3.470 \(\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x^3} \, dx\)

Optimal. Leaf size=224 \[ \frac{b d \left (b c-a \sqrt{c+d x}\right ) \sqrt{a+b \sqrt{c+d x}}}{8 c x \left (a^2-b^2 c\right )}-\frac{b d^2 \left (2 a-3 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{16 c^{3/2} \left (a-b \sqrt{c}\right )^{3/2}}+\frac{b d^2 \left (2 a+3 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{16 c^{3/2} \left (a+b \sqrt{c}\right )^{3/2}}-\frac{\sqrt{a+b \sqrt{c+d x}}}{2 x^2} \]

[Out]

-Sqrt[a + b*Sqrt[c + d*x]]/(2*x^2) + (b*d*(b*c - a*Sqrt[c + d*x])*Sqrt[a + b*Sqr
t[c + d*x]])/(8*c*(a^2 - b^2*c)*x) - (b*(2*a - 3*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a +
b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(16*(a - b*Sqrt[c])^(3/2)*c^(3/2)) + (b*
(2*a + 3*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/
(16*(a + b*Sqrt[c])^(3/2)*c^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.848323, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{b d \left (b c-a \sqrt{c+d x}\right ) \sqrt{a+b \sqrt{c+d x}}}{8 c x \left (a^2-b^2 c\right )}-\frac{b d^2 \left (2 a-3 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{16 c^{3/2} \left (a-b \sqrt{c}\right )^{3/2}}+\frac{b d^2 \left (2 a+3 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{16 c^{3/2} \left (a+b \sqrt{c}\right )^{3/2}}-\frac{\sqrt{a+b \sqrt{c+d x}}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c + d*x]]/x^3,x]

[Out]

-Sqrt[a + b*Sqrt[c + d*x]]/(2*x^2) + (b*d*(b*c - a*Sqrt[c + d*x])*Sqrt[a + b*Sqr
t[c + d*x]])/(8*c*(a^2 - b^2*c)*x) - (b*(2*a - 3*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a +
b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(16*(a - b*Sqrt[c])^(3/2)*c^(3/2)) + (b*
(2*a + 3*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/
(16*(a + b*Sqrt[c])^(3/2)*c^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 81.5484, size = 228, normalized size = 1.02 \[ - \frac{b d \sqrt{a + b \sqrt{c + d x}} \left (a \sqrt{c + d x} - b c\right )}{8 c x \left (a^{2} - b^{2} c\right )} + \frac{b d^{2} \left (2 a^{2} + a b \sqrt{c} - 3 b^{2} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a + b \sqrt{c}}} \right )}}{16 c^{\frac{3}{2}} \sqrt{a + b \sqrt{c}} \left (a^{2} - b^{2} c\right )} - \frac{b d^{2} \left (2 a^{2} - a b \sqrt{c} - 3 b^{2} c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a - b \sqrt{c}}} \right )}}{16 c^{\frac{3}{2}} \sqrt{a - b \sqrt{c}} \left (a^{2} - b^{2} c\right )} - \frac{\sqrt{a + b \sqrt{c + d x}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(d*x+c)**(1/2))**(1/2)/x**3,x)

[Out]

-b*d*sqrt(a + b*sqrt(c + d*x))*(a*sqrt(c + d*x) - b*c)/(8*c*x*(a**2 - b**2*c)) +
b*d**2*(2*a**2 + a*b*sqrt(c) - 3*b**2*c)*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a
+ b*sqrt(c)))/(16*c**(3/2)*sqrt(a + b*sqrt(c))*(a**2 - b**2*c)) - b*d**2*(2*a**
2 - a*b*sqrt(c) - 3*b**2*c)*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c)))
/(16*c**(3/2)*sqrt(a - b*sqrt(c))*(a**2 - b**2*c)) - sqrt(a + b*sqrt(c + d*x))/(
2*x**2)

_______________________________________________________________________________________

Mathematica [A]  time = 1.96749, size = 230, normalized size = 1.03 \[ \frac{1}{16} d^2 \left (\frac{2 b \sqrt{a+b \sqrt{c+d x}} \left (a \sqrt{c+d x}-b c\right )}{c d x \left (b^2 c-a^2\right )}+\frac{b \left (2 a+3 b \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{-a-b \sqrt{c}}}\right )}{c^{3/2} \left (-a-b \sqrt{c}\right )^{3/2}}-\frac{b \left (2 a-3 b \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{b \sqrt{c}-a}}\right )}{c^{3/2} \left (b \sqrt{c}-a\right )^{3/2}}-\frac{8 \sqrt{a+b \sqrt{c+d x}}}{d^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x^3,x]

[Out]

