Optimal. Leaf size=1202 \[ -\frac {33 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}}{\sqrt {c}}\right ) b^4}{8192 a c^{13/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}}{\sqrt {c}}\right ) b^3}{16 a c^{5/2}}+\frac {\sqrt {a x+\sqrt {a^2 x^2+b}} \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}} \left (1879048192 a^3 x^3 c^{15}+1409286144 a b x c^{15}+1409286144 a^4 x^4 c^{13}+176160768 b^2 c^{13}+1409286144 a^2 b x^2 c^{13}+2055208960 a^5 x^5 c^{11}+20180893696 a^3 b x^3 c^{11}+13851164672 a b^2 x c^{11}+3391094784 a^6 x^6 c^9+18295554048 a^4 b x^4 c^9+1757085696 b^3 c^9+15116402688 a^2 b^2 x^2 c^9+5904138240 a^7 x^7 c^7+29595238400 a^5 b x^5 c^7+267624448000 a^3 b^2 x^3 c^7+185450137600 a b^3 x c^7-5143607040 b^4 c^5-9932482560 a^2 b^3 x^2 c^5-219490128 a b^4 x c^3-320089770 b^5 c-640179540 a^2 b^4 x^2 c\right )+\sqrt {a^2 x^2+b} \left (\sqrt {a x+\sqrt {a^2 x^2+b}} \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}} \left (1879048192 a^2 x^2 c^{15}+469762048 b c^{15}+1409286144 a^3 x^3 c^{13}+704643072 a b x c^{13}+2055208960 a^4 x^4 c^{11}+4531421184 b^2 c^{11}+19153289216 a^2 b x^2 c^{11}+3391094784 a^5 x^5 c^9+16600006656 a^3 b x^3 c^9+7240286208 a b^2 x c^9+5904138240 a^6 x^6 c^7+26643169280 a^4 b x^4 c^7+60891084800 b^3 c^7+255040880640 a^2 b^2 x^2 c^7-9932482560 a b^3 x c^5-219490128 b^4 c^3-640179540 a b^4 x c\right )+\left (-1879048192 a^3 x^3 c^{14}-939524096 a b x c^{14}-2348810240 a^4 x^4 c^{12}-146800640 b^2 c^{12}-1761607680 a^2 b x^2 c^{12}-3699376128 a^5 x^5 c^{10}-21311258624 a^3 b x^3 c^{10}-9499574272 a b^2 x c^{10}-6297747456 a^6 x^6 c^8-29887037440 a^4 b x^4 c^8-1474330624 b^3 c^8-18872795136 a^2 b^2 x^2 c^8+200740700160 a^7 x^7 c^6+647844986880 a^5 b x^5 c^6+948956037120 a^3 b^2 x^3 c^6+276208558080 a b^3 x c^6+7644464256 b^4 c^4+29797447680 a^2 b^3 x^2 c^4+512143632 a b^4 x c^2+480134655 b^5+1920538620 a^2 b^4 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}\right )+\left (-1879048192 a^4 x^4 c^{14}-234881024 b^2 c^{14}-1879048192 a^2 b x^2 c^{14}-2348810240 a^5 x^5 c^{12}-2936012800 a^3 b x^3 c^{12}-734003200 a b^2 x c^{12}-3699376128 a^6 x^6 c^{10}-23160946688 a^4 b x^4 c^{10}-2317090816 b^3 c^{10}-19692781568 a^2 b^2 x^2 c^{10}-6297747456 a^7 x^7 c^8-33035911168 a^5 b x^5 c^8-33029095424 a^3 b^2 x^3 c^8-7568457728 a b^3 x c^8+200740700160 a^8 x^8 c^6+748215336960 a^6 b x^6 c^6+50005263360 b^4 c^6+1247785943040 a^4 b^2 x^4 c^6+682252247040 a^2 b^3 x^2 c^6+29797447680 a^3 b^3 x^3 c^4+22543188096 a b^4 x c^4+256071816 b^5 c^2+512143632 a^2 b^4 x^2 c^2+1920538620 a^3 b^4 x^3+1440403965 a b^5 x\right ) \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}}{119189790720 a c^6 \left (a x+\sqrt {a^2 x^2+b}\right )^{7/2}} \]
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Rubi [F] time = 0.88, 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 \left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ \end {align*}
