3.115 \(\int \frac{1}{x^2 \left (b \sqrt{x}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{512 a^3 \sqrt{a x+b \sqrt{x}}}{35 b^5 \sqrt{x}}-\frac{256 a^2 \sqrt{a x+b \sqrt{x}}}{35 b^4 x}+\frac{192 a \sqrt{a x+b \sqrt{x}}}{35 b^3 x^{3/2}}-\frac{32 \sqrt{a x+b \sqrt{x}}}{7 b^2 x^2}+\frac{4}{b x^{3/2} \sqrt{a x+b \sqrt{x}}} \]

[Out]

4/(b*x^(3/2)*Sqrt[b*Sqrt[x] + a*x]) - (32*Sqrt[b*Sqrt[x] + a*x])/(7*b^2*x^2) + (
192*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^3*x^(3/2)) - (256*a^2*Sqrt[b*Sqrt[x] + a*x])/
(35*b^4*x) + (512*a^3*Sqrt[b*Sqrt[x] + a*x])/(35*b^5*Sqrt[x])

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Rubi [A]  time = 0.344037, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{512 a^3 \sqrt{a x+b \sqrt{x}}}{35 b^5 \sqrt{x}}-\frac{256 a^2 \sqrt{a x+b \sqrt{x}}}{35 b^4 x}+\frac{192 a \sqrt{a x+b \sqrt{x}}}{35 b^3 x^{3/2}}-\frac{32 \sqrt{a x+b \sqrt{x}}}{7 b^2 x^2}+\frac{4}{b x^{3/2} \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

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

[Out]

4/(b*x^(3/2)*Sqrt[b*Sqrt[x] + a*x]) - (32*Sqrt[b*Sqrt[x] + a*x])/(7*b^2*x^2) + (
192*a*Sqrt[b*Sqrt[x] + a*x])/(35*b^3*x^(3/2)) - (256*a^2*Sqrt[b*Sqrt[x] + a*x])/
(35*b^4*x) + (512*a^3*Sqrt[b*Sqrt[x] + a*x])/(35*b^5*Sqrt[x])

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Rubi in Sympy [A]  time = 33.1533, size = 126, normalized size = 0.92 \[ \frac{512 a^{3} \sqrt{a x + b \sqrt{x}}}{35 b^{5} \sqrt{x}} - \frac{256 a^{2} \sqrt{a x + b \sqrt{x}}}{35 b^{4} x} + \frac{192 a \sqrt{a x + b \sqrt{x}}}{35 b^{3} x^{\frac{3}{2}}} + \frac{4}{b x^{\frac{3}{2}} \sqrt{a x + b \sqrt{x}}} - \frac{32 \sqrt{a x + b \sqrt{x}}}{7 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

512*a**3*sqrt(a*x + b*sqrt(x))/(35*b**5*sqrt(x)) - 256*a**2*sqrt(a*x + b*sqrt(x)
)/(35*b**4*x) + 192*a*sqrt(a*x + b*sqrt(x))/(35*b**3*x**(3/2)) + 4/(b*x**(3/2)*s
qrt(a*x + b*sqrt(x))) - 32*sqrt(a*x + b*sqrt(x))/(7*b**2*x**2)

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Mathematica [A]  time = 0.0487053, size = 81, normalized size = 0.59 \[ \frac{4 \sqrt{a x+b \sqrt{x}} \left (128 a^4 x^2+64 a^3 b x^{3/2}-16 a^2 b^2 x+8 a b^3 \sqrt{x}-5 b^4\right )}{35 b^5 x^2 \left (a \sqrt{x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[b*Sqrt[x] + a*x]*(-5*b^4 + 8*a*b^3*Sqrt[x] - 16*a^2*b^2*x + 64*a^3*b*x^(
3/2) + 128*a^4*x^2))/(35*b^5*(b + a*Sqrt[x])*x^2)