(d^2*((-8*Sqrt[a + b*Sqrt[c + d*x]])/(d^2*x^2) + (2*b*(-(b*c) + a*Sqrt[c + d*x])
*Sqrt[a + b*Sqrt[c + d*x]])/(c*(-a^2 + b^2*c)*d*x) + (b*(2*a + 3*b*Sqrt[c])*ArcT
an[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a - b*Sqrt[c]]])/((-a - b*Sqrt[c])^(3/2)*c^(3
/2)) - (b*(2*a - 3*b*Sqrt[c])*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a + b*Sqrt[
c]]])/((-a + b*Sqrt[c])^(3/2)*c^(3/2))))/16

_______________________________________________________________________________________

Maple [B]  time = 0.14, size = 2530, normalized size = 11.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(d*x+c)^(1/2))^(1/2)/x^3,x)

[Out]

-1/8*b^2*d^2/(b^2*(d*x+c)-b^2*c)^2*a/c/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(7/2)+1/
8*b^4*d^2/(b^2*(d*x+c)-b^2*c)^2/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(5/2)+3/8*b^2*d
^2/(b^2*(d*x+c)-b^2*c)^2/c/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(5/2)*a^2-1/8*b^4*d^
2/(b^2*(d*x+c)-b^2*c)^2*a/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(3/2)-3/8*b^2*d^2/(b^
2*(d*x+c)-b^2*c)^2*a^3/c/(-b^2*c+a^2)*(a+b*(d*x+c)^(1/2))^(3/2)-3/8*b^4*d^2/(b^2
*(d*x+c)-b^2*c)^2*(a+b*(d*x+c)^(1/2))^(1/2)+1/8*b^2*d^2/(b^2*(d*x+c)-b^2*c)^2/c*
(a+b*(d*x+c)^(1/2))^(1/2)*a^2+3/16*b^11*d^2*c^4/(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)/(
-b^2*c+a^2)/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)
))^(1/2)*arctanh((-b^4*c^2+a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-c*(-b^2*c+a^
2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)))^(1/2))-1/2*b^9*d^2*c^3/
(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)/(-b^2*c+a^2)/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*
c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)))^(1/2)*arctanh((-b^4*c^2+a^2*b^2*c)*(a+b*(d*x+
c)^(1/2))^(1/2)/b/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)
^(1/2)))^(1/2))*a^2+7/16*b^7*d^2*c^2/(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)/(-b^2*c+a^2)
/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)))^(1/2)*ar
ctanh((-b^4*c^2+a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-c*(-b^2*c+a^2)*(a*b^4*c
^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)))^(1/2))*a^4-1/8*b^5*d^2*c/(b^6*c^3*
(-b^2*c+a^2)^2)^(1/2)/(-b^2*c+a^2)/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^
3*(-b^2*c+a^2)^2)^(1/2)))^(1/2)*arctanh((-b^4*c^2+a^2*b^2*c)*(a+b*(d*x+c)^(1/2))
^(1/2)/b/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)))^
(1/2))*a^6-1/16*b^5*d^2*c/(-b^2*c+a^2)/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^
6*c^3*(-b^2*c+a^2)^2)^(1/2)))^(1/2)*arctanh((-b^4*c^2+a^2*b^2*c)*(a+b*(d*x+c)^(1
/2))^(1/2)/b/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2
)))^(1/2))*a+1/16*b^3*d^2/(-b^2*c+a^2)/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^
6*c^3*(-b^2*c+a^2)^2)^(1/2)))^(1/2)*arctanh((-b^4*c^2+a^2*b^2*c)*(a+b*(d*x+c)^(1
/2))^(1/2)/b/(-c*(-b^2*c+a^2)*(a*b^4*c^2-a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2
)))^(1/2))*a^3-3/16*b^11*d^2*c^4/(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)/(-b^2*c+a^2)/(-(
-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2)*arcta
n((b^4*c^2-a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c
^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2))+1/2*b^9*d^2*c^3/(b^6*c^3*(-b^2*
c+a^2)^2)^(1/2)/(-b^2*c+a^2)/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1
/2))*c*(-b^2*c+a^2))^(1/2)*arctan((b^4*c^2-a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/
b/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2))
*a^2-7/16*b^7*d^2*c^2/(b^6*c^3*(-b^2*c+a^2)^2)^(1/2)/(-b^2*c+a^2)/(-(-a*b^4*c^2+
a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2)*arctan((b^4*c^2-
a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+
a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2))*a^4+1/8*b^5*d^2*c/(b^6*c^3*(-b^2*c+a^2)^2)
^(1/2)/(-b^2*c+a^2)/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-
b^2*c+a^2))^(1/2)*arctan((b^4*c^2-a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-(-a*b
^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2))*a^6-1/16
*b^5*d^2*c/(-b^2*c+a^2)/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*
c*(-b^2*c+a^2))^(1/2)*arctan((b^4*c^2-a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-(
-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2))*a+1/
16*b^3*d^2/(-b^2*c+a^2)/(-(-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*
c*(-b^2*c+a^2))^(1/2)*arctan((b^4*c^2-a^2*b^2*c)*(a+b*(d*x+c)^(1/2))^(1/2)/b/(-(
-a*b^4*c^2+a^3*b^2*c+(b^6*c^3*(-b^2*c+a^2)^2)^(1/2))*c*(-b^2*c+a^2))^(1/2))*a^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{d x + c} b + a}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x^3,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.474253, size = 3856, normalized size = 17.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x^3,x, algorithm="fricas")

