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

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

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Rubi [A]  time = 0.30, antiderivative size = 200, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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])

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*}

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Mathematica [A]  time = 0.06, size = 96, normalized size = 0.48 \begin {gather*} -\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}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.23, size = 96, normalized size = 0.48 \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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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fricas [A]  time = 0.77, size = 86, normalized size = 0.43 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.20, size = 208, normalized size = 1.04 \begin {gather*} \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 {gather*}

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

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maple [C]  time = 0.07, size = 306, normalized size = 1.53 \begin {gather*} \frac {\sqrt {a x +b \sqrt {x}}\, \left (-3003 a^{7} b \,x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+3003 a^{7} b \,x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+6006 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}} x^{\frac {15}{2}}+6006 \sqrt {a x +b \sqrt {x}}\, a^{\frac {15}{2}} x^{\frac {15}{2}}-12012 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{\frac {13}{2}}+7916 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{6}-5868 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x^{\frac {11}{2}}+4332 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{5}-3052 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4} x^{\frac {9}{2}}+1932 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x^{4}-924 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{6} x^{\frac {7}{2}}\right )}{3003 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{8} x^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b \sqrt {x}} x^{4}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}

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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