3.32.42 \(\int (b+a^2 x^2)^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [3142]

Optimal. Leaf size=1202 \[ \frac {\left (256071816 b^5 c^2+50005263360 b^4 c^6-2317090816 b^3 c^{10}-234881024 b^2 c^{14}[phone] a b^5 x+22543188096 a b^4 c^4 x-7568457728 a b^3 c^8 x-734003200 a b^2 c^{12} x+512143632 a^2 b^4 c^2 x^2+682252247040 a^2 b^3 c^6 x^2-19692781568 a^2 b^2 c^{10} x^2-1879048192 a^2 b c^{14} x^2+1920538620 a^3 b^4 x^3+29797447680 a^3 b^3 c^4 x^3-33029095424 a^3 b^2 c^8 x^3-2936012800 a^3 b c^{12} x^3+1247785943040 a^4 b^2 c^6 x^4-23160946688 a^4 b c^{10} x^4-1879048192 a^4 c^{14} x^4-33035911168 a^5 b c^8 x^5-2348810240 a^5 c^{12} x^5+748215336960 a^6 b c^6 x^6-3699376128 a^6 c^{10} x^6-6297747456 a^7 c^8 x^7+200740700160 a^8 c^6 x^8\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-320089770 b^5 c-5143607040 b^4 c^5+1757085696 b^3 c^9+176160768 b^2 c^{13}-219490128 a b^4 c^3 x+185450137600 a b^3 c^7 x+13851164672 a b^2 c^{11} x+1409286144 a b c^{15} x-640179540 a^2 b^4 c x^2-9932482560 a^2 b^3 c^5 x^2+15116402688 a^2 b^2 c^9 x^2+1409286144 a^2 b c^{13} x^2+267624448000 a^3 b^2 c^7 x^3+20180893696 a^3 b c^{11} x^3+1879048192 a^3 c^{15} x^3+18295554048 a^4 b c^9 x^4+1409286144 a^4 c^{13} x^4+29595238400 a^5 b c^7 x^5+2055208960 a^5 c^{11} x^5+3391094784 a^6 c^9 x^6+5904138240 a^7 c^7 x^7\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (480134655 b^5+7644464256 b^4 c^4-1474330624 b^3 c^8-146800640 b^2 c^{12}[phone] a b^4 c^2 x+276208558080 a b^3 c^6 x-9499574272 a b^2 c^{10} x-939524096 a b c^{14} x+1920538620 a^2 b^4 x^2+29797447680 a^2 b^3 c^4 x^2-18872795136 a^2 b^2 c^8 x^2-1761607680 a^2 b c^{12} x^2+948956037120 a^3 b^2 c^6 x^3-21311258624 a^3 b c^{10} x^3-1879048192 a^3 c^{14} x^3-29887037440 a^4 b c^8 x^4-2348810240 a^4 c^{12} x^4+647844986880 a^5 b c^6 x^5-3699376128 a^5 c^{10} x^5-6297747456 a^6 c^8 x^6+200740700160 a^7 c^6 x^7\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-219490128 b^4 c^3+60891084800 b^3 c^7+4531421184 b^2 c^{11}[phone] b c^{15}-640179540 a b^4 c x-9932482560 a b^3 c^5 x+7240286208 a b^2 c^9 x+704643072 a b c^{13} x+255040880640 a^2 b^2 c^7 x^2+19153289216 a^2 b c^{11} x^2+1879048192 a^2 c^{15} x^2+16600006656 a^3 b c^9 x^3+1409286144 a^3 c^{13} x^3+26643169280 a^4 b c^7 x^4+2055208960 a^4 c^{11} x^4+3391094784 a^5 c^9 x^5+5904138240 a^6 c^7 x^6\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{119189790720 a c^6 \left (a x+\sqrt {b+a^2 x^2}\right )^{7/2}}-\frac {33 b^4 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{8192 a c^{13/2}}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{5/2}} \]

[Out]

