3.3059 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3} \, dx\)

Optimal. Leaf size=165 \[ -\frac{b \sqrt{d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{7/2}}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \left (16 a c-15 b^2 d+10 b c \sqrt{\frac{d}{x}}\right )}{12 c^3}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x} \]

[Out]

((16*a*c - 15*b^2*d + 10*b*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(12*c^3) -
(2*Sqrt[a + b*Sqrt[d/x] + c/x])/(3*c*x) - (b*Sqrt[d]*(12*a*c - 5*b^2*d)*ArcTanh[
(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(8*c^(7/
2))

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Rubi [A]  time = 0.509205, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b \sqrt{d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac{b d+2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{8 c^{7/2}}+\frac{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} \left (16 a c-15 b^2 d+10 b c \sqrt{\frac{d}{x}}\right )}{12 c^3}-\frac{2 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 c x} \]

Antiderivative was successfully verified.

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

[Out]

((16*a*c - 15*b^2*d + 10*b*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(12*c^3) -
(2*Sqrt[a + b*Sqrt[d/x] + c/x])/(3*c*x) - (b*Sqrt[d]*(12*a*c - 5*b^2*d)*ArcTanh[
(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(8*c^(7/
2))

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Rubi in Sympy [A]  time = 38.2803, size = 141, normalized size = 0.85 \[ - \frac{b \sqrt{d} \left (12 a c - 5 b^{2} d\right ) \operatorname{atanh}{\left (\frac{b d + 2 c \sqrt{\frac{d}{x}}}{2 \sqrt{c} \sqrt{d} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{8 c^{\frac{7}{2}}} - \frac{2 \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}{3 c x} + \frac{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}} \left (4 a c - \frac{15 b^{2} d}{4} + \frac{5 b c \sqrt{\frac{d}{x}}}{2}\right )}{3 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(d)*(12*a*c - 5*b**2*d)*atanh((b*d + 2*c*sqrt(d/x))/(2*sqrt(c)*sqrt(d)*sq
rt(a + b*sqrt(d/x) + c/x)))/(8*c**(7/2)) - 2*sqrt(a + b*sqrt(d/x) + c/x)/(3*c*x)
+ sqrt(a + b*sqrt(d/x) + c/x)*(4*a*c - 15*b**2*d/4 + 5*b*c*sqrt(d/x)/2)/(3*c**3
)

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Mathematica [A]  time = 0.203212, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.045, size = 267, normalized size = 1.6 \[{\frac{1}{24\,x}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }} \left ( 15\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3}{b}^{3}c-30\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{c}^{3/2}x{b}^{2}-36\,\ln \left ({\frac{1}{\sqrt{x}} \left ( 2\,c+b\sqrt{{\frac{d}{x}}}x+2\,\sqrt{c}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \right ) } \right ) \sqrt{{\frac{d}{x}}}{x}^{2}ab{c}^{2}+20\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{5/2}\sqrt{{\frac{d}{x}}}xb+32\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{5/2}xa-16\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{c}^{7/2} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)/x*(15*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(
b*(d/x)^(1/2)*x+a*x+c)^(1/2))/x^(1/2))*(d/x)^(3/2)*x^3*b^3*c-30*(b*(d/x)^(1/2)*x
+a*x+c)^(1/2)*d*c^(3/2)*x*b^2-36*ln((2*c+b*(d/x)^(1/2)*x+2*c^(1/2)*(b*(d/x)^(1/2
)*x+a*x+c)^(1/2))/x^(1/2))*(d/x)^(1/2)*x^2*a*b*c^2+20*(b*(d/x)^(1/2)*x+a*x+c)^(1
/2)*c^(5/2)*(d/x)^(1/2)*x*b+32*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*c^(5/2)*x*a-16*(b*(
d/x)^(1/2)*x+a*x+c)^(1/2)*c^(7/2))/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/c^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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