3.29.31 \(\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{(b+a^2 x^2)^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [2831]

Optimal. Leaf size=287 \[ \frac {a x \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a^2 b x \sqrt {b+a^2 x^2}+a b \left (b+a^2 x^2\right )}+\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {-c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c \text {$\#$1}-\text {$\#$1}^3}\& \right ]}{4 a b} \]

[Out]

Unintegrable

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Rubi [F]
time = 0.47, 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 {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx &=\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 210, normalized size = 0.73 \begin {gather*} -\frac {-\frac {4 \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a x \left (a x+\sqrt {b+a^2 x^2}\right )}+\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c \text {$\#$1}-\text {$\#$1}^3}\&\right ]}{4 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((-4*(a*x + Sqrt[b + a^2*x^2])^(3/2)*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a*x*(a*x + Sqrt[b + a^
2*x^2])) + RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b +
a^2*x^2]]] - #1] + Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1]*#1^2)/(c*#1 - #1^3) & ])/(a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 51.45, size = 1089073, normalized size = 3794.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(1/2)*sqrt(1/6)*(a^3*b*x^2 + a*b^2)*sqrt((24*c^3 + 6*sqrt(1/2)*sqrt(1/6)*(a^2*b^2*c^4 + a^2*b^3)*sqr
t(-(1536*c^10 + 2400*b*c^6 + 1056*b^2*c^2 + (a^4*b^5*c^8 + 2*a^4*b^6*c^4 + a^4*b^7)*(2*(1/2)^(2/3)*(-I*sqrt(3)
+ 1)*((4*c^5 - 3*b*c^3*sqrt(-1/b) - (b^3*c^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2
*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))) + b^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1
/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))))*a^2*sqrt(-1/b) + 3*b*c)^2*b/((b^3*c^4 + b^4)^2*a^4) - 3
*(8*c^4 + 2*(4*b^2*c^5*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b)
+ a^4*b^6*sqrt(-1/b))) - b^3*c^3*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4
*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))*sqrt(-1/b) + 3*b^3*c*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt
(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))))*a^2 - (16*c^6 + 11*b*c^2)*sqrt(-1/b) - b)/((b^5*c^4
+ b^6)*a^4*sqrt(-1/b)))/(9*(4*c^5 - 3*b*c^3*sqrt(-1/b) - (b^3*c^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^
4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))) + b^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/
(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))))*a^2*sqrt(-1/b) + 3*b*c)*(8*c^4 + 2*
(4*b^2*c^5*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*s
qrt(-1/b))) - b^3*c^3*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b)
+ a^4*b^6*sqrt(-1/b)))*sqrt(-1/b) + 3*b^3*c*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a
^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))))*a^2 - (16*c^6 + 11*b*c^2)*sqrt(-1/b) - b)*b/((b^5*c^4 + b^6)*(b^
3*c^4 + b^4)*a^6) - 27*(64*c^11 + 84*b*c^7 - (b^8*c^8*(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b)
+ 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))^(3/2) + 2*b^9*c^4*(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4
*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))^(3/2) + b^10*(-(c^4 + 2*b*c^2*sqrt(-1/b) - b
)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))^(3/2))*a^6*sqrt(-1/b) + 20*b^2*c^3
+ (8*b^3*c^8*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^
6*sqrt(-1/b))) + 6*b^4*c^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-
1/b) + a^4*b^6*sqrt(-1/b))) + (16*b^3*c^10*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^
4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))) + 23*b^4*c^6*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqr
t(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))) + 5*b^5*c^2*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^
4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))))*sqrt(-1/b))*a^2 - (16*b*c^9 + 13*b^2*c
^5 - 3*b^3*c)*sqrt(-1/b))/((b^8*c^8 + 2*b^9*c^4 + b^10)*a^6*sqrt(-1/b)) - 2*(4*c^5 - 3*b*c^3*sqrt(-1/b) - (b^3
*c^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1
/b))) + b^4*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*
sqrt(-1/b))))*a^2*sqrt(-1/b) + 3*b*c)^3/((b^3*c^4 + b^4)^3*a^6*(-1/b)^(3/2)) + sqrt(837*(c^4 + 2*b*c^2*sqrt(-1
/b) - b)^3*a^12*b^21/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))^3 - 49152*(8*a^2
*b^6*(-1/b)^(7/2) - 3*a^2*b^5*(-1/b)^(5/2) - 12*a^2*b^4*(-1/b)^(3/2) - a^2*b^3*sqrt(-1/b))*c^29*sqrt(-(c^4 + 2
*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))) - 256*(56*a^6
*b^12*(-1/b)^(9/2) + 216*a^6*b^11*(-1/b)^(7/2) + 693*a^6*b^10*(-1/b)^(5/2) + 317*a^6*b^9*(-1/b)^(3/2))*c^27*(-
(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))^(3/2)
- 1179648*(b^4*(-1/b)^(7/2) + b^3*(-1/b)^(5/2))*c^28 + 2304*(9*(c^4 + 2*b*c^2*sqrt(-1/b) - b)^2*a^8*b^10/(a^4
*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))^2 + 8*(7*a^4*b^10*(-1/b)^(9/2) + 25*a^4*b
^9*(-1/b)^(7/2) + 27*a^4*b^8*(-1/b)^(5/2) + 9*a^4*b^7*(-1/b)^(3/2))*(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^
8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))*c^26 - 96*((a^10*b^18*(-1/b)^(11/2) + 3*a^10*b^
17*(-1/b)^(9/2) - 29*a^10*b^15*(-1/b)^(5/2))*(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*
b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b)))^(5/2) + 64*(63*a^2*b^8*(-1/b)^(9/2) + 493*a^2*b^7*(-1/b)^(7/2) + 39*
a^2*b^6*(-1/b)^(5/2) - 429*a^2*b^5*(-1/b)^(3/2) - 38*a^2*b^4*sqrt(-1/b))*sqrt(-(c^4 + 2*b*c^2*sqrt(-1/b) - b)/
(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))))*c^25 + 9*(93*(c^4 + 2*b*c^2*sqrt(-1
/b) - b)^3*a^12*b^15/(a^4*b^4*c^8*sqrt(-1/b) + 2*a^4*b^5*c^4*sqrt(-1/b) + a^4*b^6*sqrt(-1/b))^3 - 43008*b^6*(-
1/b)^(9/2) - 583680*b^5*(-1/b)^(7/2) - 590848*b^4*(-1/b)^(5/2) + 1024*b^2*sqrt(-1/b) - 32*(5*a^8*b^16*(-1/b)^(
11/2) + 17*a^8*b^15*(-1/b)^(9/2) + 12*a^8*b^14*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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((c+(a*x+(a**2*x**2+b)**(1/2))**(1/2))**(1/2)/(a**2*x**2+b)**(3/2)/(a*x+(a**2*x**2+b)**(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((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(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 \frac {\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}}}{\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,{\left (a^2\,x^2+b\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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