Optimal. Leaf size=167 \[ \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^4 a-2 \text {$\#$1}^4 b+b^2\& ,\frac {\text {$\#$1}^4 \left (-\log \left (\sqrt {\sqrt {a^2 x^2+b}+a x}-\text {$\#$1}\right )\right )-b \log \left (\sqrt {\sqrt {a^2 x^2+b}+a x}-\text {$\#$1}\right )}{-\text {$\#$1}^7+2 \text {$\#$1}^3 a+\text {$\#$1}^3 b}\& \right ]-\frac {b}{3 a \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{a} \]
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Rubi [A] time = 1.26, antiderivative size = 319, normalized size of antiderivative = 1.91, number of steps used = 23, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6725, 2117, 14, 2119, 1628, 828, 826, 1166, 208, 205} \begin {gather*} -\frac {b}{3 a \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {a} \sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b}+a x}}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {a} \sqrt {\sqrt {a+b}+\sqrt {a}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 205
Rule 208
Rule 826
Rule 828
Rule 1166
Rule 1628
Rule 2117
Rule 2119
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+a x^2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {a x+\sqrt {b+a^2 x^2}}}+\frac {2}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}}\right ) \, dx\\ &=2 \int \frac {1}{\left (-1+a x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx+\int \frac {1}{\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ &=2 \int \left (-\frac {1}{2 \left (1-\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}}-\frac {1}{2 \left (1+\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}}\right ) \, dx+\frac {\operatorname {Subst}\left (\int \frac {b+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{2 a}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{2 a}-\int \frac {1}{\left (1-\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx-\int \frac {1}{\left (1+\sqrt {a} x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\operatorname {Subst}\left (\int \frac {b+x^2}{x^{3/2} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )-\operatorname {Subst}\left (\int \frac {b+x^2}{x^{3/2} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {a} x^{3/2}}+\frac {2 \left (b+\sqrt {a} x\right )}{x^{3/2} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )}\right ) \, dx,x,a x+\sqrt {b+a^2 x^2}\right )-\operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {a} x^{3/2}}+\frac {2 \left (b-\sqrt {a} x\right )}{x^{3/2} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )}\right ) \, dx,x,a x+\sqrt {b+a^2 x^2}\right )\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-2 \operatorname {Subst}\left (\int \frac {b+\sqrt {a} x}{x^{3/2} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )-2 \operatorname {Subst}\left (\int \frac {b-\sqrt {a} x}{x^{3/2} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {-a b+\sqrt {a} b x}{\sqrt {x} \left (\sqrt {a} b+2 a x-\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{\sqrt {a} b}+\frac {2 \operatorname {Subst}\left (\int \frac {-a b-\sqrt {a} b x}{\sqrt {x} \left (-\sqrt {a} b+2 a x+\sqrt {a} x^2\right )} \, dx,x,a x+\sqrt {b+a^2 x^2}\right )}{\sqrt {a} b}\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\frac {4 \operatorname {Subst}\left (\int \frac {-a b+\sqrt {a} b x^2}{\sqrt {a} b+2 a x^2-\sqrt {a} x^4} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )}{\sqrt {a} b}+\frac {4 \operatorname {Subst}\left (\int \frac {-a b-\sqrt {a} b x^2}{-\sqrt {a} b+2 a x^2+\sqrt {a} x^4} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )}{\sqrt {a} b}\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-2 \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {a+b}-\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {a+b}-\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {a+b}+\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {a+b}+\sqrt {a} x^2} \, dx,x,\sqrt {a x+\sqrt {b+a^2 x^2}}\right )\\ &=-\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}+\sqrt {a+b}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}-\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}-\sqrt {a+b}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a}+\sqrt {a+b}}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 166, normalized size = 0.99 \begin {gather*} \frac {-a \text {RootSum}\left [\text {$\#$1}^8 \left (-b^2\right )+4 \text {$\#$1}^4 a+2 \text {$\#$1}^4 b-1\&,\frac {\text {$\#$1}^4 b \log \left (\frac {1}{\sqrt {\sqrt {a^2 x^2+b}+a x}}-\text {$\#$1}\right )+\log \left (\frac {1}{\sqrt {\sqrt {a^2 x^2+b}+a x}}-\text {$\#$1}\right )}{\text {$\#$1}^5 b^2-2 \text {$\#$1} a-\text {$\#$1} b}\&\right ]-\frac {b}{3 \left (\sqrt {a^2 x^2+b}+a x\right )^{3/2}}+\sqrt {\sqrt {a^2 x^2+b}+a x}}{a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 167, normalized size = 1.00 \begin {gather*} -\frac {b}{3 a \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b+a^2 x^2}}}{a}+\text {RootSum}\left [b^2-4 a \text {$\#$1}^4-2 b \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-b \log \left (\sqrt {a x+\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 a \text {$\#$1}^3+b \text {$\#$1}^3-\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1097, normalized size = 6.57
result too large to display
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 \begin {gather*} \int \frac {a x^{2} + 1}{{\left (a x^{2} - 1\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}+1}{\left (a \,x^{2}-1\right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]
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*} \int \frac {a x^{2} + 1}{{\left (a x^{2} - 1\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\,{d x} \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 {a\,x^2+1}{\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,\left (a\,x^2-1\right )} \,d x \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 {a x^{2} + 1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b}} \left (a x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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