Optimal. Leaf size=100 \[ \frac {(b c-2 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}}+\frac {b x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.05, antiderivative size = 100, 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 = {382, 377, 205} \[ \frac {(b c-2 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}}+\frac {b x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 382
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx &=\frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a (b c-a d)}\\ &=\frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a (b c-a d)}\\ &=\frac {b x \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {(b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.78, size = 405, normalized size = 4.05 \[ \frac {x \sqrt {c+d x^2} \left (-30 d x^2 \sqrt {\frac {a x^2 \left (c+d x^2\right ) (b c-a d)}{c^2 \left (a+b x^2\right )^2}}-45 c \sqrt {\frac {a x^2 \left (c+d x^2\right ) (b c-a d)}{c^2 \left (a+b x^2\right )^2}}+16 d x^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+16 c \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )+30 d x^2 \sin ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )+45 c \sin ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )\right )}{30 c^2 \left (a+b x^2\right )^2 \left (\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}\right )^{3/2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.98, size = 459, normalized size = 4.59 \[ \left [\frac {4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x - {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x + \sqrt {a b c - a^{2} d} {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 225, normalized size = 2.25 \[ -\frac {1}{2} \, d^{\frac {3}{2}} {\left (\frac {{\left (b c - 2 \, a d\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a b c d - a^{2} d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d - b c^{2}\right )}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} {\left (a b c d - a^{2} d^{2}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 823, normalized size = 8.23 \[ \frac {\sqrt {-a b}\, d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, a b}-\frac {\sqrt {-a b}\, d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}\, a b}-\frac {\ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a}+\frac {\ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}\, a}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{4 \left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right ) a}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{4 \left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right ) a} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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