Optimal. Leaf size=101 \[ -\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {390, 385, 214}
\begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}-\frac {d x \sqrt {a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 385
Rule 390
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx &=-\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=-\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 c (b c-a d)}\\ &=-\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 118, normalized size = 1.17 \begin {gather*} -\frac {d x \sqrt {a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {(2 b c-a d) \tan ^{-1}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{2 c^{3/2} (-b c+a d)^{3/2}} \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. \(809\) vs.
\(2(85)=170\).
time = 0.06, size = 810, normalized size = 8.02
method | result | size |
default | \(-\frac {\ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{4 c \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}}-\frac {-\frac {d \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-c d}}{d}\right )}+\frac {b \sqrt {-c d}\, \ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{\left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}}{4 d c}+\frac {\ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{4 c \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}}-\frac {-\frac {d \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {b \sqrt {-c d}\, \ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{\left (a d -b c \right ) \sqrt {\frac {a d -b c}{d}}}}{4 d c}\) | \(810\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs.
\(2 (85) = 170\).
time = 0.58, size = 463, normalized size = 4.58 \begin {gather*} \left [-\frac {4 \, {\left (b c^{2} d - a c d^{2}\right )} \sqrt {b x^{2} + a} x - {\left (2 \, b c^{2} - a c d + {\left (2 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{8 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}, -\frac {2 \, {\left (b c^{2} d - a c d^{2}\right )} \sqrt {b x^{2} + a} x + {\left (2 \, b c^{2} - a c d + {\left (2 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{2}}\, dx \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. 242 vs.
\(2 (85) = 170\).
time = 0.70, size = 242, normalized size = 2.40 \begin {gather*} \frac {1}{2} \, b^{\frac {3}{2}} {\left (\frac {{\left (2 \, b c - a d\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{{\left (b^{2} c^{2} - a b c d\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {2 \, {\left (2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )} {\left (b^{2} c^{2} - a b c d\right )}}\right )} \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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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