1/119189790720*((200740700160*a^8*c^6*x^8-6297747456*a^7*c^8*x^7-3699376128*a^6*c^10*x^6-2348810240*a^5*c^12*x
^5-1879048192*a^4*c^14*x^4+748215336960*a^6*b*c^6*x^6-33035911168*a^5*b*c^8*x^5-23160946688*a^4*b*c^10*x^4-293
6012800*a^3*b*c^12*x^3-1879048192*a^2*b*c^14*x^2+1247785943040*a^4*b^2*c^6*x^4-33029095424*a^3*b^2*c^8*x^3-196
92781568*a^2*b^2*c^10*x^2-734003200*a*b^2*c^12*x-234881024*b^2*c^14+29797447680*a^3*b^3*c^4*x^3+682252247040*a
^2*b^3*c^6*x^2-7568457728*a*b^3*c^8*x-2317090816*b^3*c^10+1920538620*a^3*b^4*x^3+512143632*a^2*b^4*c^2*x^2+225
43188096*a*b^4*c^4*x+50005263360*b^4*c^6+1440403965*a*b^5*x+256071816*b^5*c^2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2
))^(1/2)+(5904138240*a^7*c^7*x^7+3391094784*a^6*c^9*x^6+2055208960*a^5*c^11*x^5+1409286144*a^4*c^13*x^4+187904
8192*a^3*c^15*x^3+29595238400*a^5*b*c^7*x^5+18295554048*a^4*b*c^9*x^4+20180893696*a^3*b*c^11*x^3+1409286144*a^
2*b*c^13*x^2+1409286144*a*b*c^15*x+267624448000*a^3*b^2*c^7*x^3+15116402688*a^2*b^2*c^9*x^2+13851164672*a*b^2*
c^11*x+176160768*b^2*c^13-9932482560*a^2*b^3*c^5*x^2+185450137600*a*b^3*c^7*x+1757085696*b^3*c^9-640179540*a^2
*b^4*c*x^2-219490128*a*b^4*c^3*x-5143607040*b^4*c^5-320089770*b^5*c)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^
2*x^2+b)^(1/2))^(1/2))^(1/2)+(a^2*x^2+b)^(1/2)*((200740700160*a^7*c^6*x^7-6297747456*a^6*c^8*x^6-3699376128*a^
5*c^10*x^5-2348810240*a^4*c^12*x^4-1879048192*a^3*c^14*x^3+647844986880*a^5*b*c^6*x^5-29887037440*a^4*b*c^8*x^
4-21311258624*a^3*b*c^10*x^3-1761607680*a^2*b*c^12*x^2-939524096*a*b*c^14*x+948956037120*a^3*b^2*c^6*x^3-18872
795136*a^2*b^2*c^8*x^2-9499574272*a*b^2*c^10*x-146800640*b^2*c^12+29797447680*a^2*b^3*c^4*x^2+276208558080*a*b
^3*c^6*x-1474330624*b^3*c^8+1920538620*a^2*b^4*x^2+512143632*a*b^4*c^2*x+7644464256*b^4*c^4+480134655*b^5)*(c+
(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(5904138240*a^6*c^7*x^6+3391094784*a^5*c^9*x^5+2055208960*a^4*c^11*x^4+14
09286144*a^3*c^13*x^3+1879048192*a^2*c^15*x^2+26643169280*a^4*b*c^7*x^4+16600006656*a^3*b*c^9*x^3+19153289216*
a^2*b*c^11*x^2+704643072*a*b*c^13*x+469762048*b*c^15+255040880640*a^2*b^2*c^7*x^2+7240286208*a*b^2*c^9*x+45314
21184*b^2*c^11-9932482560*a*b^3*c^5*x+60891084800*b^3*c^7-640179540*a*b^4*c*x-219490128*b^4*c^3)*(a*x+(a^2*x^2
+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)))/a/c^6/(a*x+(a^2*x^2+b)^(1/2))^(7/2)-33/8192*b^4*arc
tanh((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(13/2)-1/16*b^3*arctanh((c+(a*x+(a^2*x^2+b)^(1/2))^(
1/2))^(1/2)/c^(1/2))/a/c^(5/2)

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Rubi [F]
time = 0.68, 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.