[Out]

1/32*((b^2*c^2 - a^2*c)*x^2*sqrt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4
+ (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^
2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6
*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^
5 + 3*a^4*b^2*c^4 - a^6*c^3))*log((81*b^10*c^2 - 81*a^2*b^8*c + 20*a^4*b^6)*sqrt
(sqrt(d*x + c)*b + a)*d^6 + ((27*b^10*c^4 - 24*a^2*b^8*c^3 + 5*a^4*b^6*c^2)*d^4
- 2*(2*a*b^8*c^7 - 7*a^3*b^6*c^6 + 9*a^5*b^4*c^5 - 5*a^7*b^2*c^4 + a^9*c^3)*sqrt
((81*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15
*a^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))*sq
rt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 + (b^6*c^6 - 3*a^2*b^4*c^5 +
3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b
^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*
a^10*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)))
- (b^2*c^2 - a^2*c)*x^2*sqrt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 +
(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b
^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^
6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 +
3*a^4*b^2*c^4 - a^6*c^3))*log((81*b^10*c^2 - 81*a^2*b^8*c + 20*a^4*b^6)*sqrt(sq
rt(d*x + c)*b + a)*d^6 - ((27*b^10*c^4 - 24*a^2*b^8*c^3 + 5*a^4*b^6*c^2)*d^4 - 2
*(2*a*b^8*c^7 - 7*a^3*b^6*c^6 + 9*a^5*b^4*c^5 - 5*a^7*b^2*c^4 + a^9*c^3)*sqrt((8
1*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^
4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))*sqrt(
-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 + (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a
^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12
*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^1
0*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3))) +
(b^2*c^2 - a^2*c)*x^2*sqrt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 - (b^
6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12
*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c
^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*
a^4*b^2*c^4 - a^6*c^3))*log((81*b^10*c^2 - 81*a^2*b^8*c + 20*a^4*b^6)*sqrt(sqrt(
d*x + c)*b + a)*d^6 + ((27*b^10*c^4 - 24*a^2*b^8*c^3 + 5*a^4*b^6*c^2)*d^4 + 2*(2
*a*b^8*c^7 - 7*a^3*b^6*c^6 + 9*a^5*b^4*c^5 - 5*a^7*b^2*c^4 + a^9*c^3)*sqrt((81*b
^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b
^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))*sqrt(-((
15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 - (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*
b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^
9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b
^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3))) - (b^
2*c^2 - a^2*c)*x^2*sqrt(-((15*a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 - (b^6*c
^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c
+ 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c^6
+ 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4
*b^2*c^4 - a^6*c^3))*log((81*b^10*c^2 - 81*a^2*b^8*c + 20*a^4*b^6)*sqrt(sqrt(d*x
+ c)*b + a)*d^6 - ((27*b^10*c^4 - 24*a^2*b^8*c^3 + 5*a^4*b^6*c^2)*d^4 + 2*(2*a*
b^8*c^7 - 7*a^3*b^6*c^6 + 9*a^5*b^4*c^5 - 5*a^7*b^2*c^4 + a^9*c^3)*sqrt((81*b^14
*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 - 6*a^2*b^10*c^8 + 15*a^4*b^8*
c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*c^4 + a^12*c^3)))*sqrt(-((15*
a*b^6*c^2 - 15*a^3*b^4*c + 4*a^5*b^2)*d^4 - (b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2
*c^4 - a^6*c^3)*sqrt((81*b^14*c^2 - 90*a^2*b^12*c + 25*a^4*b^10)*d^8/(b^12*c^9 -
6*a^2*b^10*c^8 + 15*a^4*b^8*c^7 - 20*a^6*b^6*c^6 + 15*a^8*b^4*c^5 - 6*a^10*b^2*
c^4 + a^12*c^3)))/(b^6*c^6 - 3*a^2*b^4*c^5 + 3*a^4*b^2*c^4 - a^6*c^3))) - 4*(b^2
*c*d*x - sqrt(d*x + c)*a*b*d*x + 4*b^2*c^2 - 4*a^2*c)*sqrt(sqrt(d*x + c)*b + a))
/((b^2*c^2 - a^2*c)*x^2)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c + d x}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(d*x+c)**(1/2))**(1/2)/x**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a)/x^3,x, algorithm="giac")

[Out]

Exception raised: TypeError