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Mathematica [F] time = 22.26, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 4.57, size = 1202, normalized size = 1.00 \begin {gather*} -\frac {33 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}}{\sqrt {c}}\right ) b^4}{8192 a c^{13/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}}{\sqrt {c}}\right ) b^3}{16 a c^{5/2}}+\frac {\sqrt {a x+\sqrt {a^2 x^2+b}} \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}} \left (1879048192 a^3 x^3 c^{15}+1409286144 a b x c^{15}+1409286144 a^4 x^4 c^{13}+176160768 b^2 c^{13}+1409286144 a^2 b x^2 c^{13}+2055208960 a^5 x^5 c^{11}+20180893696 a^3 b x^3 c^{11}+13851164672 a b^2 x c^{11}+3391094784 a^6 x^6 c^9+18295554048 a^4 b x^4 c^9+1757085696 b^3 c^9+15116402688 a^2 b^2 x^2 c^9+5904138240 a^7 x^7 c^7+29595238400 a^5 b x^5 c^7+267624448000 a^3 b^2 x^3 c^7+185450137600 a b^3 x c^7-5143607040 b^4 c^5-9932482560 a^2 b^3 x^2 c^5-219490128 a b^4 x c^3-320089770 b^5 c-640179540 a^2 b^4 x^2 c\right )+\sqrt {a^2 x^2+b} \left (\sqrt {a x+\sqrt {a^2 x^2+b}} \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}} \left (1879048192 a^2 x^2 c^{15}+469762048 b c^{15}+1409286144 a^3 x^3 c^{13}+704643072 a b x c^{13}+2055208960 a^4 x^4 c^{11}+4531421184 b^2 c^{11}+19153289216 a^2 b x^2 c^{11}+3391094784 a^5 x^5 c^9+16600006656 a^3 b x^3 c^9+7240286208 a b^2 x c^9+5904138240 a^6 x^6 c^7+26643169280 a^4 b x^4 c^7+60891084800 b^3 c^7+255040880640 a^2 b^2 x^2 c^7-9932482560 a b^3 x c^5-219490128 b^4 c^3-640179540 a b^4 x c\right )+\left (-1879048192 a^3 x^3 c^{14}-939524096 a b x c^{14}-2348810240 a^4 x^4 c^{12}-146800640 b^2 c^{12}-1761607680 a^2 b x^2 c^{12}-3699376128 a^5 x^5 c^{10}-21311258624 a^3 b x^3 c^{10}-9499574272 a b^2 x c^{10}-6297747456 a^6 x^6 c^8-29887037440 a^4 b x^4 c^8-1474330624 b^3 c^8-18872795136 a^2 b^2 x^2 c^8+200740700160 a^7 x^7 c^6+647844986880 a^5 b x^5 c^6+948956037120 a^3 b^2 x^3 c^6+276208558080 a b^3 x c^6+7644464256 b^4 c^4+29797447680 a^2 b^3 x^2 c^4+512143632 a b^4 x c^2+480134655 b^5+1920538620 a^2 b^4 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}\right )+\left (-1879048192 a^4 x^4 c^{14}-234881024 b^2 c^{14}-1879048192 a^2 b x^2 c^{14}-2348810240 a^5 x^5 c^{12}-2936012800 a^3 b x^3 c^{12}-734003200 a b^2 x c^{12}-3699376128 a^6 x^6 c^{10}-23160946688 a^4 b x^4 c^{10}-2317090816 b^3 c^{10}-19692781568 a^2 b^2 x^2 c^{10}-6297747456 a^7 x^7 c^8-33035911168 a^5 b x^5 c^8-33029095424 a^3 b^2 x^3 c^8-7568457728 a b^3 x c^8+200740700160 a^8 x^8 c^6+748215336960 a^6 b x^6 c^6+50005263360 b^4 c^6+1247785943040 a^4 b^2 x^4 c^6+682252247040 a^2 b^3 x^2 c^6+29797447680 a^3 b^3 x^3 c^4+22543188096 a b^4 x c^4+256071816 b^5 c^2+512143632 a^2 b^4 x^2 c^2+1920538620 a^3 b^4 x^3+1440403965 a b^5 x\right ) \sqrt {c+\sqrt {a x+\sqrt {a^2 x^2+b}}}}{119189790720 a c^6 \left (a x+\sqrt {a^2 x^2+b}\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 1193, normalized size = 0.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a^{2} x^{2}+b \right )^{\frac {3}{2}} \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx \end {gather*}
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 {\left (a^{2} x^{2} + b\right )}^{\frac {3}{2}} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \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 = -0.00 \begin {gather*} \int \sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,{\left (a^2\,x^2+b\right )}^{3/2}\,\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}} \,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 \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \left (a^{2} x^{2} + b\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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