Optimal. Leaf size=103 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} d x}{\sqrt {a+b x^2} \sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2156, 205, 377} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} d x}{\sqrt {a+b x^2} \sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 2156
Rubi steps
\begin {align*} \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx &=(a c) \int \frac {1}{a^2 c^2-a d^2+a b c^2 x^2} \, dx-(a d) \int \frac {1}{\sqrt {a+b x^2} \left (a^2 c^2-a d^2+a b c^2 x^2\right )} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-(a d) \operatorname {Subst}\left (\int \frac {1}{a^2 c^2-a d^2-\left (-a^2 b c^2+b \left (a^2 c^2-a d^2\right )\right ) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} d x}{\sqrt {a c^2-d^2} \sqrt {a+b x^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 83, normalized size = 0.81 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )-\tan ^{-1}\left (\frac {\sqrt {b} d x}{\sqrt {a+b x^2} \sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 510, normalized size = 4.95 \[ \left [-\frac {\sqrt {-a b c^{2} + b d^{2}} \log \left (\frac {a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} + {\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2} - 4 \, \sqrt {-a b c^{2} + b d^{2}} {\left ({\left (a b c^{2} d - 2 \, b d^{3}\right )} x^{3} + {\left (a^{2} c^{2} d - a d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{b^{2} c^{4} x^{4} + a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4} + 2 \, {\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right ) + 2 \, \sqrt {-a b c^{2} + b d^{2}} \log \left (\frac {b c^{2} x^{2} - a c^{2} - 2 \, \sqrt {-a b c^{2} + b d^{2}} c x + d^{2}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right )}{4 \, {\left (a b c^{2} - b d^{2}\right )}}, -\frac {2 \, \sqrt {a b c^{2} - b d^{2}} \arctan \left (-\frac {\sqrt {a b c^{2} - b d^{2}} c x}{a c^{2} - d^{2}}\right ) - \sqrt {a b c^{2} - b d^{2}} \arctan \left (\frac {{\left (a^{2} c^{2} - a d^{2} + {\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}\right )} \sqrt {a b c^{2} - b d^{2}} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (a b^{2} c^{2} d - b^{2} d^{3}\right )} x^{3} + {\left (a^{2} b c^{2} d - a b d^{3}\right )} x\right )}}\right )}{2 \, {\left (a b c^{2} - b d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 107, normalized size = 1.04 \[ \frac {\arctan \left (\frac {b c x}{\sqrt {a b c^{2} - b d^{2}}}\right )}{\sqrt {a b c^{2} - b d^{2}}} + \frac {\arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt {a c^{2} - d^{2}} d}\right )}{\sqrt {a c^{2} - d^{2}} \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1995, normalized size = 19.37 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{b c x^{2} + a c + \sqrt {b x^{2} + a} d}\,{d x} \]
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 \[ \left \{\begin {array}{cl} \frac {\mathrm {atan}\left (\frac {b\,c\,x}{\sqrt {a\,b\,c^2-b\,d^2}}\right )}{\sqrt {a\,b\,c^2-b\,d^2}}-\frac {d\,x}{\sqrt {a}\,\left (a\,c^2-d^2\right )} & \text {\ if\ \ }b=0\vee d=0\\ \frac {\mathrm {atan}\left (\frac {b\,c\,x}{\sqrt {a\,b\,c^2-b\,d^2}}\right )}{\sqrt {a\,b\,c^2-b\,d^2}}-\frac {d\,\mathrm {atan}\left (\frac {x\,\sqrt {a\,b\,c^2-b\,\left (a\,c^2-d^2\right )}}{\sqrt {a\,c^2-d^2}\,\sqrt {b\,x^2+a}}\right )}{\sqrt {-\left (a\,c^2-d^2\right )\,\left (b\,\left (a\,c^2-d^2\right )-a\,b\,c^2\right )}} & \text {\ if\ \ }0<b\,d^2\\ \frac {\mathrm {atan}\left (\frac {b\,c\,x}{\sqrt {a\,b\,c^2-b\,d^2}}\right )}{\sqrt {a\,b\,c^2-b\,d^2}}-\frac {d\,\ln \left (\frac {\sqrt {\left (a\,c^2-d^2\right )\,\left (b\,x^2+a\right )}+x\,\sqrt {b\,\left (a\,c^2-d^2\right )-a\,b\,c^2}}{\sqrt {\left (a\,c^2-d^2\right )\,\left (b\,x^2+a\right )}-x\,\sqrt {b\,\left (a\,c^2-d^2\right )-a\,b\,c^2}}\right )}{2\,\sqrt {\left (a\,c^2-d^2\right )\,\left (b\,\left (a\,c^2-d^2\right )-a\,b\,c^2\right )}} & \text {\ if\ \ }b\,d^2<0\\ \int \frac {1}{a\,c+d\,\sqrt {b\,x^2+a}+b\,c\,x^2} \,d x & \text {\ if\ \ }b\,d^2\notin \mathbb {R} \end {array}\right . \]
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 \[ \int \frac {1}{a c + b c x^{2} + d \sqrt {a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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