3.491 \(\int \frac{1}{x^3 \sqrt{a+b \sqrt{c+d x}}} \, dx\)

Optimal. Leaf size=261 \[ -\frac{\left (a-b \sqrt{c+d x}\right ) \sqrt{a+b \sqrt{c+d x}}}{2 x^2 \left (a^2-b^2 c\right )}-\frac{b d \sqrt{a+b \sqrt{c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt{c+d x}\right )}{8 c x \left (a^2-b^2 c\right )^2}+\frac{b d^2 \left (2 a-5 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{16 c^{3/2} \left (a-b \sqrt{c}\right )^{5/2}}-\frac{b d^2 \left (2 a+5 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{16 c^{3/2} \left (a+b \sqrt{c}\right )^{5/2}} \]

[Out]

-((a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/(2*(a^2 - b^2*c)*x^2) - (b*d*
Sqrt[a + b*Sqrt[c + d*x]]*(6*a*b*c - (a^2 + 5*b^2*c)*Sqrt[c + d*x]))/(8*c*(a^2 -
b^2*c)^2*x) + (b*(2*a - 5*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt
[a - b*Sqrt[c]]])/(16*(a - b*Sqrt[c])^(5/2)*c^(3/2)) - (b*(2*a + 5*b*Sqrt[c])*d^
2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(16*(a + b*Sqrt[c])^(5
/2)*c^(3/2))

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Rubi [A]  time = 1.01265, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{\left (a-b \sqrt{c+d x}\right ) \sqrt{a+b \sqrt{c+d x}}}{2 x^2 \left (a^2-b^2 c\right )}-\frac{b d \sqrt{a+b \sqrt{c+d x}} \left (6 a b c-\left (a^2+5 b^2 c\right ) \sqrt{c+d x}\right )}{8 c x \left (a^2-b^2 c\right )^2}+\frac{b d^2 \left (2 a-5 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{16 c^{3/2} \left (a-b \sqrt{c}\right )^{5/2}}-\frac{b d^2 \left (2 a+5 b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{16 c^{3/2} \left (a+b \sqrt{c}\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/(2*(a^2 - b^2*c)*x^2) - (b*d*
Sqrt[a + b*Sqrt[c + d*x]]*(6*a*b*c - (a^2 + 5*b^2*c)*Sqrt[c + d*x]))/(8*c*(a^2 -
b^2*c)^2*x) + (b*(2*a - 5*b*Sqrt[c])*d^2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt
[a - b*Sqrt[c]]])/(16*(a - b*Sqrt[c])^(5/2)*c^(3/2)) - (b*(2*a + 5*b*Sqrt[c])*d^
2*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(16*(a + b*Sqrt[c])^(5
/2)*c^(3/2))

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Rubi in Sympy [A]  time = 91.3238, size = 289, normalized size = 1.11 \[ \frac{b d \sqrt{a + b \sqrt{c + d x}} \left (- 3 a b c + \left (\frac{a^{2}}{2} + \frac{5 b^{2} c}{2}\right ) \sqrt{c + d x}\right )}{4 c x \left (a^{2} - b^{2} c\right )^{2}} - \frac{b d^{2} \left (2 a \left (a^{2} - 4 b^{2} c\right ) + b \sqrt{c} \left (a^{2} + 5 b^{2} c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a + b \sqrt{c}}} \right )}}{16 c^{\frac{3}{2}} \sqrt{a + b \sqrt{c}} \left (a^{2} - b^{2} c\right )^{2}} + \frac{b d^{2} \left (2 a \left (a^{2} - 4 b^{2} c\right ) - b \sqrt{c} \left (a^{2} + 5 b^{2} c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a - b \sqrt{c}}} \right )}}{16 c^{\frac{3}{2}} \sqrt{a - b \sqrt{c}} \left (a^{2} - b^{2} c\right )^{2}} - \frac{\left (a - b \sqrt{c + d x}\right ) \sqrt{a + b \sqrt{c + d x}}}{2 x^{2} \left (a^{2} - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d*sqrt(a + b*sqrt(c + d*x))*(-3*a*b*c + (a**2/2 + 5*b**2*c/2)*sqrt(c + d*x))/(
4*c*x*(a**2 - b**2*c)**2) - b*d**2*(2*a*(a**2 - 4*b**2*c) + b*sqrt(c)*(a**2 + 5*
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)**2) + b*d**2*(2*a*(a**2 - 4*b**2*c) - b*sqrt(c)*(
a**2 + 5*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)**2) - (a - b*sqrt(c + d*x))*sqrt(a + b*s
qrt(c + d*x))/(2*x**2*(a**2 - b**2*c))

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Mathematica [A]  time = 2.83799, size = 410, normalized size = 1.57 \[ \frac{1}{16} \left (-\frac{b d^2 \left (2 a-5 b \sqrt{c}\right ) \sqrt{a^2-b^2 c} \tan ^{-1}\left (\frac{\sqrt{a^2-b^2 c}}{\sqrt{-a-b \sqrt{c}} \sqrt{a+b \sqrt{c+d x}}}\right )}{c^{3/2} \sqrt{-a-b \sqrt{c}} \left (a-b \sqrt{c}\right )^3}+\frac{b d^2 \left (2 a+5 b \sqrt{c}\right ) \sqrt{a^2-b^2 c} \tan ^{-1}\left (\frac{\sqrt{a^2-b^2 c}}{\sqrt{b \sqrt{c}-a} \sqrt{a+b \sqrt{c+d x}}}\right )}{c^{3/2} \sqrt{b \sqrt{c}-a} \left (a+b \sqrt{c}\right )^3}-\frac{8 \left (a^3-3 a^2 b \sqrt{c+d x}+3 a b^2 c-b^3 c \sqrt{c+d x}\right ) \left (a+b \sqrt{c+d x}\right )^{5/2}}{x^2 \left (a^2-b^2 c\right )^3}+\frac{2 b d \left (a^4 \left (-\sqrt{c+d x}\right )+26 a^3 b c+8 a^2 b^2 c \sqrt{c+d x}-10 a b^3 c^2+9 b^4 c^2 \sqrt{c+d x}\right ) \sqrt{a+b \sqrt{c+d x}}}{c x \left (b^2 c-a^2\right )^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-8*(a + b*Sqrt[c + d*x])^(5/2)*(a^3 + 3*a*b^2*c - 3*a^2*b*Sqrt[c + d*x] - b^3*
c*Sqrt[c + d*x]))/((a^2 - b^2*c)^3*x^2) + (2*b*d*Sqrt[a + b*Sqrt[c + d*x]]*(26*a
^3*b*c - 10*a*b^3*c^2 - a^4*Sqrt[c + d*x] + 8*a^2*b^2*c*Sqrt[c + d*x] + 9*b^4*c^
2*Sqrt[c + d*x]))/(c*(-a^2 + b^2*c)^3*x) - (b*(2*a - 5*b*Sqrt[c])*Sqrt[a^2 - b^2
*c]*d^2*ArcTan[Sqrt[a^2 - b^2*c]/(Sqrt[-a - b*Sqrt[c]]*Sqrt[a + b*Sqrt[c + d*x]]
)])/(Sqrt[-a - b*Sqrt[c]]*(a - b*Sqrt[c])^3*c^(3/2)) + (b*(2*a + 5*b*Sqrt[c])*Sq
rt[a^2 - b^2*c]*d^2*ArcTan[Sqrt[a^2 - b^2*c]/(Sqrt[-a + b*Sqrt[c]]*Sqrt[a + b*Sq
rt[c + d*x]])])/(Sqrt[-a + b*Sqrt[c]]*(a + b*Sqrt[c])^3*c^(3/2)))/16