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Maple [C]  time = 0.017, size = 562, normalized size = 4.1 \[ -{\frac{1}{35\,{b}^{6}}\sqrt{b\sqrt{x}+ax} \left ( -105\,{x}^{11/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{11/2}b+105\,{x}^{11/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{11/2}b-210\,{x}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{9/2}{b}^{2}+210\,{x}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{9/2}{b}^{2}-105\,{x}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}{b}^{3}+105\,{x}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7/2}{b}^{3}+210\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{11/2}{a}^{6}-560\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{9/2}{a}^{5}+210\,\sqrt{b\sqrt{x}+ax}{x}^{11/2}{a}^{6}+140\,{a}^{5} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}{x}^{9/2}+420\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{5}{a}^{5}b-256\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{7/2}{a}^{3}{b}^{2}-932\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{4}{a}^{4}b+420\,\sqrt{b\sqrt{x}+ax}{x}^{5}{a}^{5}b+210\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{9/2}{a}^{4}{b}^{2}+64\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{3}{a}^{2}{b}^{3}+210\,\sqrt{b\sqrt{x}+ax}{x}^{9/2}{a}^{4}{b}^{2}-32\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{5/2}a{b}^{4}+20\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{2}{b}^{5} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{9}{2}}} \left ( b+\sqrt{x}a \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(b*x^(1/2)+a*x)^(1/2)*(-105*x^(11/2)*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^(11/2)*b+105*x^(11/2)*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^(11/2)*b-210*x^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^(9/2)*b^2+210*x^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^(9/2)*b^2-105*x^(9/2
)*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^(7/2
)*b^3+105*x^(9/2)*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^(7/2)*b^3+210*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(11/2)*a^6-560*(b*x^(1/2)+a*x)
^(3/2)*x^(9/2)*a^5+210*(b*x^(1/2)+a*x)^(1/2)*x^(11/2)*a^6+140*a^5*(x^(1/2)*(b+x^
(1/2)*a))^(3/2)*x^(9/2)+420*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^5*a^5*b-256*(b*x^(1/
2)+a*x)^(3/2)*x^(7/2)*a^3*b^2-932*(b*x^(1/2)+a*x)^(3/2)*x^4*a^4*b+420*(b*x^(1/2)
+a*x)^(1/2)*x^5*a^5*b+210*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(9/2)*a^4*b^2+64*(b*x^
(1/2)+a*x)^(3/2)*x^3*a^2*b^3+210*(b*x^(1/2)+a*x)^(1/2)*x^(9/2)*a^4*b^2-32*(b*x^(
1/2)+a*x)^(3/2)*x^(5/2)*a*b^4+20*(b*x^(1/2)+a*x)^(3/2)*x^2*b^5)/(x^(1/2)*(b+x^(1
/2)*a))^(1/2)/b^6/x^(9/2)/(b+x^(1/2)*a)^2

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Maxima [A]  time = 1.46145, size = 78, normalized size = 0.57 \[ \frac{4 \,{\left (128 \, a^{4} x^{2} + 64 \, a^{3} b x^{\frac{3}{2}} - 16 \, a^{2} b^{2} x + 8 \, a b^{3} \sqrt{x} - 5 \, b^{4}\right )}}{35 \, \sqrt{a \sqrt{x} + b} b^{5} x^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*sqrt(x))^(3/2)*x^2),x, algorithm="maxima")

[Out]

4/35*(128*a^4*x^2 + 64*a^3*b*x^(3/2) - 16*a^2*b^2*x + 8*a*b^3*sqrt(x) - 5*b^4)/(
sqrt(a*sqrt(x) + b)*b^5*x^(7/4))

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Fricas [A]  time = 0.262559, size = 117, normalized size = 0.85 \[ -\frac{4 \,{\left (64 \, a^{4} b x^{2} - 24 \, a^{2} b^{3} x - 5 \, b^{5} -{\left (128 \, a^{5} x^{2} - 80 \, a^{3} b^{2} x - 13 \, a b^{4}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{35 \,{\left (a^{2} b^{5} x^{3} - b^{7} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*sqrt(x))^(3/2)*x^2),x, algorithm="fricas")

[Out]

-4/35*(64*a^4*b*x^2 - 24*a^2*b^3*x - 5*b^5 - (128*a^5*x^2 - 80*a^3*b^2*x - 13*a*
b^4)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^5*x^3 - b^7*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*(a*x + b*sqrt(x))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x + b*sqrt(x))^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^2), x)