Optimal. Leaf size=650 \[ \frac {231 b^3 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{2048 a^3}+\frac {3 b^2 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{16 a^3}+\frac {\sqrt {a^2 x^2-b} \left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-5734400 a^4 x^4-3932160 a^3 x^3-5160960 a^2 b x^2-5242880 a^2 x^2+1067220 a b^3 x+1774080 a b^2 x+1966080 a b x+365904 b^3+2007040 b^2+1310720 b\right )+\left (5160960 a^4 x^4+3276800 a^3 x^3+860160 a^2 b x^2+2621440 a^2 x^2-1600830 a b^3 x-2661120 a b^2 x-1638400 a b x-426888 b^3-860160 b^2-655360 b\right ) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )+\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-5734400 a^5 x^5-3932160 a^4 x^4-2293760 a^3 b x^3-5242880 a^3 x^3+1067220 a^2 b^3 x^2+1774080 a^2 b^2 x^2+3932160 a^2 b x^2+365904 a b^3 x+5304320 a b^2 x+3932160 a b x-533610 b^4-591360 b^3-491520 b^2\right )+\left (5160960 a^5 x^5+3276800 a^4 x^4-1720320 a^3 b x^3+2621440 a^3 x^3-1600830 a^2 b^3 x^2-2661120 a^2 b^2 x^2-3276800 a^2 b x^2-426888 a b^3 x-1935360 a b^2 x-1966080 a b x+800415 b^4+1005312 b^3+409600 b^2\right ) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{7096320 a^3 \left (4 a^3 x^3-3 a b x\right )+7096320 a^3 \sqrt {a^2 x^2-b} \left (4 a^2 x^2-b\right )} \]
________________________________________________________________________________________
Rubi [F] time = 0.28, 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 {x^2}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {x^2}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.83, size = 655, normalized size = 1.01 \begin {gather*} -\frac {-\frac {b^3 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{11/2}}{12 \left (\sqrt {a^2 x^2-b}+a x\right )^3}+\frac {61 b^3 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{9/2}}{120 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/2}}-\frac {417 b^3 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{7/2}}{320 \left (\sqrt {a^2 x^2-b}+a x\right )^2}+\frac {3481 b^3 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{5/2}}{1920 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}}-\frac {b^2 (2279 b+384) \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{3/2}}{1536 \left (\sqrt {a^2 x^2-b}+a x\right )}+\frac {b^2 (793 b+640) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{1024 \sqrt {\sqrt {a^2 x^2-b}+a x}}+\frac {3 b^2 (77 b+128) \log \left (1-\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}\right )}{2048}-\frac {3 b^2 (77 b+128) \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}+1\right )}{2048}-\frac {1}{11} \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{11/2}+\frac {5}{9} \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{9/2}-\frac {10}{7} \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{7/2}+2 \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{5/2}-\frac {1}{3} (b+5) \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+1\right )^{3/2}+(b+1) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.21, size = 650, normalized size = 1.00 \begin {gather*} \frac {231 b^3 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{2048 a^3}+\frac {3 b^2 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{16 a^3}+\frac {\sqrt {a^2 x^2-b} \left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-5734400 a^4 x^4-3932160 a^3 x^3-5160960 a^2 b x^2-5242880 a^2 x^2+1067220 a b^3 x+1774080 a b^2 x+1966080 a b x+365904 b^3+2007040 b^2+1310720 b\right )+\left (5160960 a^4 x^4+3276800 a^3 x^3+860160 a^2 b x^2+2621440 a^2 x^2-1600830 a b^3 x-2661120 a b^2 x-1638400 a b x-426888 b^3-860160 b^2-655360 b\right ) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )+\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-5734400 a^5 x^5-3932160 a^4 x^4-2293760 a^3 b x^3-5242880 a^3 x^3+1067220 a^2 b^3 x^2+1774080 a^2 b^2 x^2+3932160 a^2 b x^2+365904 a b^3 x+5304320 a b^2 x+3932160 a b x-533610 b^4-591360 b^3-491520 b^2\right )+\left (5160960 a^5 x^5+3276800 a^4 x^4-1720320 a^3 b x^3+2621440 a^3 x^3-1600830 a^2 b^3 x^2-2661120 a^2 b^2 x^2-3276800 a^2 b x^2-426888 a b^3 x-1935360 a b^2 x-1966080 a b x+800415 b^4+1005312 b^3+409600 b^2\right ) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{7096320 a^3 \left (4 a^3 x^3-3 a b x\right )+7096320 a^3 \sqrt {a^2 x^2-b} \left (4 a^2 x^2-b\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 317, normalized size = 0.49 \begin {gather*} \frac {10395 \, {\left (77 \, b^{3} + 128 \, b^{2}\right )} \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} + 1\right ) - 10395 \, {\left (77 \, b^{3} + 128 \, b^{2}\right )} \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} - 1\right ) + 2 \, {\left (1182720 \, a^{3} x^{3} + 224 \, {\left (3267 \, a^{2} b - 3200 \, a^{2}\right )} x^{2} - 365904 \, b^{2} + 30 \, {\left (17787 \, a b^{2} - 16384 \, a\right )} x - 2 \, {\left (591360 \, a^{2} x^{2} + 266805 \, b^{2} + 112 \, {\left (3267 \, a b + 3200 \, a\right )} x + 295680 \, b + 245760\right )} \sqrt {a^{2} x^{2} - b} - {\left (1300992 \, a^{3} x^{3} + 1008 \, {\left (847 \, a^{2} b - 640 \, a^{2}\right )} x^{2} - 426888 \, b^{2} + {\left (800415 \, a b^{2} + 354816 \, a b - 409600 \, a\right )} x - {\left (1300992 \, a^{2} x^{2} + 800415 \, b^{2} + 1008 \, {\left (847 \, a b + 640 \, a\right )} x + 1005312 \, b + 409600\right )} \sqrt {a^{2} x^{2} - b} - 860160 \, b - 655360\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 2007040 \, b - 1310720\right )} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}{14192640 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {1+\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx \end {gather*}
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 {x^{2}}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\,{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 = -0.00 \begin {gather*} \int \frac {x^2}{\sqrt {\sqrt {a\,x+\sqrt {a^2\,x^2-b}}+1}} \,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 {x^{2}}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________