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Maple [B]  time = 0.117, size = 834, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/16*b^2*d^2/c/(b*(d*x+c)^(1/2)+(b^2*c)^(1/2))^2/(b^2*c-2*a*(b^2*c)^(1/2)+a^2)*(
a+b*(d*x+c)^(1/2))^(3/2)-1/8*b^2*d^2/c/(b^2*c)^(1/2)/(b*(d*x+c)^(1/2)+(b^2*c)^(1
/2))^2/(b^2*c-2*a*(b^2*c)^(1/2)+a^2)*(a+b*(d*x+c)^(1/2))^(3/2)*a-7/16*b^2*d^2/c/
(b*(d*x+c)^(1/2)+(b^2*c)^(1/2))^2/(-(b^2*c)^(1/2)+a)*(a+b*(d*x+c)^(1/2))^(1/2)+1
/8*b^2*d^2/c/(b^2*c)^(1/2)/(b*(d*x+c)^(1/2)+(b^2*c)^(1/2))^2/(-(b^2*c)^(1/2)+a)*
(a+b*(d*x+c)^(1/2))^(1/2)*a+5/16*b^2*d^2/c/(b^2*c-2*a*(b^2*c)^(1/2)+a^2)/((b^2*c
)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2))-1/8*b
^2*d^2/c/(b^2*c)^(1/2)/(b^2*c-2*a*(b^2*c)^(1/2)+a^2)/((b^2*c)^(1/2)-a)^(1/2)*arc
tan((a+b*(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2))*a+5/16*b^2*d^2/c/(b*(d*x+
c)^(1/2)-(b^2*c)^(1/2))^2/(b^2*c+2*a*(b^2*c)^(1/2)+a^2)*(a+b*(d*x+c)^(1/2))^(3/2
)+1/8*b^2*d^2/c/(b^2*c)^(1/2)/(b*(d*x+c)^(1/2)-(b^2*c)^(1/2))^2/(b^2*c+2*a*(b^2*
c)^(1/2)+a^2)*(a+b*(d*x+c)^(1/2))^(3/2)*a-7/16*b^2*d^2/c/(b*(d*x+c)^(1/2)-(b^2*c
)^(1/2))^2/((b^2*c)^(1/2)+a)*(a+b*(d*x+c)^(1/2))^(1/2)-1/8*b^2*d^2/c/(b^2*c)^(1/
2)/(b*(d*x+c)^(1/2)-(b^2*c)^(1/2))^2/((b^2*c)^(1/2)+a)*(a+b*(d*x+c)^(1/2))^(1/2)
*a+5/16*b^2*d^2/c/(b^2*c+2*a*(b^2*c)^(1/2)+a^2)/(-(b^2*c)^(1/2)-a)^(1/2)*arctan(
(a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))+1/8*b^2*d^2/c/(b^2*c)^(1/2)/
(b^2*c+2*a*(b^2*c)^(1/2)+a^2)/(-(b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2)
)^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\sqrt{d x + c} b + a} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(sqrt(d*x + c)*b + a)*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(sqrt(d*x + c)*b + a)*x^3), x)

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Fricas [A]  time = 0.956999, size = 5927, normalized size = 22.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(sqrt(d*x + c)*b + a)*x^3),x, algorithm="fricas")

[Out]

