Optimal. Leaf size=87 \[ \frac {\sqrt {d x-1} \sqrt {d x+1} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac {b \cosh ^{-1}(d x)}{2 d^3}+\frac {c x^2 \sqrt {d x-1} \sqrt {d x+1}}{3 d^2} \]
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Rubi [A] time = 0.15, antiderivative size = 151, normalized size of antiderivative = 1.74, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1610, 1809, 780, 217, 206} \[ -\frac {\left (1-d^2 x^2\right ) \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4 \sqrt {d x-1} \sqrt {d x+1}}+\frac {b \sqrt {d^2 x^2-1} \tanh ^{-1}\left (\frac {d x}{\sqrt {d^2 x^2-1}}\right )}{2 d^3 \sqrt {d x-1} \sqrt {d x+1}}-\frac {c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt {d x-1} \sqrt {d x+1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 780
Rule 1610
Rule 1809
Rubi steps
\begin {align*} \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {-1+d x} \sqrt {1+d x}} \, dx &=\frac {\sqrt {-1+d^2 x^2} \int \frac {x \left (a+b x+c x^2\right )}{\sqrt {-1+d^2 x^2}} \, dx}{\sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\sqrt {-1+d^2 x^2} \int \frac {x \left (2 c+3 a d^2+3 b d^2 x\right )}{\sqrt {-1+d^2 x^2}} \, dx}{3 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \left (1-d^2 x^2\right )}{6 d^4 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (b \sqrt {-1+d^2 x^2}\right ) \int \frac {1}{\sqrt {-1+d^2 x^2}} \, dx}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \left (1-d^2 x^2\right )}{6 d^4 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {\left (b \sqrt {-1+d^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-d^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+d^2 x^2}}\right )}{2 d^2 \sqrt {-1+d x} \sqrt {1+d x}}\\ &=-\frac {c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt {-1+d x} \sqrt {1+d x}}-\frac {\left (2 \left (2 c+3 a d^2\right )+3 b d^2 x\right ) \left (1-d^2 x^2\right )}{6 d^4 \sqrt {-1+d x} \sqrt {1+d x}}+\frac {b \sqrt {-1+d^2 x^2} \tanh ^{-1}\left (\frac {d x}{\sqrt {-1+d^2 x^2}}\right )}{2 d^3 \sqrt {-1+d x} \sqrt {1+d x}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 149, normalized size = 1.71 \[ \frac {\sqrt {-(d x-1)^2} \sqrt {d x+1} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )+6 \sqrt {d x-1} \sin ^{-1}\left (\frac {\sqrt {1-d x}}{\sqrt {2}}\right ) (d (2 a d-b)+2 c)-12 \sqrt {1-d x} \tanh ^{-1}\left (\sqrt {\frac {d x-1}{d x+1}}\right ) (d (a d-b)+c)}{6 d^4 \sqrt {1-d x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 73, normalized size = 0.84 \[ -\frac {3 \, b d \log \left (-d x + \sqrt {d x + 1} \sqrt {d x - 1}\right ) - {\left (2 \, c d^{2} x^{2} + 3 \, b d^{2} x + 6 \, a d^{2} + 4 \, c\right )} \sqrt {d x + 1} \sqrt {d x - 1}}{6 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 105, normalized size = 1.21 \[ \frac {\sqrt {d x + 1} \sqrt {d x - 1} {\left ({\left (d x + 1\right )} {\left (\frac {2 \, {\left (d x + 1\right )} c}{d^{3}} + \frac {3 \, b d^{10} - 4 \, c d^{9}}{d^{12}}\right )} + \frac {3 \, {\left (2 \, a d^{11} - b d^{10} + 2 \, c d^{9}\right )}}{d^{12}}\right )} - \frac {6 \, b \log \left (\sqrt {d x + 1} - \sqrt {d x - 1}\right )}{d^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 137, normalized size = 1.57 \[ \frac {\sqrt {d x -1}\, \sqrt {d x +1}\, \left (2 \sqrt {d^{2} x^{2}-1}\, c \,d^{2} x^{2} \mathrm {csgn}\relax (d )+3 \sqrt {d^{2} x^{2}-1}\, b \,d^{2} x \,\mathrm {csgn}\relax (d )+6 \sqrt {d^{2} x^{2}-1}\, a \,d^{2} \mathrm {csgn}\relax (d )+3 b d \ln \left (\left (d x +\sqrt {d^{2} x^{2}-1}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+4 \sqrt {d^{2} x^{2}-1}\, c \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{6 \sqrt {d^{2} x^{2}-1}\, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 100, normalized size = 1.15 \[ \frac {\sqrt {d^{2} x^{2} - 1} c x^{2}}{3 \, d^{2}} + \frac {\sqrt {d^{2} x^{2} - 1} b x}{2 \, d^{2}} + \frac {\sqrt {d^{2} x^{2} - 1} a}{d^{2}} + \frac {b \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - 1} d\right )}{2 \, d^{3}} + \frac {2 \, \sqrt {d^{2} x^{2} - 1} c}{3 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.35, size = 318, normalized size = 3.66 \[ \frac {\sqrt {d\,x-1}\,\left (\frac {2\,c}{3\,d^4}+\frac {c\,x^3}{3\,d}+\frac {c\,x^2}{3\,d^2}+\frac {2\,c\,x}{3\,d^3}\right )}{\sqrt {d\,x+1}}+\frac {2\,b\,\mathrm {atanh}\left (\frac {\sqrt {d\,x-1}-\mathrm {i}}{\sqrt {d\,x+1}-1}\right )}{d^3}-\frac {\frac {14\,b\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {d\,x+1}-1\right )}^3}+\frac {14\,b\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {d\,x+1}-1\right )}^5}+\frac {2\,b\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {d\,x+1}-1\right )}^7}+\frac {2\,b\,\left (\sqrt {d\,x-1}-\mathrm {i}\right )}{\sqrt {d\,x+1}-1}}{d^3-\frac {4\,d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {d\,x+1}-1\right )}^2}+\frac {6\,d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {d\,x+1}-1\right )}^4}-\frac {4\,d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {d\,x+1}-1\right )}^6}+\frac {d^3\,{\left (\sqrt {d\,x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {d\,x+1}-1\right )}^8}}+\frac {a\,\sqrt {d\,x-1}\,\sqrt {d\,x+1}}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 78.83, size = 308, normalized size = 3.54 \[ \frac {a {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{2}} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} - \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} + \frac {c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} + \frac {i c {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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