3.32.15 \(\int \frac {x^2}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3115]

Optimal. Leaf size=650 \[ \frac {\left (-491520 b^2-591360 b^3-533610 b^4+3932160 a b x+5304320 a b^2 x+365904 a b^3 x+3932160 a^2 b x^2+1774080 a^2 b^2 x^2+1067220 a^2 b^3 x^2-5242880 a^3 x^3-2293760 a^3 b x^3-3932160 a^4 x^4-5734400 a^5 x^5\right ) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (409600 b^2+1005312 b^3+800415 b^4-1966080 a b x-1935360 a b^2 x-426888 a b^3 x-3276800 a^2 b x^2-2661120 a^2 b^2 x^2-1600830 a^2 b^3 x^2+2621440 a^3 x^3-1720320 a^3 b x^3+3276800 a^4 x^4+5160960 a^5 x^5\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (1310720 b+2007040 b^2+365904 b^3+1966080 a b x+1774080 a b^2 x+1067220 a b^3 x-5242880 a^2 x^2-5160960 a^2 b x^2-3932160 a^3 x^3-5734400 a^4 x^4\right ) \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (-655360 b-860160 b^2-426888 b^3-1638400 a b x-2661120 a b^2 x-1600830 a b^3 x+2621440 a^2 x^2+860160 a^2 b x^2+3276800 a^3 x^3+5160960 a^4 x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{7096320 a^3 \sqrt {-b+a^2 x^2} \left (-b+4 a^2 x^2\right )+7096320 a^3 \left (-3 a b x+4 a^3 x^3\right )}+\frac {3 b^2 \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{16 a^3}+\frac {231 b^3 \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{2048 a^3} \]

[Out]

((-5734400*a^5*x^5-3932160*a^4*x^4-2293760*a^3*b*x^3+1067220*a^2*b^3*x^2-5242880*a^3*x^3+1774080*a^2*b^2*x^2+3
932160*a^2*b*x^2+365904*a*b^3*x+5304320*a*b^2*x-533610*b^4+3932160*a*b*x-591360*b^3-491520*b^2)*(1+(a*x+(a^2*x
^2-b)^(1/2))^(1/2))^(1/2)+(5160960*a^5*x^5+3276800*a^4*x^4-1720320*a^3*b*x^3-1600830*a^2*b^3*x^2+2621440*a^3*x
^3-2661120*a^2*b^2*x^2-3276800*a^2*b*x^2-426888*a*b^3*x-1935360*a*b^2*x+800415*b^4-1966080*a*b*x+1005312*b^3+4
09600*b^2)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(a^2*x^2-b)^(1/2)*((-5734400*
a^4*x^4-3932160*a^3*x^3-5160960*a^2*b*x^2+1067220*a*b^3*x-5242880*a^2*x^2+1774080*a*b^2*x+1966080*a*b*x+365904
*b^3+2007040*b^2+1310720*b)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(5160960*a^4*x^4+3276800*a^3*x^3+860160*a^
2*b*x^2-1600830*a*b^3*x+2621440*a^2*x^2-2661120*a*b^2*x-1638400*a*b*x-426888*b^3-860160*b^2-655360*b)*(a*x+(a^
2*x^2-b)^(1/2))^(1/2)*(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)))/(7096320*a^3*(a^2*x^2-b)^(1/2)*(4*a^2*x^2-b)+7
096320*a^3*(4*a^3*x^3-3*a*b*x))+3/16*b^2*arctanh((1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2))/a^3+231/2048*b^3*arc
tanh((1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2))/a^3

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Rubi [F]
time = 0.20, 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]

Int[x^2/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

Defer[Int][x^2/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]], x]