[In]

Int[(b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]

[Out]

Defer[Int][(b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]], x]

Rubi steps

\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 [A]
time = 5.43, size = 1202, normalized size = 1.00 \begin {gather*} \frac {\left (256071816 b^5 c^2+50005263360 b^4 c^6-2317090816 b^3 c^{10}-234881024 b^2 c^{14}[phone] a b^5 x+22543188096 a b^4 c^4 x-7568457728 a b^3 c^8 x-734003200 a b^2 c^{12} x+512143632 a^2 b^4 c^2 x^2+682252247040 a^2 b^3 c^6 x^2-19692781568 a^2 b^2 c^{10} x^2-1879048192 a^2 b c^{14} x^2+1920538620 a^3 b^4 x^3+29797447680 a^3 b^3 c^4 x^3-33029095424 a^3 b^2 c^8 x^3-2936012800 a^3 b c^{12} x^3+1247785943040 a^4 b^2 c^6 x^4-23160946688 a^4 b c^{10} x^4-1879048192 a^4 c^{14} x^4-33035911168 a^5 b c^8 x^5-2348810240 a^5 c^{12} x^5+748215336960 a^6 b c^6 x^6-3699376128 a^6 c^{10} x^6-6297747456 a^7 c^8 x^7+200740700160 a^8 c^6 x^8\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-320089770 b^5 c-5143607040 b^4 c^5+1757085696 b^3 c^9+176160768 b^2 c^{13}-219490128 a b^4 c^3 x+185450137600 a b^3 c^7 x+13851164672 a b^2 c^{11} x+1409286144 a b c^{15} x-640179540 a^2 b^4 c x^2-9932482560 a^2 b^3 c^5 x^2+15116402688 a^2 b^2 c^9 x^2+1409286144 a^2 b c^{13} x^2+267624448000 a^3 b^2 c^7 x^3+20180893696 a^3 b c^{11} x^3+1879048192 a^3 c^{15} x^3+18295554048 a^4 b c^9 x^4+1409286144 a^4 c^{13} x^4+29595238400 a^5 b c^7 x^5+2055208960 a^5 c^{11} x^5+3391094784 a^6 c^9 x^6+5904138240 a^7 c^7 x^7\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (480134655 b^5+7644464256 b^4 c^4-1474330624 b^3 c^8-146800640 b^2 c^{12}[phone] a b^4 c^2 x+276208558080 a b^3 c^6 x-9499574272 a b^2 c^{10} x-939524096 a b c^{14} x+1920538620 a^2 b^4 x^2+29797447680 a^2 b^3 c^4 x^2-18872795136 a^2 b^2 c^8 x^2-1761607680 a^2 b c^{12} x^2+948956037120 a^3 b^2 c^6 x^3-21311258624 a^3 b c^{10} x^3-1879048192 a^3 c^{14} x^3-29887037440 a^4 b c^8 x^4-2348810240 a^4 c^{12} x^4+647844986880 a^5 b c^6 x^5-3699376128 a^5 c^{10} x^5-6297747456 a^6 c^8 x^6+200740700160 a^7 c^6 x^7\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-219490128 b^4 c^3+60891084800 b^3 c^7+4531421184 b^2 c^{11}[phone] b c^{15}-640179540 a b^4 c x-9932482560 a b^3 c^5 x+7240286208 a b^2 c^9 x+704643072 a b c^{13} x+255040880640 a^2 b^2 c^7 x^2+19153289216 a^2 b c^{11} x^2+1879048192 a^2 c^{15} x^2+16600006656 a^3 b c^9 x^3+1409286144 a^3 c^{13} x^3+26643169280 a^4 b c^7 x^4+2055208960 a^4 c^{11} x^4+3391094784 a^5 c^9 x^5+5904138240 a^6 c^7 x^6\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{119189790720 a c^6 \left (a x+\sqrt {b+a^2 x^2}\right )^{7/2}}-\frac {33 b^4 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{8192 a c^{13/2}}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + a^2*x^2)^(3/2)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]

