3.109 \(\int \frac{1}{x^4 \sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{4096 a^6 \sqrt{a x+b \sqrt{x}}}{3003 b^7 \sqrt{x}}+\frac{2048 a^5 \sqrt{a x+b \sqrt{x}}}{3003 b^6 x}-\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{1001 b^5 x^{3/2}}+\frac{1280 a^3 \sqrt{a x+b \sqrt{x}}}{3003 b^4 x^2}-\frac{160 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^3 x^{5/2}}+\frac{48 a \sqrt{a x+b \sqrt{x}}}{143 b^2 x^3}-\frac{4 \sqrt{a x+b \sqrt{x}}}{13 b x^{7/2}} \]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(13*b*x^(7/2)) + (48*a*Sqrt[b*Sqrt[x] + a*x])/(143*b^
2*x^3) - (160*a^2*Sqrt[b*Sqrt[x] + a*x])/(429*b^3*x^(5/2)) + (1280*a^3*Sqrt[b*Sq
rt[x] + a*x])/(3003*b^4*x^2) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/(1001*b^5*x^(3/2)
) + (2048*a^5*Sqrt[b*Sqrt[x] + a*x])/(3003*b^6*x) - (4096*a^6*Sqrt[b*Sqrt[x] + a
*x])/(3003*b^7*Sqrt[x])

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Rubi [A]  time = 0.51757, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{4096 a^6 \sqrt{a x+b \sqrt{x}}}{3003 b^7 \sqrt{x}}+\frac{2048 a^5 \sqrt{a x+b \sqrt{x}}}{3003 b^6 x}-\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{1001 b^5 x^{3/2}}+\frac{1280 a^3 \sqrt{a x+b \sqrt{x}}}{3003 b^4 x^2}-\frac{160 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^3 x^{5/2}}+\frac{48 a \sqrt{a x+b \sqrt{x}}}{143 b^2 x^3}-\frac{4 \sqrt{a x+b \sqrt{x}}}{13 b x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(13*b*x^(7/2)) + (48*a*Sqrt[b*Sqrt[x] + a*x])/(143*b^
2*x^3) - (160*a^2*Sqrt[b*Sqrt[x] + a*x])/(429*b^3*x^(5/2)) + (1280*a^3*Sqrt[b*Sq
rt[x] + a*x])/(3003*b^4*x^2) - (512*a^4*Sqrt[b*Sqrt[x] + a*x])/(1001*b^5*x^(3/2)
) + (2048*a^5*Sqrt[b*Sqrt[x] + a*x])/(3003*b^6*x) - (4096*a^6*Sqrt[b*Sqrt[x] + a
*x])/(3003*b^7*Sqrt[x])

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Rubi in Sympy [A]  time = 49.3499, size = 187, normalized size = 0.94 \[ - \frac{4096 a^{6} \sqrt{a x + b \sqrt{x}}}{3003 b^{7} \sqrt{x}} + \frac{2048 a^{5} \sqrt{a x + b \sqrt{x}}}{3003 b^{6} x} - \frac{512 a^{4} \sqrt{a x + b \sqrt{x}}}{1001 b^{5} x^{\frac{3}{2}}} + \frac{1280 a^{3} \sqrt{a x + b \sqrt{x}}}{3003 b^{4} x^{2}} - \frac{160 a^{2} \sqrt{a x + b \sqrt{x}}}{429 b^{3} x^{\frac{5}{2}}} + \frac{48 a \sqrt{a x + b \sqrt{x}}}{143 b^{2} x^{3}} - \frac{4 \sqrt{a x + b \sqrt{x}}}{13 b x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**(1/2)+a*x)**(1/2),x)

[Out]

-4096*a**6*sqrt(a*x + b*sqrt(x))/(3003*b**7*sqrt(x)) + 2048*a**5*sqrt(a*x + b*sq
rt(x))/(3003*b**6*x) - 512*a**4*sqrt(a*x + b*sqrt(x))/(1001*b**5*x**(3/2)) + 128
0*a**3*sqrt(a*x + b*sqrt(x))/(3003*b**4*x**2) - 160*a**2*sqrt(a*x + b*sqrt(x))/(
429*b**3*x**(5/2)) + 48*a*sqrt(a*x + b*sqrt(x))/(143*b**2*x**3) - 4*sqrt(a*x + b
*sqrt(x))/(13*b*x**(7/2))

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Mathematica [A]  time = 0.0538285, size = 96, normalized size = 0.48 \[ -\frac{4 \sqrt{a x+b \sqrt{x}} \left (1024 a^6 x^3-512 a^5 b x^{5/2}+384 a^4 b^2 x^2-320 a^3 b^3 x^{3/2}+280 a^2 b^4 x-252 a b^5 \sqrt{x}+231 b^6\right )}{3003 b^7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x]*(231*b^6 - 252*a*b^5*Sqrt[x] + 280*a^2*b^4*x - 320*a^3
*b^3*x^(3/2) + 384*a^4*b^2*x^2 - 512*a^5*b*x^(5/2) + 1024*a^6*x^3))/(3003*b^7*x^
(7/2))

