3.108 \(\int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{315 b^5 \sqrt{x}}+\frac{256 a^3 \sqrt{a x+b \sqrt{x}}}{315 b^4 x}-\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{105 b^3 x^{3/2}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{63 b^2 x^2}-\frac{4 \sqrt{a x+b \sqrt{x}}}{9 b x^{5/2}} \]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(9*b*x^(5/2)) + (32*a*Sqrt[b*Sqrt[x] + a*x])/(63*b^2*
x^2) - (64*a^2*Sqrt[b*Sqrt[x] + a*x])/(105*b^3*x^(3/2)) + (256*a^3*Sqrt[b*Sqrt[x
] + a*x])/(315*b^4*x) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/(315*b^5*Sqrt[x])

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Rubi [A]  time = 0.348738, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{315 b^5 \sqrt{x}}+\frac{256 a^3 \sqrt{a x+b \sqrt{x}}}{315 b^4 x}-\frac{64 a^2 \sqrt{a x+b \sqrt{x}}}{105 b^3 x^{3/2}}+\frac{32 a \sqrt{a x+b \sqrt{x}}}{63 b^2 x^2}-\frac{4 \sqrt{a x+b \sqrt{x}}}{9 b x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(9*b*x^(5/2)) + (32*a*Sqrt[b*Sqrt[x] + a*x])/(63*b^2*
x^2) - (64*a^2*Sqrt[b*Sqrt[x] + a*x])/(105*b^3*x^(3/2)) + (256*a^3*Sqrt[b*Sqrt[x
] + a*x])/(315*b^4*x) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/(315*b^5*Sqrt[x])

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Rubi in Sympy [A]  time = 32.0236, size = 131, normalized size = 0.92 \[ - \frac{512 a^{4} \sqrt{a x + b \sqrt{x}}}{315 b^{5} \sqrt{x}} + \frac{256 a^{3} \sqrt{a x + b \sqrt{x}}}{315 b^{4} x} - \frac{64 a^{2} \sqrt{a x + b \sqrt{x}}}{105 b^{3} x^{\frac{3}{2}}} + \frac{32 a \sqrt{a x + b \sqrt{x}}}{63 b^{2} x^{2}} - \frac{4 \sqrt{a x + b \sqrt{x}}}{9 b x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-512*a**4*sqrt(a*x + b*sqrt(x))/(315*b**5*sqrt(x)) + 256*a**3*sqrt(a*x + b*sqrt(
x))/(315*b**4*x) - 64*a**2*sqrt(a*x + b*sqrt(x))/(105*b**3*x**(3/2)) + 32*a*sqrt
(a*x + b*sqrt(x))/(63*b**2*x**2) - 4*sqrt(a*x + b*sqrt(x))/(9*b*x**(5/2))

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Mathematica [A]  time = 0.0443692, size = 72, normalized size = 0.51 \[ -\frac{4 \sqrt{a x+b \sqrt{x}} \left (128 a^4 x^2-64 a^3 b x^{3/2}+48 a^2 b^2 x-40 a b^3 \sqrt{x}+35 b^4\right )}{315 b^5 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x]*(35*b^4 - 40*a*b^3*Sqrt[x] + 48*a^2*b^2*x - 64*a^3*b*x
^(3/2) + 128*a^4*x^2))/(315*b^5*x^(5/2))

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Maple [C]  time = 0.016, size = 254, normalized size = 1.8 \[{\frac{1}{315\,{b}^{6}}\sqrt{b\sqrt{x}+ax} \left ( 315\,{a}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{11/2}-315\,{a}^{9/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{11/2}+630\,{a}^{5}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{11/2}-1260\,{a}^{4} \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{9/2}+630\,{a}^{5}\sqrt{b\sqrt{x}+ax}{x}^{11/2}-492\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{7/2}{a}^{2}{b}^{2}-140\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{5/2}{b}^{4}+748\,{a}^{3} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{4}+300\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{3}a{b}^{3} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(b*x^(1/2)+a*x)^(1/2)*(315*a^(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))*b*x^(11/2)-315*a^(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))*b*x^(11/2)+630*a^5*(x^(1/2)*(b+x^(1/2)*a
))^(1/2)*x^(11/2)-1260*a^4*(b*x^(1/2)+a*x)^(3/2)*x^(9/2)+630*a^5*(b*x^(1/2)+a*x)
^(1/2)*x^(11/2)-492*(b*x^(1/2)+a*x)^(3/2)*x^(7/2)*a^2*b^2-140*(b*x^(1/2)+a*x)^(3
/2)*x^(5/2)*b^4+748*a^3*(b*x^(1/2)+a*x)^(3/2)*b*x^4+300*(b*x^(1/2)+a*x)^(3/2)*x^
3*a*b^3)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/b^6/x^(11/2)

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Maxima [A]  time = 5.31708, size = 116, normalized size = 0.82 \[ -\frac{4 \,{\left (\frac{315 \, \sqrt{a \sqrt{x} + b} a^{4}}{x^{\frac{1}{4}}} - \frac{420 \,{\left (a \sqrt{x} + b\right )}^{\frac{3}{2}} a^{3}}{x^{\frac{3}{4}}} + \frac{378 \,{\left (a \sqrt{x} + b\right )}^{\frac{5}{2}} a^{2}}{x^{\frac{5}{4}}} - \frac{180 \,{\left (a \sqrt{x} + b\right )}^{\frac{7}{2}} a}{x^{\frac{7}{4}}} + \frac{35 \,{\left (a \sqrt{x} + b\right )}^{\frac{9}{2}}}{x^{\frac{9}{4}}}\right )}}{315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/315*(315*sqrt(a*sqrt(x) + b)*a^4/x^(1/4) - 420*(a*sqrt(x) + b)^(3/2)*a^3/x^(3
/4) + 378*(a*sqrt(x) + b)^(5/2)*a^2/x^(5/4) - 180*(a*sqrt(x) + b)^(7/2)*a/x^(7/4
) + 35*(a*sqrt(x) + b)^(9/2)/x^(9/4))/b^5

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Fricas [A]  time = 0.269194, size = 86, normalized size = 0.61 \[ \frac{4 \,{\left (64 \, a^{3} b x^{2} + 40 \, a b^{3} x -{\left (128 \, a^{4} x^{2} + 48 \, a^{2} b^{2} x + 35 \, b^{4}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{315 \, b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/315*(64*a^3*b*x^2 + 40*a*b^3*x - (128*a^4*x^2 + 48*a^2*b^2*x + 35*b^4)*sqrt(x)
)*sqrt(a*x + b*sqrt(x))/(b^5*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226749, size = 197, normalized size = 1.39 \[ \frac{4 \,{\left (1008 \, a^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{4} + 1680 \, a^{\frac{3}{2}} b{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3} + 1080 \, a b^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{2} + 315 \, \sqrt{a} b^{3}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + 35 \, b^{4}\right )}}{315 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/315*(1008*a^2*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^4 + 1680*a^(3/2)*b*(sq
rt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 1080*a*b^2*(sqrt(a)*sqrt(x) - sqrt(a*
x + b*sqrt(x)))^2 + 315*sqrt(a)*b^3*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) +
35*b^4)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^9