[Out]

((256071816*b^5*c^2 + 50005263360*b^4*c^6 - 2317090816*b^3*c^10 - 234881024*b^2*c^14 + 1440403965*a*b^5*x + 22
543188096*a*b^4*c^4*x - 7568457728*a*b^3*c^8*x - 734003200*a*b^2*c^12*x + 512143632*a^2*b^4*c^2*x^2 + 68225224
7040*a^2*b^3*c^6*x^2 - 19692781568*a^2*b^2*c^10*x^2 - 1879048192*a^2*b*c^14*x^2 + 1920538620*a^3*b^4*x^3 + 297
97447680*a^3*b^3*c^4*x^3 - 33029095424*a^3*b^2*c^8*x^3 - 2936012800*a^3*b*c^12*x^3 + 1247785943040*a^4*b^2*c^6
*x^4 - 23160946688*a^4*b*c^10*x^4 - 1879048192*a^4*c^14*x^4 - 33035911168*a^5*b*c^8*x^5 - 2348810240*a^5*c^12*
x^5 + 748215336960*a^6*b*c^6*x^6 - 3699376128*a^6*c^10*x^6 - 6297747456*a^7*c^8*x^7 + 200740700160*a^8*c^6*x^8
)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-320089770*b^5*c - 5143607040*b^4*c^5 + 1757085696*b^3*c^9 + 1761
60768*b^2*c^13 - 219490128*a*b^4*c^3*x + 185450137600*a*b^3*c^7*x + 13851164672*a*b^2*c^11*x + 1409286144*a*b*
c^15*x - 640179540*a^2*b^4*c*x^2 - 9932482560*a^2*b^3*c^5*x^2 + 15116402688*a^2*b^2*c^9*x^2 + 1409286144*a^2*b
*c^13*x^2 + 267624448000*a^3*b^2*c^7*x^3 + 20180893696*a^3*b*c^11*x^3 + 1879048192*a^3*c^15*x^3 + 18295554048*
a^4*b*c^9*x^4 + 1409286144*a^4*c^13*x^4 + 29595238400*a^5*b*c^7*x^5 + 2055208960*a^5*c^11*x^5 + 3391094784*a^6
*c^9*x^6 + 5904138240*a^7*c^7*x^7)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + Sqr
t[b + a^2*x^2]*((480134655*b^5 + 7644464256*b^4*c^4 - 1474330624*b^3*c^8 - 146800640*b^2*c^12 + 512143632*a*b^
4*c^2*x + 276208558080*a*b^3*c^6*x - 9499574272*a*b^2*c^10*x - 939524096*a*b*c^14*x + 1920538620*a^2*b^4*x^2 +
29797447680*a^2*b^3*c^4*x^2 - 18872795136*a^2*b^2*c^8*x^2 - 1761607680*a^2*b*c^12*x^2 + 948956037120*a^3*b^2*
c^6*x^3 - 21311258624*a^3*b*c^10*x^3 - 1879048192*a^3*c^14*x^3 - 29887037440*a^4*b*c^8*x^4 - 2348810240*a^4*c^
12*x^4 + 647844986880*a^5*b*c^6*x^5 - 3699376128*a^5*c^10*x^5 - 6297747456*a^6*c^8*x^6 + 200740700160*a^7*c^6*
x^7)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] + (-219490128*b^4*c^3 + 60891084800*b^3*c^7 + 4531421184*b^2*c^11
+ 469762048*b*c^15 - 640179540*a*b^4*c*x - 9932482560*a*b^3*c^5*x + 7240286208*a*b^2*c^9*x + 704643072*a*b*c^
13*x + 255040880640*a^2*b^2*c^7*x^2 + 19153289216*a^2*b*c^11*x^2 + 1879048192*a^2*c^15*x^2 + 16600006656*a^3*b
*c^9*x^3 + 1409286144*a^3*c^13*x^3 + 26643169280*a^4*b*c^7*x^4 + 2055208960*a^4*c^11*x^4 + 3391094784*a^5*c^9*
x^5 + 5904138240*a^6*c^7*x^6)*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]))/(1191897
90720*a*c^6*(a*x + Sqrt[b + a^2*x^2])^(7/2)) - (33*b^4*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]
])/(8192*a*c^(13/2)) - (b^3*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(16*a*c^(5/2))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2} x^{2}+b \right )^{\frac {3}{2}} \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2+b)^(3/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2),x)

