3.3046 \(\int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2 \, dx\)

Optimal. Leaf size=333 \[ \frac{7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{480 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{80 a^3}-\frac{3 b d^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{10 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{11/2}}+\frac{x \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{256 a^5}+\frac{x^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 a} \]

[Out]

(-3*b*d^3*(a + b*Sqrt[d/x] + c/x)^(3/2))/(10*a^2*(d/x)^(5/2)) + (7*b*d^2*(28*a*c
- 15*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/(480*a^4*(d/x)^(3/2)) + ((16*a^2*c^2
- 56*a*b^2*c*d + 21*b^4*d^2)*(2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)
/(256*a^5) - ((20*a*c - 21*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2)*x^2)/(80*a^3) +
((a + b*Sqrt[d/x] + c/x)^(3/2)*x^3)/(3*a) + ((4*a*c - b^2*d)*(16*a^2*c^2 - 56*a*
b^2*c*d + 21*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x
] + c/x])])/(512*a^(11/2))

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Rubi [A]  time = 1.44905, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{7 b d^2 \left (28 a c-15 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{480 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (20 a c-21 b^2 d\right ) \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{80 a^3}-\frac{3 b d^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{10 a^2 \left (\frac{d}{x}\right )^{5/2}}+\frac{\left (4 a c-b^2 d\right ) \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{11/2}}+\frac{x \left (16 a^2 c^2-56 a b^2 c d+21 b^4 d^2\right ) \left (2 a+b \sqrt{\frac{d}{x}}\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{256 a^5}+\frac{x^3 \left (a+b \sqrt{\frac{d}{x}}+\frac{c}{x}\right )^{3/2}}{3 a} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*d^3*(a + b*Sqrt[d/x] + c/x)^(3/2))/(10*a^2*(d/x)^(5/2)) + (7*b*d^2*(28*a*c
- 15*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/(480*a^4*(d/x)^(3/2)) + ((16*a^2*c^2
- 56*a*b^2*c*d + 21*b^4*d^2)*(2*a + b*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x]*x)
/(256*a^5) - ((20*a*c - 21*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2)*x^2)/(80*a^3) +
((a + b*Sqrt[d/x] + c/x)^(3/2)*x^3)/(3*a) + ((4*a*c - b^2*d)*(16*a^2*c^2 - 56*a*
b^2*c*d + 21*b^4*d^2)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a + b*Sqrt[d/x
] + c/x])])/(512*a^(11/2))

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Rubi in Sympy [A]  time = 112.808, size = 287, normalized size = 0.86 \[ \frac{x^{3} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{3 a} - \frac{3 b d^{3} \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{10 a^{2} \left (\frac{d}{x}\right )^{\frac{5}{2}}} - \frac{x^{2} \left (20 a c - 21 b^{2} d\right ) \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{80 a^{3}} + \frac{7 b d^{2} \left (28 a c - 15 b^{2} d\right ) \left (a + b \sqrt{\frac{d}{x}} + \frac{c}{x}\right )^{\frac{3}{2}}}{480 a^{4} \left (\frac{d}{x}\right )^{\frac{3}{2}}} + \frac{x \left (2 a + b \sqrt{\frac{d}{x}}\right ) \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}} \left (16 a^{2} c^{2} - 56 a b^{2} c d + 21 b^{4} d^{2}\right )}{256 a^{5}} + \frac{\left (4 a c - b^{2} d\right ) \left (16 a^{2} c^{2} - 56 a b^{2} c d + 21 b^{4} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}} \right )}}{512 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(a + b*sqrt(d/x) + c/x)**(3/2)/(3*a) - 3*b*d**3*(a + b*sqrt(d/x) + c/x)**(3
/2)/(10*a**2*(d/x)**(5/2)) - x**2*(20*a*c - 21*b**2*d)*(a + b*sqrt(d/x) + c/x)**
(3/2)/(80*a**3) + 7*b*d**2*(28*a*c - 15*b**2*d)*(a + b*sqrt(d/x) + c/x)**(3/2)/(
480*a**4*(d/x)**(3/2)) + x*(2*a + b*sqrt(d/x))*sqrt(a + b*sqrt(d/x) + c/x)*(16*a
**2*c**2 - 56*a*b**2*c*d + 21*b**4*d**2)/(256*a**5) + (4*a*c - b**2*d)*(16*a**2*
c**2 - 56*a*b**2*c*d + 21*b**4*d**2)*atanh((2*a + b*sqrt(d/x))/(2*sqrt(a)*sqrt(a
+ b*sqrt(d/x) + c/x)))/(512*a**(11/2))

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Mathematica [A]  time = 0.094889, size = 0, normalized size = 0. \[ \int \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}} x^2 \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]*x^2, x]

