Optimal. Leaf size=149 \[ \frac {35 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{64 \sqrt {a} b^{9/2}}+\frac {7 x \left (15 a x^2+23 b^2\right )}{192 b^4 \left (a x^2+b^2\right )^{3/2} \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {x \left (35 a x^2+59 b^2\right )}{96 b^3 \left (a x^2+b^2\right )^2 \sqrt {\sqrt {a x^2+b^2}+b}} \]
________________________________________________________________________________________
Rubi [F] time = 1.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx &=\int \left (-\frac {a^3 \sqrt {b+\sqrt {b^2+a x^2}}}{8 (-a)^{3/2} b^3 \left (\sqrt {-a} b-a x\right )^3}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{16 b^4 \left (\sqrt {-a} b-a x\right )^2}-\frac {a^3 \sqrt {b+\sqrt {b^2+a x^2}}}{8 (-a)^{3/2} b^3 \left (\sqrt {-a} b+a x\right )^3}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{16 b^4 \left (\sqrt {-a} b+a x\right )^2}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{8 b^4 \left (-a b^2-a^2 x^2\right )}\right ) \, dx\\ &=-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{-a b^2-a^2 x^2} \, dx}{8 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3}\\ &=-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \left (-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b-\sqrt {-a} x\right )}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b+\sqrt {-a} x\right )}\right ) \, dx}{8 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3}\\ &=\frac {3 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {-a} x} \, dx}{16 b^5}+\frac {3 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {-a} x} \, dx}{16 b^5}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.34, size = 149, normalized size = 1.00 \begin {gather*} \frac {7 x \left (23 b^2+15 a x^2\right )}{192 b^4 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {x \left (59 b^2+35 a x^2\right )}{96 b^3 \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {35 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{64 \sqrt {a} b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{{\left (b^2+a\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\left (a x^{2} + b^{2}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________