Optimal. Leaf size=82 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d} \]
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Rubi [A] time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {402, 217, 206, 377, 208} \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 217
Rule 377
Rule 402
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{c+d x^2} \, dx &=\frac {b \int \frac {1}{\sqrt {a+b x^2}} \, dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{d}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{d}\\ &=\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d}-\frac {\sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 84, normalized size = 1.02 \[ \frac {\sqrt {a d-b c} \tan ^{-1}\left (\frac {x \sqrt {a d-b c}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d}+\frac {\sqrt {b} \log \left (\sqrt {b} \sqrt {a+b x^2}+b x\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 596, normalized size = 7.27 \[ \left [\frac {2 \, \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d}, -\frac {4 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d}, \frac {\sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right ) + \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, d}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right )}{2 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 932, normalized size = 11.37 \[ -\frac {a \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {2 a d -2 b c}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}}+\frac {a \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {2 a d -2 b c}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}}+\frac {b c \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {2 a d -2 b c}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}\, d}-\frac {b c \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {2 a d -2 b c}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {\frac {a d -b c}{d}}\, d}+\frac {\sqrt {b}\, \ln \left (\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) b -\frac {\sqrt {-c d}\, b}{d}}{\sqrt {b}}+\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}\right )}{2 d}+\frac {\sqrt {b}\, \ln \left (\frac {\left (x -\frac {\sqrt {-c d}}{d}\right ) b +\frac {\sqrt {-c d}\, b}{d}}{\sqrt {b}}+\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}\right )}{2 d}+\frac {\sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{2 \sqrt {-c d}}-\frac {\sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) b}{d}+\frac {a d -b c}{d}}}{2 \sqrt {-c d}} \]
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 \[ \int \frac {\sqrt {b x^{2} + a}}{d x^{2} + c}\,{d x} \]
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 \[ \left \{\begin {array}{cl} \frac {\sqrt {-b}\,\mathrm {asin}\left (x\,\sqrt {-\frac {b}{a}}\right )}{c} & \text {\ if\ \ }\left (\left (a+b\,c=0\wedge d=-1\right )\vee a\,d=b\,c\right )\wedge b<0\\ \frac {\sqrt {b}\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{d}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {a\,d-b\,c}}{\sqrt {c}\,\sqrt {b\,x^2+a}}\right )\,\sqrt {a\,d-b\,c}}{\sqrt {c}\,d} & \text {\ if\ \ }a\neq 0\wedge \left (\left (\left (a+b\,c\neq 0\vee d\neq -1\right )\wedge a\,d\neq b\,c\right )\vee \neg b<0\right )\\ \int \frac {\sqrt {b\,x^2+a}}{d\,x^2+c} \,d x & \text {\ if\ \ }\left (\left (\left (\left (a+b\,c=0\wedge d=-1\right )\vee a\,d=b\,c\right )\wedge b<0\right )\vee a=0\right )\wedge \left (\left (\left (a+b\,c\neq 0\vee d\neq -1\right )\wedge a\,d\neq b\,c\right )\vee \neg b<0\right ) \end {array}\right . \]
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 \[ \int \frac {\sqrt {a + b x^{2}}}{c + d x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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