Optimal. Leaf size=83 \[ \frac {1}{2} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt {d x-1} \sqrt {d x+1}\right )+\frac {a \sqrt {d x-1} \sqrt {d x+1}}{2 x^2}+\frac {b \sqrt {d x-1} \sqrt {d x+1}}{x} \]
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Rubi [A] time = 0.19, antiderivative size = 129, normalized size of antiderivative = 1.55, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1610, 1807, 807, 266, 63, 205} \[ \frac {\sqrt {d^2 x^2-1} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt {d^2 x^2-1}\right )}{2 \sqrt {d x-1} \sqrt {d x+1}}-\frac {a \left (1-d^2 x^2\right )}{2 x^2 \sqrt {d x-1} \sqrt {d x+1}}-\frac {b \left (1-d^2 x^2\right )}{x \sqrt {d x-1} \sqrt {d x+1}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 266
Rule 807
Rule 1610
Rule 1807
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{x^3 \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^3 \sqrt {-1+d^2 x^2}} \, dx}{\sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \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 b+\left (2 c+a d^2\right ) x}{x^2 \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 )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (\left (2 c+a d^2\right ) \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 )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (\left (2 c+a d^2\right ) \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 )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (\left (2 c+a d^2\right ) \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 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {a \left (1-d^2 x^2\right )}{2 x^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {b \left (1-d^2 x^2\right )}{x \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (2 c+a d^2\right ) \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 = 0.14, size = 82, normalized size = 0.99 \[ \frac {\left (d^2 x^2-1\right ) (a+2 b x)+x^2 \sqrt {d^2 x^2-1} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt {d^2 x^2-1}\right )}{2 x^2 \sqrt {d x-1} \sqrt {d x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 69, normalized size = 0.83 \[ \frac {2 \, b d x^{2} + 2 \, {\left (a d^{2} + 2 \, c\right )} x^{2} \arctan \left (-d x + \sqrt {d x + 1} \sqrt {d x - 1}\right ) + {\left (2 \, b x + a\right )} \sqrt {d x + 1} \sqrt {d x - 1}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.35, size = 145, normalized size = 1.75 \[ -\frac {{\left (a d^{3} + 2 \, c d\right )} \arctan \left (\frac {1}{2} \, {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{2}\right ) + \frac {2 \, {\left (a d^{3} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{6} - 4 \, b d^{2} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{4} - 4 \, a d^{3} {\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{2} - 16 \, b d^{2}\right )}}{{\left ({\left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}^{4} + 4\right )}^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 103, normalized size = 1.24 \[ -\frac {\sqrt {d x -1}\, \sqrt {d x +1}\, \left (a \,d^{2} x^{2} \arctan \left (\frac {1}{\sqrt {d^{2} x^{2}-1}}\right )+2 c \,x^{2} \arctan \left (\frac {1}{\sqrt {d^{2} x^{2}-1}}\right )-2 \sqrt {d^{2} x^{2}-1}\, b x -\sqrt {d^{2} x^{2}-1}\, a \right ) \mathrm {csgn}\relax (d )^{2}}{2 \sqrt {d^{2} x^{2}-1}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 61, normalized size = 0.73 \[ -\frac {1}{2} \, a d^{2} \arcsin \left (\frac {1}{d {\left | x \right |}}\right ) - c \arcsin \left (\frac {1}{d {\left | x \right |}}\right ) + \frac {\sqrt {d^{2} x^{2} - 1} b}{x} + \frac {\sqrt {d^{2} x^{2} - 1} a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.89, size = 316, normalized size = 3.81 \[ \frac {\frac {a\,d^2\,1{}\mathrm {i}}{32}+\frac {a\,d^2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,{\left (\sqrt {d\,x+1}-1\right )}^2}-\frac {a\,d^2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,{\left (\sqrt {d\,x+1}-1\right )}^4}}{\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {d\,x+1}-1\right )}^4}+\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {d\,x+1}-1\right )}^6}}-c\,\left (\ln \left (\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )-\ln \left (\frac {\sqrt {d\,x-1}-\mathrm {i}}{\sqrt {d\,x+1}-1}\right )\right )\,1{}\mathrm {i}-\frac {a\,d^2\,\ln \left (\frac {{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2}+\frac {a\,d^2\,\ln \left (\frac {\sqrt {d\,x-1}-\mathrm {i}}{\sqrt {d\,x+1}-1}\right )\,1{}\mathrm {i}}{2}+\frac {b\,\sqrt {d\,x-1}\,\sqrt {d\,x+1}}{x}+\frac {a\,d^2\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,{\left (\sqrt {d\,x+1}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 74.80, size = 212, normalized size = 2.55 \[ - \frac {a d^{2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & 2, 2, \frac {5}{2} \\\frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i a d^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, 1 & \\\frac {5}{4}, \frac {7}{4} & 1, \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {b d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i b d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {c {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i c {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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