b d^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, 1 & \\\frac {5}{4}, \frac {7}{4} & 1, \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {c d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i c d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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