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*}

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Mathematica [A]
time = 2.31, size = 514, normalized size = 0.79 \begin {gather*} \frac {\frac {\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}} \left (-533610 b^4-327680 a b x \left (-12-12 a x+7 a^2 x^2\right )-163840 a^3 x^3 \left (32+24 a x+35 a^2 x^2\right )+2560 b^2 \left (-192+2072 a x+693 a^2 x^2\right )+924 b^3 \left (-640+396 a x+1155 a^2 x^2\right )+\sqrt {a x+\sqrt {-b+a^2 x^2}} \left (800415 b^4-81920 a b x \left (24+40 a x+21 a^2 x^2\right )+81920 a^3 x^3 \left (32+40 a x+63 a^2 x^2\right )-1280 b^2 \left (-320+1512 a x+2079 a^2 x^2\right )-462 b^3 \left (-2176+924 a x+3465 a^2 x^2\right )\right )+2 \sqrt {-b+a^2 x^2} \left (655360 b+1003520 b^2+182952 b^3+983040 a b x+887040 a b^2 x+533610 a b^3 x-2621440 a^2 x^2-2580480 a^2 b x^2-1966080 a^3 x^3-2867200 a^4 x^4+\sqrt {a x+\sqrt {-b+a^2 x^2}} \left (-53361 b^3 (4+15 a x)-13440 b^2 (32+99 a x)+20480 b \left (-16-40 a x+21 a^2 x^2\right )+40960 a^2 x^2 \left (32+40 a x+63 a^2 x^2\right )\right )\right )\right )}{-3 a b x+4 a^3 x^3-\left (b-4 a^2 x^2\right ) \sqrt {-b+a^2 x^2}}+1330560 b^2 \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )+800415 b^3 \tanh ^{-1}\left (\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{7096320 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]],x]

[Out]

((Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]*(-533610*b^4 - 327680*a*b*x*(-12 - 12*a*x + 7*a^2*x^2) - 163840*a^3
*x^3*(32 + 24*a*x + 35*a^2*x^2) + 2560*b^2*(-192 + 2072*a*x + 693*a^2*x^2) + 924*b^3*(-640 + 396*a*x + 1155*a^
2*x^2) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(800415*b^4 - 81920*a*b*x*(24 + 40*a*x + 21*a^2*x^2) + 81920*a^3*x^3*(
32 + 40*a*x + 63*a^2*x^2) - 1280*b^2*(-320 + 1512*a*x + 2079*a^2*x^2) - 462*b^3*(-2176 + 924*a*x + 3465*a^2*x^
2)) + 2*Sqrt[-b + a^2*x^2]*(655360*b + 1003520*b^2 + 182952*b^3 + 983040*a*b*x + 887040*a*b^2*x + 533610*a*b^3
*x - 2621440*a^2*x^2 - 2580480*a^2*b*x^2 - 1966080*a^3*x^3 - 2867200*a^4*x^4 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]*
(-53361*b^3*(4 + 15*a*x) - 13440*b^2*(32 + 99*a*x) + 20480*b*(-16 - 40*a*x + 21*a^2*x^2) + 40960*a^2*x^2*(32 +
40*a*x + 63*a^2*x^2)))))/(-3*a*b*x + 4*a^3*x^3 - (b - 4*a^2*x^2)*Sqrt[-b + a^2*x^2]) + 1330560*b^2*ArcTanh[Sq
rt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]] + 800415*b^3*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(70963
20*a^3)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sqrt {1+\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/(1+(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(x^2/(1+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1), x)

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Fricas [A]
time = 0.44, 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]

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

[Out]

1/14192640*(10395*(77*b^3 + 128*b^2)*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) + 1) - 10395*(77*b^3 + 128*b^
2)*log(sqrt(sqrt(a*x + sqrt(a^2*x^2 - b)) + 1) - 1) + 2*(1182720*a^3*x^3 + 224*(3267*a^2*b - 3200*a^2)*x^2 - 3
65904*b^2 + 30*(17787*a*b^2 - 16384*a)*x - 2*(591360*a^2*x^2 + 266805*b^2 + 112*(3267*a*b + 3200*a)*x + 295680
*b + 245760)*sqrt(a^2*x^2 - b) - (1300992*a^3*x^3 + 1008*(847*a^2*b - 640*a^2)*x^2 - 426888*b^2 + (800415*a*b^
2 + 354816*a*b - 409600*a)*x - (1300992*a^2*x^2 + 800415*b^2 + 1008*(847*a*b + 640*a)*x + 1005312*b + 409600)*
sqrt(a^2*x^2 - b) - 860160*b - 655360)*sqrt(a*x + sqrt(a^2*x^2 - b)) - 2007040*b - 1310720)*sqrt(sqrt(a*x + sq
rt(a^2*x^2 - b)) + 1))/a^3

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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]

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

[Out]

Integral(x**2/sqrt(sqrt(a*x + sqrt(a**2*x**2 - b)) + 1), x)

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Giac [F(-1)] Timed out
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]

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

[Out]

Timed out

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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]

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

[Out]

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

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