3.111 \(\int \frac{x^2}{\left (b \sqrt{x}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{4 a^{9/2}}+\frac{35 b^2 \sqrt{a x+b \sqrt{x}}}{4 a^4}-\frac{35 b \sqrt{x} \sqrt{a x+b \sqrt{x}}}{6 a^3}+\frac{14 x \sqrt{a x+b \sqrt{x}}}{3 a^2}-\frac{4 x^2}{a \sqrt{a x+b \sqrt{x}}} \]

[Out]

(-4*x^2)/(a*Sqrt[b*Sqrt[x] + a*x]) + (35*b^2*Sqrt[b*Sqrt[x] + a*x])/(4*a^4) - (3
5*b*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(6*a^3) + (14*x*Sqrt[b*Sqrt[x] + a*x])/(3*a^2
) - (35*b^3*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(4*a^(9/2))

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Rubi [A]  time = 0.269441, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{4 a^{9/2}}+\frac{35 b^2 \sqrt{a x+b \sqrt{x}}}{4 a^4}-\frac{35 b \sqrt{x} \sqrt{a x+b \sqrt{x}}}{6 a^3}+\frac{14 x \sqrt{a x+b \sqrt{x}}}{3 a^2}-\frac{4 x^2}{a \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(b*Sqrt[x] + a*x)^(3/2),x]

[Out]

(-4*x^2)/(a*Sqrt[b*Sqrt[x] + a*x]) + (35*b^2*Sqrt[b*Sqrt[x] + a*x])/(4*a^4) - (3
5*b*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(6*a^3) + (14*x*Sqrt[b*Sqrt[x] + a*x])/(3*a^2
) - (35*b^3*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(4*a^(9/2))

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Rubi in Sympy [A]  time = 26.9052, size = 129, normalized size = 0.93 \[ - \frac{4 x^{2}}{a \sqrt{a x + b \sqrt{x}}} + \frac{14 x \sqrt{a x + b \sqrt{x}}}{3 a^{2}} - \frac{35 b \sqrt{x} \sqrt{a x + b \sqrt{x}}}{6 a^{3}} + \frac{35 b^{2} \sqrt{a x + b \sqrt{x}}}{4 a^{4}} - \frac{35 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{4 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*x**2/(a*sqrt(a*x + b*sqrt(x))) + 14*x*sqrt(a*x + b*sqrt(x))/(3*a**2) - 35*b*s
qrt(x)*sqrt(a*x + b*sqrt(x))/(6*a**3) + 35*b**2*sqrt(a*x + b*sqrt(x))/(4*a**4) -
35*b**3*atanh(sqrt(a)*sqrt(x)/sqrt(a*x + b*sqrt(x)))/(4*a**(9/2))

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Mathematica [A]  time = 0.158524, size = 113, normalized size = 0.81 \[ \frac{\sqrt{a x+b \sqrt{x}} \left (8 a^3 x^{3/2}-14 a^2 b x+35 a b^2 \sqrt{x}+105 b^3\right )}{12 a^4 \left (a \sqrt{x}+b\right )}-\frac{35 b^3 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{8 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(b*Sqrt[x] + a*x)^(3/2),x]

[Out]

(Sqrt[b*Sqrt[x] + a*x]*(105*b^3 + 35*a*b^2*Sqrt[x] - 14*a^2*b*x + 8*a^3*x^(3/2))
)/(12*a^4*(b + a*Sqrt[x])) - (35*b^3*Log[b + 2*a*Sqrt[x] + 2*Sqrt[a]*Sqrt[b*Sqrt
[x] + a*x]])/(8*a^(9/2))

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Maple [B]  time = 0.013, size = 507, normalized size = 3.7 \[ -{\frac{1}{24}\sqrt{b\sqrt{x}+ax} \left ( -16\, \left ( b\sqrt{x}+ax \right ) ^{3/2}x{a}^{15/2}+60\,\sqrt{b\sqrt{x}+ax}{x}^{3/2}{a}^{15/2}b-240\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }x{a}^{13/2}{b}^{2}-32\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{x}{a}^{13/2}b+150\,\sqrt{b\sqrt{x}+ax}x{a}^{13/2}{b}^{2}+96\,{b}^{2}{a}^{11/2} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}-480\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{x}{a}^{11/2}{b}^{3}-16\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{11/2}{b}^{2}+120\,\sqrt{b\sqrt{x}+ax}\sqrt{x}{a}^{11/2}{b}^{3}-240\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{a}^{9/2}{b}^{4}+30\,\sqrt{b\sqrt{x}+ax}{a}^{9/2}{b}^{4}-15\,x\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{6}{b}^{3}+120\,x\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{6}{b}^{3}-30\,\sqrt{x}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5}{b}^{4}+240\,\sqrt{x}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5}{b}^{4}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{4}{b}^{5}+120\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{4}{b}^{5} \right ){a}^{-{\frac{17}{2}}}{\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}} \left ( b+\sqrt{x}a \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(b*x^(1/2)+a*x)^(1/2)/a^(17/2)*(-16*(b*x^(1/2)+a*x)^(3/2)*x*a^(15/2)+60*(b
*x^(1/2)+a*x)^(1/2)*x^(3/2)*a^(15/2)*b-240*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x*a^(13
/2)*b^2-32*(b*x^(1/2)+a*x)^(3/2)*x^(1/2)*a^(13/2)*b+150*(b*x^(1/2)+a*x)^(1/2)*x*
a^(13/2)*b^2+96*b^2*a^(11/2)*(x^(1/2)*(b+x^(1/2)*a))^(3/2)-480*(x^(1/2)*(b+x^(1/
2)*a))^(1/2)*x^(1/2)*a^(11/2)*b^3-16*(b*x^(1/2)+a*x)^(3/2)*a^(11/2)*b^2+120*(b*x
^(1/2)+a*x)^(1/2)*x^(1/2)*a^(11/2)*b^3-240*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(9/2)
*b^4+30*(b*x^(1/2)+a*x)^(1/2)*a^(9/2)*b^4-15*x*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^6*b^3+120*x*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^6*b^3-30*x^(1/2)*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*b^4+240*x^(1/2)*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*b^4-15*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^4*b^5+120*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^4*b^5)/(x^(1/2)*(b+x^(1/2)*
a))^(1/2)/(b+x^(1/2)*a)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(a*x + b*sqrt(x))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^2/(a*x + b*sqrt(x))^(3/2), 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(x^2/(a*x + b*sqrt(x))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(a*x + b*sqrt(x))**(3/2), 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(x^2/(a*x + b*sqrt(x))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError