3.490 \(\int \frac{1}{x^2 \sqrt{a+b \sqrt{c+d x}}} \, dx\)

Optimal. Leaf size=163 \[ -\frac{\sqrt{a+b \sqrt{c+d x}} \left (a-b \sqrt{c+d x}\right )}{x \left (a^2-b^2 c\right )}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{2 \sqrt{c} \left (a-b \sqrt{c}\right )^{3/2}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{2 \sqrt{c} \left (a+b \sqrt{c}\right )^{3/2}} \]

[Out]

-(((a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/((a^2 - b^2*c)*x)) - (b*d*Ar
cTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(2*(a - b*Sqrt[c])^(3/2)*S
qrt[c]) + (b*d*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(2*(a + b
*Sqrt[c])^(3/2)*Sqrt[c])

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Rubi [A]  time = 0.471403, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{\sqrt{a+b \sqrt{c+d x}} \left (a-b \sqrt{c+d x}\right )}{x \left (a^2-b^2 c\right )}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{2 \sqrt{c} \left (a-b \sqrt{c}\right )^{3/2}}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{2 \sqrt{c} \left (a+b \sqrt{c}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(((a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/((a^2 - b^2*c)*x)) - (b*d*Ar
cTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(2*(a - b*Sqrt[c])^(3/2)*S
qrt[c]) + (b*d*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(2*(a + b
*Sqrt[c])^(3/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 41.9321, size = 138, normalized size = 0.85 \[ \frac{b d \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a + b \sqrt{c}}} \right )}}{2 \sqrt{c} \left (a + b \sqrt{c}\right )^{\frac{3}{2}}} - \frac{b d \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{c + d x}}}{\sqrt{a - b \sqrt{c}}} \right )}}{2 \sqrt{c} \left (a - b \sqrt{c}\right )^{\frac{3}{2}}} - \frac{\left (a - b \sqrt{c + d x}\right ) \sqrt{a + b \sqrt{c + d x}}}{x \left (a^{2} - b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 2.40075, size = 250, normalized size = 1.53 \[ \frac{\sqrt{a+b \sqrt{c+d x}} \left (a-b \sqrt{c+d x}\right )}{x \left (b^2 c-a^2\right )}+\frac{b d \sqrt{a^2-b^2 c} \tan ^{-1}\left (\frac{\sqrt{a^2-b^2 c}}{\sqrt{-a-b \sqrt{c}} \sqrt{a+b \sqrt{c+d x}}}\right )}{2 \sqrt{c} \sqrt{-a-b \sqrt{c}} \left (a-b \sqrt{c}\right )^2}-\frac{b d \sqrt{a^2-b^2 c} \tan ^{-1}\left (\frac{\sqrt{a^2-b^2 c}}{\sqrt{b \sqrt{c}-a} \sqrt{a+b \sqrt{c+d x}}}\right )}{2 \sqrt{c} \sqrt{b \sqrt{c}-a} \left (a+b \sqrt{c}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((a - b*Sqrt[c + d*x])*Sqrt[a + b*Sqrt[c + d*x]])/((-a^2 + b^2*c)*x) + (b*Sqrt[a
^2 - b^2*c]*d*ArcTan[Sqrt[a^2 - b^2*c]/(Sqrt[-a - b*Sqrt[c]]*Sqrt[a + b*Sqrt[c +
d*x]])])/(2*Sqrt[-a - b*Sqrt[c]]*(a - b*Sqrt[c])^2*Sqrt[c]) - (b*Sqrt[a^2 - b^2
*c]*d*ArcTan[Sqrt[a^2 - b^2*c]/(Sqrt[-a + b*Sqrt[c]]*Sqrt[a + b*Sqrt[c + d*x]])]
)/(2*Sqrt[-a + b*Sqrt[c]]*(a + b*Sqrt[c])^2*Sqrt[c])

