Optimal. Leaf size=65 \[ -\frac {\frac {a}{c^2}+\frac {b}{d^2}}{\sqrt {-c+d x} \sqrt {c+d x}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{c^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {469, 94, 211}
\begin {gather*} -\frac {a \text {ArcTan}\left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{c^3}-\frac {\frac {a}{c^2}+\frac {b}{d^2}}{\sqrt {d x-c} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 94
Rule 211
Rule 469
Rubi steps
\begin {align*} \int \frac {a+b x^2}{x (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=-\frac {\frac {a}{c^2}+\frac {b}{d^2}}{\sqrt {-c+d x} \sqrt {c+d x}}-\frac {a \int \frac {1}{x \sqrt {-c+d x} \sqrt {c+d x}} \, dx}{c^2}\\ &=-\frac {\frac {a}{c^2}+\frac {b}{d^2}}{\sqrt {-c+d x} \sqrt {c+d x}}-\frac {(a d) \text {Subst}\left (\int \frac {1}{c^2 d+d x^2} \, dx,x,\sqrt {-c+d x} \sqrt {c+d x}\right )}{c^2}\\ &=-\frac {\frac {a}{c^2}+\frac {b}{d^2}}{\sqrt {-c+d x} \sqrt {c+d x}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{c^3}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 70, normalized size = 1.08 \begin {gather*} \frac {-\frac {2 \left (b c^3+a c d^2\right )}{d^2 \sqrt {-c+d x} \sqrt {c+d x}}+4 a \tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-c+d x}}\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(193\) vs.
\(2(58)=116\).
time = 0.30, size = 194, normalized size = 2.98
method | result | size |
default | \(\frac {\sqrt {d x -c}\, \left (-\ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,d^{4} x^{2}+\ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,c^{2} d^{2}+\sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, a \,d^{2}+\sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {-c^{2}}\, b \,c^{2}\right )}{c^{2} \sqrt {-c^{2}}\, \left (-d x +c \right ) \sqrt {d^{2} x^{2}-c^{2}}\, d^{2} \sqrt {d x +c}}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 58, normalized size = 0.89 \begin {gather*} \frac {a \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{c^{3}} - \frac {a}{\sqrt {d^{2} x^{2} - c^{2}} c^{2}} - \frac {b}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.13, size = 101, normalized size = 1.55 \begin {gather*} -\frac {{\left (b c^{3} + a c d^{2}\right )} \sqrt {d x + c} \sqrt {d x - c} + 2 \, {\left (a d^{4} x^{2} - a c^{2} d^{2}\right )} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right )}{c^{3} d^{4} x^{2} - c^{5} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs.
\(2 (57) = 114\).
time = 0.59, size = 115, normalized size = 1.77 \begin {gather*} \frac {2 \, a \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} - \frac {{\left (b c^{2} + a d^{2}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{3} d^{2}} + \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^{2} d^{2}} \end {gather*}
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 \begin {gather*} \int \frac {b\,x^2+a}{x\,{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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