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

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

[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*Sqrt[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 = 0.296884, 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, Rules used = {2016, 2014} \[ -\frac{512 a^4 \sqrt{a x+b \sqrt{x}}}{1001 b^5 x^{3/2}}+\frac{1280 a^3 \sqrt{a x+b \sqrt{x}}}{3003 b^4 x^2}-\frac{160 a^2 \sqrt{a x+b \sqrt{x}}}{429 b^3 x^{5/2}}-\frac{4096 a^6 \sqrt{a x+b \sqrt{x}}}{3003 b^7 \sqrt{x}}+\frac{2048 a^5 \sqrt{a x+b \sqrt{x}}}{3003 b^6 x}+\frac{48 a \sqrt{a x+b \sqrt{x}}}{143 b^2 x^3}-\frac{4 \sqrt{a x+b \sqrt{x}}}{13 b x^{7/2}} \]

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*Sqrt[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])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^4 \sqrt{b \sqrt{x}+a x}} \, dx &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}-\frac{(12 a) \int \frac{1}{x^{7/2} \sqrt{b \sqrt{x}+a x}} \, dx}{13 b}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}+\frac{48 a \sqrt{b \sqrt{x}+a x}}{143 b^2 x^3}+\frac{\left (120 a^2\right ) \int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx}{143 b^2}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}+\frac{48 a \sqrt{b \sqrt{x}+a x}}{143 b^2 x^3}-\frac{160 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^3 x^{5/2}}-\frac{\left (320 a^3\right ) \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx}{429 b^3}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}+\frac{48 a \sqrt{b \sqrt{x}+a x}}{143 b^2 x^3}-\frac{160 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^3 x^{5/2}}+\frac{1280 a^3 \sqrt{b \sqrt{x}+a x}}{3003 b^4 x^2}+\frac{\left (640 a^4\right ) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{1001 b^4}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}+\frac{48 a \sqrt{b \sqrt{x}+a x}}{143 b^2 x^3}-\frac{160 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^3 x^{5/2}}+\frac{1280 a^3 \sqrt{b \sqrt{x}+a x}}{3003 b^4 x^2}-\frac{512 a^4 \sqrt{b \sqrt{x}+a x}}{1001 b^5 x^{3/2}}-\frac{\left (512 a^5\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{1001 b^5}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}+\frac{48 a \sqrt{b \sqrt{x}+a x}}{143 b^2 x^3}-\frac{160 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^3 x^{5/2}}+\frac{1280 a^3 \sqrt{b \sqrt{x}+a x}}{3003 b^4 x^2}-\frac{512 a^4 \sqrt{b \sqrt{x}+a x}}{1001 b^5 x^{3/2}}+\frac{2048 a^5 \sqrt{b \sqrt{x}+a x}}{3003 b^6 x}+\frac{\left (1024 a^6\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{3003 b^6}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{13 b x^{7/2}}+\frac{48 a \sqrt{b \sqrt{x}+a x}}{143 b^2 x^3}-\frac{160 a^2 \sqrt{b \sqrt{x}+a x}}{429 b^3 x^{5/2}}+\frac{1280 a^3 \sqrt{b \sqrt{x}+a x}}{3003 b^4 x^2}-\frac{512 a^4 \sqrt{b \sqrt{x}+a x}}{1001 b^5 x^{3/2}}+\frac{2048 a^5 \sqrt{b \sqrt{x}+a x}}{3003 b^6 x}-\frac{4096 a^6 \sqrt{b \sqrt{x}+a x}}{3003 b^7 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.059094, size = 96, normalized size = 0.48 \[ -\frac{4 \sqrt{a x+b \sqrt{x}} \left (384 a^4 b^2 x^2-320 a^3 b^3 x^{3/2}+280 a^2 b^4 x-512 a^5 b x^{5/2}+1024 a^6 x^3-252 a b^5 \sqrt{x}+231 b^6\right )}{3003 b^7 x^{7/2}} \]

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 = 0.013, size = 306, normalized size = 1.5 \begin{align*} -{\frac{1}{3003\,{b}^{8}}\sqrt{b\sqrt{x}+ax} \left ( 12012\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{13/2}{x}^{13/2}-6006\,\sqrt{b\sqrt{x}+ax}{a}^{15/2}{x}^{15/2}-3003\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{15/2}{a}^{7}b-6006\,{a}^{15/2}{x}^{15/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }+3003\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{15/2}{a}^{7}b+5868\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{9/2}{x}^{11/2}{b}^{2}+3052\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{x}^{9/2}{b}^{4}-7916\,{a}^{11/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}b{x}^{6}+924\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}{x}^{7/2}{b}^{6}-4332\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}{x}^{5}{b}^{3}-1932\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{4}{b}^{5} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}{x}^{-{\frac{15}{2}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b \sqrt{x}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.41402, size = 212, normalized size = 1.06 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^(1/2)+a*x)^(1/2),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. \begin{align*} \int \frac{1}{x^{4} \sqrt{a x + b \sqrt{x}}}\, dx \end{align*}

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 [A]  time = 1.27784, size = 281, normalized size = 1.4 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^(1/2)+a*x)^(1/2),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) - sqrt(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) - sqrt(a*x + b*sqrt(x)))^13