3.3054 \(\int \frac{x^2}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx\)

Optimal. Leaf size=386 \[ \frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{240 a^3}-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}-\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{x \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{640 a^5}-\frac{\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{13/2}}+\frac{x^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 a} \]

[Out]

(-11*b*d^3*Sqrt[a + b*Sqrt[d/x] + c/x])/(30*a^2*(d/x)^(5/2)) + (b*d^2*(156*a*c -
77*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(160*a^4*(d/x)^(3/2)) - (7*b*d*(528*a^2*
c^2 - 680*a*b^2*c*d + 165*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + c/x])/(1280*a^6*Sqrt[d
/x]) + ((400*a^2*c^2 - 1176*a*b^2*c*d + 385*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + c/x]
*x)/(640*a^5) - ((100*a*c - 99*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x]*x^2)/(240*a^3)
+ (Sqrt[a + b*Sqrt[d/x] + c/x]*x^3)/(3*a) - ((320*a^3*c^3 - 1680*a^2*b^2*c^2*d
+ 1260*a*b^4*c*d^2 - 231*b^6*d^3)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a
+ b*Sqrt[d/x] + c/x])])/(512*a^(13/2))

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Rubi [A]  time = 1.87481, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{b d^2 \left (156 a c-77 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{160 a^4 \left (\frac{d}{x}\right )^{3/2}}-\frac{x^2 \left (100 a c-99 b^2 d\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{240 a^3}-\frac{11 b d^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{30 a^2 \left (\frac{d}{x}\right )^{5/2}}-\frac{7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{1280 a^6 \sqrt{\frac{d}{x}}}+\frac{x \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{640 a^5}-\frac{\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac{2 a+b \sqrt{\frac{d}{x}}}{2 \sqrt{a} \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\right )}{512 a^{13/2}}+\frac{x^3 \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}{3 a} \]

Antiderivative was successfully verified.

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

[Out]

(-11*b*d^3*Sqrt[a + b*Sqrt[d/x] + c/x])/(30*a^2*(d/x)^(5/2)) + (b*d^2*(156*a*c -
77*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x])/(160*a^4*(d/x)^(3/2)) - (7*b*d*(528*a^2*
c^2 - 680*a*b^2*c*d + 165*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + c/x])/(1280*a^6*Sqrt[d
/x]) + ((400*a^2*c^2 - 1176*a*b^2*c*d + 385*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + c/x]
*x)/(640*a^5) - ((100*a*c - 99*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x]*x^2)/(240*a^3)
+ (Sqrt[a + b*Sqrt[d/x] + c/x]*x^3)/(3*a) - ((320*a^3*c^3 - 1680*a^2*b^2*c^2*d
+ 1260*a*b^4*c*d^2 - 231*b^6*d^3)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[a]*Sqrt[a
+ b*Sqrt[d/x] + c/x])])/(512*a^(13/2))

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

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]

Timed out

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Mathematica [A]  time = 0.348088, size = 0, normalized size = 0. \[ \int \frac{x^2}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.049, size = 655, normalized size = 1.7 \[{\frac{1}{7680}\sqrt{{\frac{1}{x} \left ( b\sqrt{{\frac{d}{x}}}x+ax+c \right ) }}\sqrt{x} \left ( 2560\,{x}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{a}^{13/2}-2816\,{a}^{11/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{5/2}b-3696\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{5/2}{b}^{3}-6930\,{a}^{3/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{5/2}{x}^{5/2}{b}^{5}+3168\,{a}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d{x}^{3/2}{b}^{2}+4620\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{d}^{2}\sqrt{x}{b}^{4}+3465\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{3}a{b}^{6}-3200\,{a}^{11/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}{x}^{3/2}c+7488\,{a}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}{x}^{3/2}bc+28560\,{a}^{5/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c} \left ({\frac{d}{x}} \right ) ^{3/2}{x}^{3/2}{b}^{3}c-14112\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}d\sqrt{x}{b}^{2}c-18900\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){d}^{2}{a}^{2}{b}^{4}c+4800\,{a}^{9/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{x}{c}^{2}-22176\,{a}^{7/2}\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{{\frac{d}{x}}}\sqrt{x}b{c}^{2}+25200\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ) d{a}^{3}{b}^{2}{c}^{2}-4800\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{d}{x}}}\sqrt{x}+2\,\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{4}{c}^{3} \right ){\frac{1}{\sqrt{b\sqrt{{\frac{d}{x}}}x+ax+c}}}{a}^{-{\frac{15}{2}}}} \]

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)*(2560*x^(5/2)*(b*(d/x)^(1/2)*x+
a*x+c)^(1/2)*a^(13/2)-2816*a^(11/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^
(5/2)*b-3696*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(5/2)*b^3-6930*
a^(3/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(5/2)*x^(5/2)*b^5+3168*a^(9/2)*(b*(d
/x)^(1/2)*x+a*x+c)^(1/2)*d*x^(3/2)*b^2+4620*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2
)*d^2*x^(1/2)*b^4+3465*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^6-3200*a^(11/2)*(b*(d/x)^(1/2)*x+a*x+
c)^(1/2)*x^(3/2)*c+7488*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(3/2
)*b*c+28560*a^(5/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(3/2)*x^(3/2)*b^3*c-1411
2*a^(7/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*d*x^(1/2)*b^2*c-18900*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+4800*a^(9/2)*(b*(d/x)^(1/2)*x+a*x+c)^(1/2)*x^(1/2)*c^2-22176*a^(7/2)*(b*
(d/x)^(1/2)*x+a*x+c)^(1/2)*(d/x)^(1/2)*x^(1/2)*b*c^2+25200*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^2-4800*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^(15/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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