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Maple [B]  time = 0.041, size = 655, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*((b*(d/x)^(1/2)*x+a*x+c)/x)^(1/2)*x^(1/2)*(630*a^(3/2)*(b*(d/x)^(1/2)*x+a
*x+c)^(1/2)*(d/x)^(5/2)*x^(5/2)*b^5+2560*x^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*a
^(11/2)-2304*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^(3/2)*b-1680*a^
(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(3/2)*x^(3/2)*b^3+1260*a^(5/2)*(b*(d/x
)^(1/2)*x+a*x+c)^(1/2)*d^2*x^(1/2)*b^4-315*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d
/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d^3*a*b^6+2016*a^(7/2)*(b
*(d/x)^(1/2)*x+a*x+c)^(3/2)*d*x^(1/2)*b^2-1680*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(
1/2)*(d/x)^(3/2)*x^(3/2)*b^3*c-3360*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^(1
/2)*b^2*c-1920*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(3/2)*x^(1/2)*c+3136*a^(7/2)*(b*(
d/x)^(1/2)*x+a*x+c)^(3/2)*(d/x)^(1/2)*x^(1/2)*b*c+2100*ln(1/2*(b*(d/x)^(1/2)*x^(
1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d^2*a^2*b^4*c
+960*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)*c^2+480*a^(7/2)*(b*(d/x)^(1/2
)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b*c^2-3600*ln(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*
(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*d*a^3*b^2*c^2+960*ln
(1/2*(b*(d/x)^(1/2)*x^(1/2)+2*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*a^(1/2)+2*a*x^(1/2))
/a^(1/2))*a^4*c^3)/(b*(d/x)^(1/2)*x+a*x+c)^(1/2)/a^(13/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*sqrt(a + b*sqrt(d/x) + c/x), x)

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GIAC/XCAS [A]  time = 0.339219, size = 558, normalized size = 1.68 \[ \frac{1}{7680} \,{\left (2 \, \sqrt{b \sqrt{d} \sqrt{x} + a x + c}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, \sqrt{x}{\left (\frac{b \sqrt{d}}{a} + 10 \, \sqrt{x}\right )} - \frac{9 \, a^{3} b^{2} d - 20 \, a^{4} c}{a^{5}}\right )} \sqrt{x} + \frac{21 \, a^{2} b^{3} d^{\frac{3}{2}} - 68 \, a^{3} b c \sqrt{d}}{a^{5}}\right )} \sqrt{x} - \frac{105 \, a b^{4} d^{2} - 448 \, a^{2} b^{2} c d + 240 \, a^{3} c^{2}}{a^{5}}\right )} \sqrt{x} + \frac{315 \, b^{5} d^{\frac{5}{2}} - 1680 \, a b^{3} c d^{\frac{3}{2}} + 1808 \, a^{2} b c^{2} \sqrt{d}}{a^{5}}\right )} + \frac{15 \,{\left (21 \, b^{6} d^{3} - 140 \, a b^{4} c d^{2} + 240 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3}\right )}{\rm ln}\left ({\left | -b \sqrt{d} - 2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{b \sqrt{d} \sqrt{x} + a x + c}\right )} \right |}\right )}{a^{\frac{11}{2}}}\right )}{\rm sign}\left (x\right ) - \frac{{\left (315 \, b^{6} d^{3}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 2100 \, a b^{4} c d^{2}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 630 \, \sqrt{a} b^{5} \sqrt{c} d^{\frac{5}{2}} + 3600 \, a^{2} b^{2} c^{2} d{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 3360 \, a^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} d^{\frac{3}{2}} - 960 \, a^{3} c^{3}{\rm ln}\left ({\left | -b \sqrt{d} + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 3616 \, a^{\frac{5}{2}} b c^{\frac{5}{2}} \sqrt{d}\right )}{\rm sign}\left (x\right )}{7680 \, a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*(2*sqrt(b*sqrt(d)*sqrt(x) + a*x + c)*(2*(4*(2*(8*sqrt(x)*(b*sqrt(d)/a + 1
0*sqrt(x)) - (9*a^3*b^2*d - 20*a^4*c)/a^5)*sqrt(x) + (21*a^2*b^3*d^(3/2) - 68*a^
3*b*c*sqrt(d))/a^5)*sqrt(x) - (105*a*b^4*d^2 - 448*a^2*b^2*c*d + 240*a^3*c^2)/a^
5)*sqrt(x) + (315*b^5*d^(5/2) - 1680*a*b^3*c*d^(3/2) + 1808*a^2*b*c^2*sqrt(d))/a
^5) + 15*(21*b^6*d^3 - 140*a*b^4*c*d^2 + 240*a^2*b^2*c^2*d - 64*a^3*c^3)*ln(abs(
-b*sqrt(d) - 2*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(b*sqrt(d)*sqrt(x) + a*x + c))))/a
^(11/2))*sign(x) - 1/7680*(315*b^6*d^3*ln(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) -
2100*a*b^4*c*d^2*ln(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) + 630*sqrt(a)*b^5*sqrt
(c)*d^(5/2) + 3600*a^2*b^2*c^2*d*ln(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))) - 3360*
a^(3/2)*b^3*c^(3/2)*d^(3/2) - 960*a^3*c^3*ln(abs(-b*sqrt(d) + 2*sqrt(a)*sqrt(c))
) + 3616*a^(5/2)*b*c^(5/2)*sqrt(d))*sign(x)/a^(11/2)