-1/32*((b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)*x^2*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*
c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 + (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6
- 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16
*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 1
0*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*
a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*
b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^
5 + 5*a^8*b^2*c^4 - a^10*c^3))*log((625*b^12*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*
b^8*c + 140*a^6*b^6)*sqrt(sqrt(d*x + c)*b + a)*d^6 + ((325*a*b^12*c^5 + 1977*a^3
*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4 - (5*b^14*c^10 - 16*a^2*b^12*c
^9 + 3*a^4*b^10*c^8 + 50*a^6*b^8*c^7 - 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^1
2*b^2*c^4 + 2*a^14*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*
c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a
^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^
12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*
sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 + (b^10*c
^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)
*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c
+ 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*
b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b
^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c
^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))) - (b^4*c^3 -
2*a^2*b^2*c^2 + a^4*c)*x^2*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c
+ 4*a^7*b^2)*d^4 + (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5
+ 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b
^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 +
45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 21
0*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3
)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4
- a^10*c^3))*log((625*b^12*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^
6)*sqrt(sqrt(d*x + c)*b + a)*d^6 - ((325*a*b^12*c^5 + 1977*a^3*b^10*c^4 - 609*a^
5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4 - (5*b^14*c^10 - 16*a^2*b^12*c^9 + 3*a^4*b^10*c^
8 + 50*a^6*b^8*c^7 - 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^12*b^2*c^4 + 2*a^14
*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^
12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120
*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a
^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*sqrt(-((105*a*b^8*
c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 + (b^10*c^8 - 5*a^2*b^8*c^7
+ 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^
4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d
^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^
8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b
^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^
6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))) + (b^4*c^3 - 2*a^2*b^2*c^2 + a^
4*c)*x^2*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4
- (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 -
a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^
6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 -
120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 1
20*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*
a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))*log((
625*b^12*c^3 + 3750*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^6)*sqrt(sqrt(d*x +
c)*b + a)*d^6 + ((325*a*b^12*c^5 + 1977*a^3*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7
*b^6*c^2)*d^4 + (5*b^14*c^10 - 16*a^2*b^12*c^9 + 3*a^4*b^10*c^8 + 50*a^6*b^8*c^7
- 85*a^8*b^6*c^6 + 60*a^10*b^4*c^5 - 19*a^12*b^2*c^4 + 2*a^14*c^3)*sqrt((625*b^
18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^
10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 2
10*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a
^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c
^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 - (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6
- 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*
c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10
*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a
^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b
^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5
+ 5*a^8*b^2*c^4 - a^10*c^3))) - (b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)*x^2*sqrt(-((1
05*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c + 4*a^7*b^2)*d^4 - (b^10*c^8 - 5*a^
2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3)*sqrt((62
5*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^
8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10
+ 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 +
45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a
^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 - a^10*c^3))*log((625*b^12*c^3 + 375
0*a^2*b^10*c^2 - 1491*a^4*b^8*c + 140*a^6*b^6)*sqrt(sqrt(d*x + c)*b + a)*d^6 - (
(325*a*b^12*c^5 + 1977*a^3*b^10*c^4 - 609*a^5*b^8*c^3 + 35*a^7*b^6*c^2)*d^4 + (5
*b^14*c^10 - 16*a^2*b^12*c^9 + 3*a^4*b^10*c^8 + 50*a^6*b^8*c^7 - 85*a^8*b^6*c^6
+ 60*a^10*b^4*c^5 - 19*a^12*b^2*c^4 + 2*a^14*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*
b^16*c^3 + 21966*a^4*b^14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13
- 10*a^2*b^18*c^12 + 45*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 -
252*a^10*b^10*c^8 + 210*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a
^18*b^2*c^4 + a^20*c^3)))*sqrt(-((105*a*b^8*c^3 + 70*a^3*b^6*c^2 - 35*a^5*b^4*c
+ 4*a^7*b^2)*d^4 - (b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 +
5*a^8*b^2*c^4 - a^10*c^3)*sqrt((625*b^18*c^4 + 7700*a^2*b^16*c^3 + 21966*a^4*b^
14*c^2 - 10780*a^6*b^12*c + 1225*a^8*b^10)*d^8/(b^20*c^13 - 10*a^2*b^18*c^12 + 4
5*a^4*b^16*c^11 - 120*a^6*b^14*c^10 + 210*a^8*b^12*c^9 - 252*a^10*b^10*c^8 + 210
*a^12*b^8*c^7 - 120*a^14*b^6*c^6 + 45*a^16*b^4*c^5 - 10*a^18*b^2*c^4 + a^20*c^3)
))/(b^10*c^8 - 5*a^2*b^8*c^7 + 10*a^4*b^6*c^6 - 10*a^6*b^4*c^5 + 5*a^8*b^2*c^4 -
a^10*c^3))) + 4*(6*a*b^2*c*d*x - 4*a*b^2*c^2 + 4*a^3*c + (4*b^3*c^2 - 4*a^2*b*c
- (5*b^3*c + a^2*b)*d*x)*sqrt(d*x + c))*sqrt(sqrt(d*x + c)*b + a))/((b^4*c^3 -
2*a^2*b^2*c^2 + a^4*c)*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a + b \sqrt{c + d x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt(a + b*sqrt(c + d*x))), x)

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

[Out]

Exception raised: TypeError