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Maple [C]  time = 0.02, size = 298, normalized size = 1.5 \[{\frac{1}{3003\,{b}^{8}}\sqrt{b\sqrt{x}+ax} \left ( 3003\,{a}^{13/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{15/2}-3003\,{a}^{13/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ) b{x}^{15/2}+6006\,{a}^{7}\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{x}^{15/2}-12012\,{a}^{6} \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{13/2}+6006\,{a}^{7}\sqrt{b\sqrt{x}+ax}{x}^{15/2}-5868\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{11/2}{a}^{4}{b}^{2}-3052\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{9/2}{a}^{2}{b}^{4}+7916\,{a}^{5} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{6}-924\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{7/2}{b}^{6}+4332\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{5}{a}^{3}{b}^{3}+1932\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{x}^{4}a{b}^{5} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{x}^{-{\frac{15}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^(1/2)+a*x)^(1/2),x)

[Out]

1/3003*(b*x^(1/2)+a*x)^(1/2)*(3003*a^(13/2)*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^(15/2)-3003*a^(13/2)*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^(15/2)+6006*a^7*(x^(1/2)*(b+x^(
1/2)*a))^(1/2)*x^(15/2)-12012*a^6*(b*x^(1/2)+a*x)^(3/2)*x^(13/2)+6006*a^7*(b*x^(
1/2)+a*x)^(1/2)*x^(15/2)-5868*(b*x^(1/2)+a*x)^(3/2)*x^(11/2)*a^4*b^2-3052*(b*x^(
1/2)+a*x)^(3/2)*x^(9/2)*a^2*b^4+7916*a^5*(b*x^(1/2)+a*x)^(3/2)*b*x^6-924*(b*x^(1
/2)+a*x)^(3/2)*x^(7/2)*b^6+4332*(b*x^(1/2)+a*x)^(3/2)*x^5*a^3*b^3+1932*(b*x^(1/2
)+a*x)^(3/2)*x^4*a*b^5)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/b^8/x^(15/2)

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Maxima [A]  time = 1.45266, size = 162, normalized size = 0.81 \[ -\frac{4 \,{\left (\frac{3003 \, \sqrt{a \sqrt{x} + b} a^{6}}{x^{\frac{1}{4}}} - \frac{6006 \,{\left (a \sqrt{x} + b\right )}^{\frac{3}{2}} a^{5}}{x^{\frac{3}{4}}} + \frac{9009 \,{\left (a \sqrt{x} + b\right )}^{\frac{5}{2}} a^{4}}{x^{\frac{5}{4}}} - \frac{8580 \,{\left (a \sqrt{x} + b\right )}^{\frac{7}{2}} a^{3}}{x^{\frac{7}{4}}} + \frac{5005 \,{\left (a \sqrt{x} + b\right )}^{\frac{9}{2}} a^{2}}{x^{\frac{9}{4}}} - \frac{1638 \,{\left (a \sqrt{x} + b\right )}^{\frac{11}{2}} a}{x^{\frac{11}{4}}} + \frac{231 \,{\left (a \sqrt{x} + b\right )}^{\frac{13}{2}}}{x^{\frac{13}{4}}}\right )}}{3003 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a*x + b*sqrt(x))*x^4),x, algorithm="maxima")

[Out]

-4/3003*(3003*sqrt(a*sqrt(x) + b)*a^6/x^(1/4) - 6006*(a*sqrt(x) + b)^(3/2)*a^5/x
^(3/4) + 9009*(a*sqrt(x) + b)^(5/2)*a^4/x^(5/4) - 8580*(a*sqrt(x) + b)^(7/2)*a^3
/x^(7/4) + 5005*(a*sqrt(x) + b)^(9/2)*a^2/x^(9/4) - 1638*(a*sqrt(x) + b)^(11/2)*
a/x^(11/4) + 231*(a*sqrt(x) + b)^(13/2)/x^(13/4))/b^7

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Fricas [A]  time = 0.276484, size = 116, normalized size = 0.58 \[ \frac{4 \,{\left (512 \, a^{5} b x^{3} + 320 \, a^{3} b^{3} x^{2} + 252 \, a b^{5} x -{\left (1024 \, a^{6} x^{3} + 384 \, a^{4} b^{2} x^{2} + 280 \, a^{2} b^{4} x + 231 \, b^{6}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{3003 \, b^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a*x + b*sqrt(x))*x^4),x, algorithm="fricas")

[Out]

4/3003*(512*a^5*b*x^3 + 320*a^3*b^3*x^2 + 252*a*b^5*x - (1024*a^6*x^3 + 384*a^4*
b^2*x^2 + 280*a^2*b^4*x + 231*b^6)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(b^7*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a x + b \sqrt{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**(1/2)+a*x)**(1/2),x)

[Out]

Integral(1/(x**4*sqrt(a*x + b*sqrt(x))), x)

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GIAC/XCAS [A]  time = 0.227653, size = 281, normalized size = 1.4 \[ \frac{4 \,{\left (27456 \, a^{3}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{6} + 72072 \, a^{\frac{5}{2}} b{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{5} + 80080 \, a^{2} b^{2}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{4} + 48048 \, a^{\frac{3}{2}} b^{3}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3} + 16380 \, a b^{4}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{2} + 3003 \, \sqrt{a} b^{5}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + 231 \, b^{6}\right )}}{3003 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a*x + b*sqrt(x))*x^4),x, algorithm="giac")

[Out]

4/3003*(27456*a^3*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^6 + 72072*a^(5/2)*b*
(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^5 + 80080*a^2*b^2*(sqrt(a)*sqrt(x) - s
qrt(a*x + b*sqrt(x)))^4 + 48048*a^(3/2)*b^3*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt
(x)))^3 + 16380*a*b^4*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^2 + 3003*sqrt(a)
*b^5*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + 231*b^6)/(sqrt(a)*sqrt(x) - sqr
t(a*x + b*sqrt(x)))^13