3.2.9 \(\int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx\) [109]

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

[Out]

-4/13*(b*x^(1/2)+a*x)^(1/2)/b/x^(7/2)+48/143*a*(b*x^(1/2)+a*x)^(1/2)/b^2/x^3-160/429*a^2*(b*x^(1/2)+a*x)^(1/2)
/b^3/x^(5/2)+1280/3003*a^3*(b*x^(1/2)+a*x)^(1/2)/b^4/x^2-512/1001*a^4*(b*x^(1/2)+a*x)^(1/2)/b^5/x^(3/2)+2048/3
003*a^5*(b*x^(1/2)+a*x)^(1/2)/b^6/x-4096/3003*a^6*(b*x^(1/2)+a*x)^(1/2)/b^7/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.20, 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 = {2041, 2039} \begin {gather*} -\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}} \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 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 96, normalized size = 0.48 \begin {gather*} -\frac {4 \sqrt {b \sqrt {x}+a x} \left (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\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))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.39, size = 306, normalized size = 1.53

method	result	size
derivativedivides	\(-\frac {4 \sqrt {b \sqrt {x}+a x}}{13 b \,x^{\frac {7}{2}}}-\frac {24 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{11 b \,x^{3}}-\frac {10 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{9 b \,x^{\frac {5}{2}}}-\frac {8 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{7 b \,x^{2}}-\frac {6 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{5 b \,x^{\frac {3}{2}}}-\frac {4 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {4 a \sqrt {b \sqrt {x}+a x}}{3 b^{2} \sqrt {x}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )}{13 b}\)	\(171\)
default	\(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (12012 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {13}{2}} a^{\frac {13}{2}}-6006 \sqrt {b \sqrt {x}+a x}\, x^{\frac {15}{2}} a^{\frac {15}{2}}-6006 x^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {15}{2}}-3003 x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b +3003 x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{7} b +5868 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {11}{2}} a^{\frac {9}{2}} b^{2}+3052 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {9}{2}} a^{\frac {5}{2}} b^{4}-7916 a^{\frac {11}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b \,x^{6}+924 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{\frac {7}{2}} \sqrt {a}\, b^{6}-4332 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} x^{5} a^{\frac {7}{2}} b^{3}-1932 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x^{4}\right )}{3003 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{8} x^{\frac {15}{2}} \sqrt {a}}\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

________________________________________________________________________________________

Fricas [A]
time = 3.84, 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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [A]
time = 1.15, 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

________________________________________________________________________________________

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)

________________________________________________________________________________________