[Out]

int((a^2*x^2+b)^(3/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2+b)^(3/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2),x, algorithm
="maxima")

[Out]

integrate((a^2*x^2 + b)^(3/2)*sqrt(a*x + sqrt(a^2*x^2 + b))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))), x)

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Fricas [A]
time = 0.45, size = 1193, normalized size = 0.99 \begin {gather*} \left [\frac {14549535 \, {\left (512 \, b^{3} c^{4} + 33 \, b^{4}\right )} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) + 2 \, {\left (469762048 \, c^{16} + 738017280 \, a^{4} c^{8} x^{4} + 4531421184 \, b c^{12} + 60891084800 \, b^{2} c^{8} - 219490128 \, b^{3} c^{4} + 33792 \, {\left (12544 \, a^{3} c^{10} + 20995 \, a^{3} b c^{6}\right )} x^{3} + 32 \, {\left (8028160 \, a^{2} c^{12} + 98309120 \, a^{2} b c^{8} - 13718133 \, a^{2} b^{2} c^{4}\right )} x^{2} + 6 \, {\left (29360128 \, a c^{14} + 328171520 \, a b c^{10} + 916389760 \, a b^{2} c^{6} + 53348295 \, a b^{3} c^{2}\right )} x + 2 \, {\left (88080384 \, c^{14} + 369008640 \, a^{3} c^{8} x^{3} + 878542848 \, b c^{10} - 2571803520 \, b^{2} c^{6} - 160044885 \, b^{3} c^{2} + 16896 \, {\left (12544 \, a^{2} c^{10} - 20995 \, a^{2} b c^{6}\right )} x^{2} + 16 \, {\left (8028160 \, a c^{12} + 86777600 \, a b c^{8} + 13718133 \, a b^{2} c^{4}\right )} x\right )} \sqrt {a^{2} x^{2} + b} - {\left (234881024 \, c^{15} + 4480819200 \, a^{4} c^{7} x^{4} + 2317090816 \, b c^{11} - 50005263360 \, b^{2} c^{7} - 256071816 \, b^{3} c^{3} + 219648 \, {\left (1792 \, a^{3} c^{9} + 3553 \, a^{3} b c^{5}\right )} x^{3} + 48 \, {\left (4816896 \, a^{2} c^{11} + 469580800 \, a^{2} b c^{7} - 10669659 \, a^{2} b^{2} c^{3}\right )} x^{2} + {\left (146800640 \, a c^{13} + 1671135232 \, a b c^{9} + 8034668928 \, a b^{2} c^{5} + 480134655 \, a b^{3} c\right )} x + {\left (146800640 \, c^{13} - 29573406720 \, a^{3} c^{7} x^{3} + 1474330624 \, b c^{9} - 7644464256 \, b^{2} c^{5} - 480134655 \, b^{3} c + 219648 \, {\left (1792 \, a^{2} c^{9} - 3553 \, a^{2} b c^{5}\right )} x^{2} + 48 \, {\left (4816896 \, a c^{11} - 1587239680 \, a b c^{7} + 10669659 \, a b^{2} c^{3}\right )} x\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{238379581440 \, a c^{7}}, \frac {14549535 \, {\left (512 \, b^{3} c^{4} + 33 \, b^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + {\left (469762048 \, c^{16} + 738017280 \, a^{4} c^{8} x^{4} + 4531421184 \, b c^{12} + 60891084800 \, b^{2} c^{8} - 219490128 \, b^{3} c^{4} + 33792 \, {\left (12544 \, a^{3} c^{10} + 20995 \, a^{3} b c^{6}\right )} x^{3} + 32 \, {\left (8028160 \, a^{2} c^{12} + 98309120 \, a^{2} b c^{8} - 13718133 \, a^{2} b^{2} c^{4}\right )} x^{2} + 6 \, {\left (29360128 \, a c^{14} + 328171520 \, a b c^{10} + 916389760 \, a b^{2} c^{6} + 53348295 \, a b^{3} c^{2}\right )} x + 2 \, {\left (88080384 \, c^{14} + 369008640 \, a^{3} c^{8} x^{3} + 878542848 \, b c^{10} - 2571803520 \, b^{2} c^{6} - 160044885 \, b^{3} c^{2} + 16896 \, {\left (12544 \, a^{2} c^{10} - 20995 \, a^{2} b c^{6}\right )} x^{2} + 16 \, {\left (8028160 \, a c^{12} + 86777600 \, a b c^{8} + 13718133 \, a b^{2} c^{4}\right )} x\right )} \sqrt {a^{2} x^{2} + b} - {\left (234881024 \, c^{15} + 4480819200 \, a^{4} c^{7} x^{4} + 2317090816 \, b c^{11} - 50005263360 \, b^{2} c^{7} - 256071816 \, b^{3} c^{3} + 219648 \, {\left (1792 \, a^{3} c^{9} + 3553 \, a^{3} b c^{5}\right )} x^{3} + 48 \, {\left (4816896 \, a^{2} c^{11} + 469580800 \, a^{2} b c^{7} - 10669659 \, a^{2} b^{2} c^{3}\right )} x^{2} + {\left (146800640 \, a c^{13} + 1671135232 \, a b c^{9} + 8034668928 \, a b^{2} c^{5} + 480134655 \, a b^{3} c\right )} x + {\left (146800640 \, c^{13} - 29573406720 \, a^{3} c^{7} x^{3} + 1474330624 \, b c^{9} - 7644464256 \, b^{2} c^{5} - 480134655 \, b^{3} c + 219648 \, {\left (1792 \, a^{2} c^{9} - 3553 \, a^{2} b c^{5}\right )} x^{2} + 48 \, {\left (4816896 \, a c^{11} - 1587239680 \, a b c^{7} + 10669659 \, a b^{2} c^{3}\right )} x\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{119189790720 \, a c^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2+b)^(3/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2),x, algorithm
="fricas")

