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

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

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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]

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 = 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]

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

[Out]

-1/2*((1 + b)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (b^2*(640 + 793*b)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*
x^2]]])/(1024*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - ((5 + b)*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(3/2))/3 - (b^2*
(384 + 2279*b)*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(3/2))/(1536*(a*x + Sqrt[-b + a^2*x^2])) + 2*(1 + Sqrt[a*x
+ Sqrt[-b + a^2*x^2]])^(5/2) + (3481*b^3*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(5/2))/(1920*(a*x + Sqrt[-b + a
^2*x^2])^(3/2)) - (10*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(7/2))/7 - (417*b^3*(1 + Sqrt[a*x + Sqrt[-b + a^2*x
^2]])^(7/2))/(320*(a*x + Sqrt[-b + a^2*x^2])^2) + (5*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(9/2))/9 + (61*b^3*(
1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(9/2))/(120*(a*x + Sqrt[-b + a^2*x^2])^(5/2)) - (1 + Sqrt[a*x + Sqrt[-b +
a^2*x^2]])^(11/2)/11 - (b^3*(1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]])^(11/2))/(12*(a*x + Sqrt[-b + a^2*x^2])^3) + (
3*b^2*(128 + 77*b)*Log[1 - 1/Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/2048 - (3*b^2*(128 + 77*b)*Log[1 + 1/S
qrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/2048)/a^3

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IntegrateAlgebraic [A]  time = 2.14, size = 650, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-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)*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (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)*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]*((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)*Sqrt[1 +
Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (-655360*b - 860160*b^2 - 426888*b^3 - 1638400*a*b*x - 2661120*a*b^2*x - 160
0830*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)*Sqrt[a*x + Sqrt[-b + a^
2*x^2]]*Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]))/(7096320*a^3*Sqrt[-b + a^2*x^2]*(-b + 4*a^2*x^2) + 7096320*
a^3*(-3*a*b*x + 4*a^3*x^3)) + (3*b^2*ArcTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(16*a^3) + (231*b^3*Ar
cTanh[Sqrt[1 + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]])/(2048*a^3)

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fricas [A]  time = 0.65, 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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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]

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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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*} \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]

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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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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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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