Optimal. Leaf size=68 \[ \frac {\left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}+\frac {b x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2} \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {389, 63, 217, 206} \[ \frac {\left (2 a d^2+b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}+\frac {b x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 217
Rule 389
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx &=\frac {b x \sqrt {-c+d x} \sqrt {c+d x}}{2 d^2}-\frac {\left (-b c^2-2 a d^2\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{2 d^2}\\ &=\frac {b x \sqrt {-c+d x} \sqrt {c+d x}}{2 d^2}+\frac {\left (b c^2+2 a d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{d^3}\\ &=\frac {b x \sqrt {-c+d x} \sqrt {c+d x}}{2 d^2}+\frac {\left (b c^2+2 a d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3}\\ &=\frac {b x \sqrt {-c+d x} \sqrt {c+d x}}{2 d^2}+\frac {\left (b c^2+2 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 119, normalized size = 1.75 \[ \frac {4 \left (a d^2+b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )-\frac {2 b c^{5/2} \sqrt {\frac {d x}{c}+1} \sinh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c+d x}}+b d x \sqrt {d x-c} \sqrt {c+d x}}{2 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 63, normalized size = 0.93 \[ \frac {\sqrt {d x + c} \sqrt {d x - c} b d x - {\left (b c^{2} + 2 \, a d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 79, normalized size = 1.16 \[ \frac {\sqrt {d x + c} \sqrt {d x - c} {\left (\frac {{\left (d x + c\right )} b}{d^{2}} - \frac {b c}{d^{2}}\right )} - \frac {2 \, {\left (b c^{2} + 2 \, a d^{2}\right )} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 124, normalized size = 1.82 \[ \frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (2 a \,d^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+b \,c^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+\sqrt {d^{2} x^{2}-c^{2}}\, b d x \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{2 \sqrt {d^{2} x^{2}-c^{2}}\, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 89, normalized size = 1.31 \[ \frac {b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{2 \, d^{3}} + \frac {a \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{d} + \frac {\sqrt {d^{2} x^{2} - c^{2}} b x}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.80, size = 417, normalized size = 6.13 \[ \frac {\frac {2\,b\,c^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {-c}-\sqrt {d\,x-c}}+\frac {14\,b\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}+\frac {14\,b\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^5}+\frac {2\,b\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^7}}{d^3-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {6\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}}+\frac {4\,a\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}{\sqrt {-d^2}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {-d^2}}-\frac {2\,b\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 41.78, size = 199, normalized size = 2.93 \[ \frac {a {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} - \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d} + \frac {b c^{2} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} - \frac {i b c^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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