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

Optimal. Leaf size=174 \[ -\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{64 a^{11/2}}+\frac{63 b^4 \sqrt{a x+b \sqrt{x}}}{64 a^5}-\frac{21 b^3 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{32 a^4}+\frac{21 b^2 x \sqrt{a x+b \sqrt{x}}}{40 a^3}-\frac{9 b x^{3/2} \sqrt{a x+b \sqrt{x}}}{20 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt{x}}}{5 a} \]

[Out]

(63*b^4*Sqrt[b*Sqrt[x] + a*x])/(64*a^5) - (21*b^3*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])
/(32*a^4) + (21*b^2*x*Sqrt[b*Sqrt[x] + a*x])/(40*a^3) - (9*b*x^(3/2)*Sqrt[b*Sqrt
[x] + a*x])/(20*a^2) + (2*x^2*Sqrt[b*Sqrt[x] + a*x])/(5*a) - (63*b^5*ArcTanh[(Sq
rt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(64*a^(11/2))

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Rubi [A]  time = 0.334354, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{64 a^{11/2}}+\frac{63 b^4 \sqrt{a x+b \sqrt{x}}}{64 a^5}-\frac{21 b^3 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{32 a^4}+\frac{21 b^2 x \sqrt{a x+b \sqrt{x}}}{40 a^3}-\frac{9 b x^{3/2} \sqrt{a x+b \sqrt{x}}}{20 a^2}+\frac{2 x^2 \sqrt{a x+b \sqrt{x}}}{5 a} \]

Antiderivative was successfully verified.

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

[Out]

(63*b^4*Sqrt[b*Sqrt[x] + a*x])/(64*a^5) - (21*b^3*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])
/(32*a^4) + (21*b^2*x*Sqrt[b*Sqrt[x] + a*x])/(40*a^3) - (9*b*x^(3/2)*Sqrt[b*Sqrt
[x] + a*x])/(20*a^2) + (2*x^2*Sqrt[b*Sqrt[x] + a*x])/(5*a) - (63*b^5*ArcTanh[(Sq
rt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(64*a^(11/2))

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Rubi in Sympy [A]  time = 33.2406, size = 163, normalized size = 0.94 \[ \frac{2 x^{2} \sqrt{a x + b \sqrt{x}}}{5 a} - \frac{9 b x^{\frac{3}{2}} \sqrt{a x + b \sqrt{x}}}{20 a^{2}} + \frac{21 b^{2} x \sqrt{a x + b \sqrt{x}}}{40 a^{3}} - \frac{21 b^{3} \sqrt{x} \sqrt{a x + b \sqrt{x}}}{32 a^{4}} + \frac{63 b^{4} \sqrt{a x + b \sqrt{x}}}{64 a^{5}} - \frac{63 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{64 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*sqrt(a*x + b*sqrt(x))/(5*a) - 9*b*x**(3/2)*sqrt(a*x + b*sqrt(x))/(20*a**2
) + 21*b**2*x*sqrt(a*x + b*sqrt(x))/(40*a**3) - 21*b**3*sqrt(x)*sqrt(a*x + b*sqr
t(x))/(32*a**4) + 63*b**4*sqrt(a*x + b*sqrt(x))/(64*a**5) - 63*b**5*atanh(sqrt(a
)*sqrt(x)/sqrt(a*x + b*sqrt(x)))/(64*a**(11/2))

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Mathematica [A]  time = 0.163727, size = 115, normalized size = 0.66 \[ \frac{2 \sqrt{a} \sqrt{a x+b \sqrt{x}} \left (128 a^4 x^2-144 a^3 b x^{3/2}+168 a^2 b^2 x-210 a b^3 \sqrt{x}+315 b^4\right )-315 b^5 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{640 a^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*Sqrt[b*Sqrt[x] + a*x]*(315*b^4 - 210*a*b^3*Sqrt[x] + 168*a^2*b^2*x -
144*a^3*b*x^(3/2) + 128*a^4*x^2) - 315*b^5*Log[b + 2*a*Sqrt[x] + 2*Sqrt[a]*Sqrt[
b*Sqrt[x] + a*x]])/(640*a^(11/2))

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Maple [A]  time = 0.023, size = 227, normalized size = 1.3 \[{\frac{1}{640}\sqrt{b\sqrt{x}+ax} \left ( -544\,b\sqrt{x} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{15/2}+256\,x \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{17/2}+880\,{b}^{2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{13/2}-1300\,{b}^{3}\sqrt{b\sqrt{x}+ax}\sqrt{x}{a}^{13/2}+1280\,{b}^{4}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{a}^{11/2}-650\,{b}^{4}\sqrt{b\sqrt{x}+ax}{a}^{11/2}+325\,{b}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5}-640\,{b}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{a}^{-{\frac{21}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640*(b*x^(1/2)+a*x)^(1/2)*(-544*b*x^(1/2)*(b*x^(1/2)+a*x)^(3/2)*a^(15/2)+256*x
*(b*x^(1/2)+a*x)^(3/2)*a^(17/2)+880*b^2*(b*x^(1/2)+a*x)^(3/2)*a^(13/2)-1300*b^3*
(b*x^(1/2)+a*x)^(1/2)*x^(1/2)*a^(13/2)+1280*b^4*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^
(11/2)-650*b^4*(b*x^(1/2)+a*x)^(1/2)*a^(11/2)+325*b^5*ln(1/2*(2*(b*x^(1/2)+a*x)^
(1/2)*a^(1/2)+2*x^(1/2)*a+b)/a^(1/2))*a^5-640*b^5*ln(1/2*(2*(x^(1/2)*(b+x^(1/2)*
a))^(1/2)*a^(1/2)+2*x^(1/2)*a+b)/a^(1/2))*a^5)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/a^(
21/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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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(a*x + b*sqrt(x)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.279174, size = 150, normalized size = 0.86 \[ \frac{1}{320} \, \sqrt{a x + b \sqrt{x}}{\left (2 \,{\left (4 \,{\left (2 \, \sqrt{x}{\left (\frac{8 \, \sqrt{x}}{a} - \frac{9 \, b}{a^{2}}\right )} + \frac{21 \, b^{2}}{a^{3}}\right )} \sqrt{x} - \frac{105 \, b^{3}}{a^{4}}\right )} \sqrt{x} + \frac{315 \, b^{4}}{a^{5}}\right )} + \frac{63 \, b^{5}{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{128 \, a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/320*sqrt(a*x + b*sqrt(x))*(2*(4*(2*sqrt(x)*(8*sqrt(x)/a - 9*b/a^2) + 21*b^2/a^
3)*sqrt(x) - 105*b^3/a^4)*sqrt(x) + 315*b^4/a^5) + 63/128*b^5*ln(abs(-2*sqrt(a)*
(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/a^(11/2)