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Maple [B]  time = 0.033, size = 265, normalized size = 1.6 \[ -2\,{\frac{d\sqrt{{b}^{2}c}\sqrt{a+b\sqrt{dx+c}}}{c \left ( 4\,\sqrt{{b}^{2}c}-4\,a \right ) \left ( b\sqrt{dx+c}+\sqrt{{b}^{2}c} \right ) }}-2\,{\frac{d\sqrt{{b}^{2}c}}{c \left ( 4\,\sqrt{{b}^{2}c}-4\,a \right ) \sqrt{\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{\sqrt{{b}^{2}c}-a}}} \right ) }-2\,{\frac{d\sqrt{{b}^{2}c}\sqrt{a+b\sqrt{dx+c}}}{c \left ( -4\,\sqrt{{b}^{2}c}-4\,a \right ) \left ( -b\sqrt{dx+c}+\sqrt{{b}^{2}c} \right ) }}+2\,{\frac{d\sqrt{{b}^{2}c}}{c \left ( -4\,\sqrt{{b}^{2}c}-4\,a \right ) \sqrt{-\sqrt{{b}^{2}c}-a}}\arctan \left ({\frac{\sqrt{a+b\sqrt{dx+c}}}{\sqrt{-\sqrt{{b}^{2}c}-a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d*(b^2*c)^(1/2)/c*(a+b*(d*x+c)^(1/2))^(1/2)/(4*(b^2*c)^(1/2)-4*a)/(b*(d*x+c)^
(1/2)+(b^2*c)^(1/2))-2*d*(b^2*c)^(1/2)/c/(4*(b^2*c)^(1/2)-4*a)/((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))-2*d*(b^2*c)^(1/
2)/c*(a+b*(d*x+c)^(1/2))^(1/2)/(-4*(b^2*c)^(1/2)-4*a)/(-b*(d*x+c)^(1/2)+(b^2*c)^
(1/2))+2*d*(b^2*c)^(1/2)/c/(-4*(b^2*c)^(1/2)-4*a)/(-(b^2*c)^(1/2)-a)^(1/2)*arcta
n((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\sqrt{d x + c} b + a} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(sqrt(d*x + c)*b + a)*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.387123, size = 3366, normalized size = 20.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(sqrt(d*x + c)*b + a)*x^2),x, algorithm="fricas")

[Out]

1/4*((b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*a^2*b^4*c^3
+ 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^
7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b
^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c))*log((b^6*c
+ 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 + (2*(a*b^6*c^2 + 3*a^3*b^4*c)*d^2 -
(b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - a^8*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c
+ 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 +
15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (
b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c +
9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15
*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c
^2 - a^6*c))) - (b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 + (b^6*c^4 - 3*
a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d
^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3
- 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c))
*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 - (2*(a*b^6*c^2 + 3*a^3*b
^4*c)*d^2 - (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - a^8*c)*sqrt((b^10*c^2 + 6
*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6
*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))*sqrt(-((3*a*b^4*c + a^3*b
^2)*d^2 + (b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a
^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b
^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 +
3*a^4*b^2*c^2 - a^6*c))) + (b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^2)*d^2 - (b
^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9
*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*
a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^
2 - a^6*c))*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 + (2*(a*b^6*c^
2 + 3*a^3*b^4*c)*d^2 + (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - a^8*c)*sqrt((b
^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c
^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))*sqrt(-((3*a*b^
4*c + a^3*b^2)*d^2 - (b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^1
0*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5
- 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2
*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c))) - (b^2*c - a^2)*x*sqrt(-((3*a*b^4*c + a^3*b^
2)*d^2 - (b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt((b^10*c^2 + 6*a^
2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*a^4*b^8*c^5 - 20*a^6*b^
6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*c^4 - 3*a^2*b^4*c^3 + 3
*a^4*b^2*c^2 - a^6*c))*log((b^6*c + 3*a^2*b^4)*sqrt(sqrt(d*x + c)*b + a)*d^3 - (
2*(a*b^6*c^2 + 3*a^3*b^4*c)*d^2 + (b^8*c^5 - 2*a^2*b^6*c^4 + 2*a^6*b^2*c^2 - a^8
*c)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 1
5*a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))*sqr
t(-((3*a*b^4*c + a^3*b^2)*d^2 - (b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c
)*sqrt((b^10*c^2 + 6*a^2*b^8*c + 9*a^4*b^6)*d^4/(b^12*c^7 - 6*a^2*b^10*c^6 + 15*
a^4*b^8*c^5 - 20*a^6*b^6*c^4 + 15*a^8*b^4*c^3 - 6*a^10*b^2*c^2 + a^12*c)))/(b^6*
c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c))) - 4*sqrt(sqrt(d*x + c)*b + a)*(sq
rt(d*x + c)*b - a))/((b^2*c - a^2)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b \sqrt{c + d x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*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^2),x, algorithm="giac")

[Out]

Exception raised: TypeError