[Out]

[1/238379581440*(14549535*(512*b^3*c^4 + 33*b^4)*sqrt(c)*log(2*(a*sqrt(c)*x - sqrt(a^2*x^2 + b)*sqrt(c))*sqrt(
a*x + sqrt(a^2*x^2 + b))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))) - 2*(a*c*x - sqrt(a^2*x^2 + b)*c)*sqrt(a*x +
sqrt(a^2*x^2 + b)) + b) + 2*(469762048*c^16 + 738017280*a^4*c^8*x^4 + 4531421184*b*c^12 + 60891084800*b^2*c^8
- 219490128*b^3*c^4 + 33792*(12544*a^3*c^10 + 20995*a^3*b*c^6)*x^3 + 32*(8028160*a^2*c^12 + 98309120*a^2*b*c^8
- 13718133*a^2*b^2*c^4)*x^2 + 6*(29360128*a*c^14 + 328171520*a*b*c^10 + 916389760*a*b^2*c^6 + 53348295*a*b^3*
c^2)*x + 2*(88080384*c^14 + 369008640*a^3*c^8*x^3 + 878542848*b*c^10 - 2571803520*b^2*c^6 - 160044885*b^3*c^2
+ 16896*(12544*a^2*c^10 - 20995*a^2*b*c^6)*x^2 + 16*(8028160*a*c^12 + 86777600*a*b*c^8 + 13718133*a*b^2*c^4)*x
)*sqrt(a^2*x^2 + b) - (234881024*c^15 + 4480819200*a^4*c^7*x^4 + 2317090816*b*c^11 - 50005263360*b^2*c^7 - 256
071816*b^3*c^3 + 219648*(1792*a^3*c^9 + 3553*a^3*b*c^5)*x^3 + 48*(4816896*a^2*c^11 + 469580800*a^2*b*c^7 - 106
69659*a^2*b^2*c^3)*x^2 + (146800640*a*c^13 + 1671135232*a*b*c^9 + 8034668928*a*b^2*c^5 + 480134655*a*b^3*c)*x
+ (146800640*c^13 - 29573406720*a^3*c^7*x^3 + 1474330624*b*c^9 - 7644464256*b^2*c^5 - 480134655*b^3*c + 219648
*(1792*a^2*c^9 - 3553*a^2*b*c^5)*x^2 + 48*(4816896*a*c^11 - 1587239680*a*b*c^7 + 10669659*a*b^2*c^3)*x)*sqrt(a
^2*x^2 + b))*sqrt(a*x + sqrt(a^2*x^2 + b)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^7), 1/119189790720*(
14549535*(512*b^3*c^4 + 33*b^4)*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))/c) + (4697620
48*c^16 + 738017280*a^4*c^8*x^4 + 4531421184*b*c^12 + 60891084800*b^2*c^8 - 219490128*b^3*c^4 + 33792*(12544*a
^3*c^10 + 20995*a^3*b*c^6)*x^3 + 32*(8028160*a^2*c^12 + 98309120*a^2*b*c^8 - 13718133*a^2*b^2*c^4)*x^2 + 6*(29
360128*a*c^14 + 328171520*a*b*c^10 + 916389760*a*b^2*c^6 + 53348295*a*b^3*c^2)*x + 2*(88080384*c^14 + 36900864
0*a^3*c^8*x^3 + 878542848*b*c^10 - 2571803520*b^2*c^6 - 160044885*b^3*c^2 + 16896*(12544*a^2*c^10 - 20995*a^2*
b*c^6)*x^2 + 16*(8028160*a*c^12 + 86777600*a*b*c^8 + 13718133*a*b^2*c^4)*x)*sqrt(a^2*x^2 + b) - (234881024*c^1
5 + 4480819200*a^4*c^7*x^4 + 2317090816*b*c^11 - 50005263360*b^2*c^7 - 256071816*b^3*c^3 + 219648*(1792*a^3*c^
9 + 3553*a^3*b*c^5)*x^3 + 48*(4816896*a^2*c^11 + 469580800*a^2*b*c^7 - 10669659*a^2*b^2*c^3)*x^2 + (146800640*
a*c^13 + 1671135232*a*b*c^9 + 8034668928*a*b^2*c^5 + 480134655*a*b^3*c)*x + (146800640*c^13 - 29573406720*a^3*
c^7*x^3 + 1474330624*b*c^9 - 7644464256*b^2*c^5 - 480134655*b^3*c + 219648*(1792*a^2*c^9 - 3553*a^2*b*c^5)*x^2
+ 48*(4816896*a*c^11 - 1587239680*a*b*c^7 + 10669659*a*b^2*c^3)*x)*sqrt(a^2*x^2 + b))*sqrt(a*x + sqrt(a^2*x^2
+ b)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^7)]

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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.

[In]

integrate((a**2*x**2+b)**(3/2)*(a*x+(a**2*x**2+b)**(1/2))**(1/2)*(c+(a*x+(a**2*x**2+b)**(1/2))**(1/2))**(1/2),
x)

[Out]

Integral(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 + b)))*sqrt(a*x + sqrt(a**2*x**2 + b))*(a**2*x**2 + b)**(3/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2+b)^(3/2)*(a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2),x, algorithm
="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(4*

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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.

[In]

int(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(b + a^2*x^2)^(3/2)*(c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2),x)

[Out]

int(((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(b + a^2*x^2)^(3/2)*(c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2), x)

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