Optimal. Leaf size=63 \[ \frac {2 b \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}-\frac {x \left (\frac {a}{c^2}+\frac {b}{d^2}\right )}{\sqrt {d x-c} \sqrt {c+d x}} \]
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Rubi [A] time = 0.03, antiderivative size = 63, 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 = {386, 63, 217, 206} \[ \frac {2 b \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}-\frac {x \left (\frac {a}{c^2}+\frac {b}{d^2}\right )}{\sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 217
Rule 386
Rubi steps
\begin {align*} \int \frac {a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {b \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{d^2}\\ &=-\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{d^3}\\ &=-\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3}\\ &=-\frac {\left (\frac {a}{c^2}+\frac {b}{d^2}\right ) x}{\sqrt {-c+d x} \sqrt {c+d x}}+\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 86, normalized size = 1.37 \[ \frac {2 b c^{5/2} \sqrt {\frac {d x}{c}+1} \sinh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {2} \sqrt {c}}\right )-\frac {d x \left (a d^2+b c^2\right )}{\sqrt {d x-c}}}{c^2 d^3 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.41, size = 129, normalized size = 2.05 \[ \frac {b c^{4} + a c^{2} d^{2} - {\left (b c^{2} d + a d^{3}\right )} \sqrt {d x + c} \sqrt {d x - c} x - {\left (b c^{2} d^{2} + a d^{4}\right )} x^{2} - {\left (b c^{2} d^{2} x^{2} - b c^{4}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{c^{2} d^{5} x^{2} - c^{4} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 113, normalized size = 1.79 \[ -\frac {b \log \left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}{d^{3}} - \frac {2 \, {\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} c d^{3}} - \frac {{\left (b c^{2} d^{3} + a d^{5}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{2} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 160, normalized size = 2.54 \[ \frac {\left (b \,c^{2} d^{2} x^{2} \ln \left (\left (d x +\sqrt {\left (d x -c \right ) \left (d x +c \right )}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )-\sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{3} x \,\mathrm {csgn}\relax (d )-b \,c^{4} \ln \left (\left (d x +\sqrt {\left (d x -c \right ) \left (d x +c \right )}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )-\sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} d x \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{\sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {d x +c}\, \sqrt {d x -c}\, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 76, normalized size = 1.21 \[ -\frac {a x}{\sqrt {d^{2} x^{2} - c^{2}} c^{2}} - \frac {b x}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} + \frac {b \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {b\,x^2+a}{{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 112.36, size = 182, normalized size = 2.89 \[ a \left (- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{2} d} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{2} d}\right ) + b \left (\frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, \frac {1}{2}, 1, 1 \\- \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{3}} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 1 & \\- \frac {3}{4}, - \frac {1}{4} & - \frac {3}{2}, -1, 0, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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