3.2.23 \(\int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx\) [123]

3.2.23.1 Optimal result

Integrand size = 21, antiderivative size = 170 \[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b x^3}+\frac {40 a \sqrt {b \sqrt {x}+a x}}{99 b^2 x^{5/2}}-\frac {320 a^2 \sqrt {b \sqrt {x}+a x}}{693 b^3 x^2}+\frac {128 a^3 \sqrt {b \sqrt {x}+a x}}{231 b^4 x^{3/2}}-\frac {512 a^4 \sqrt {b \sqrt {x}+a x}}{693 b^5 x}+\frac {1024 a^5 \sqrt {b \sqrt {x}+a x}}{693 b^6 \sqrt {x}} \]

output

-4/11*(b*x^(1/2)+a*x)^(1/2)/b/x^3+40/99*a*(b*x^(1/2)+a*x)^(1/2)/b^2/x^(5/2 
)-320/693*a^2*(b*x^(1/2)+a*x)^(1/2)/b^3/x^2+128/231*a^3*(b*x^(1/2)+a*x)^(1 
/2)/b^4/x^(3/2)-512/693*a^4*(b*x^(1/2)+a*x)^(1/2)/b^5/x+1024/693*a^5*(b*x^ 
(1/2)+a*x)^(1/2)/b^6/x^(1/2)
 

3.2.23.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (63 b^5-70 a b^4 \sqrt {x}+80 a^2 b^3 x-96 a^3 b^2 x^{3/2}+128 a^4 b x^2-256 a^5 x^{5/2}\right )}{693 b^6 x^3} \]

input

Integrate[1/(x^(7/2)*Sqrt[b*Sqrt[x] + a*x]),x]
 

output

(-4*Sqrt[b*Sqrt[x] + a*x]*(63*b^5 - 70*a*b^4*Sqrt[x] + 80*a^2*b^3*x - 96*a 
^3*b^2*x^(3/2) + 128*a^4*b*x^2 - 256*a^5*x^(5/2)))/(693*b^6*x^3)
 

3.2.23.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1922, 1922, 1922, 1922, 1922, 1920}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^{7/2} \sqrt {a x+b \sqrt {x}}} \, dx\)

\(\Big \downarrow \) 1922

\(\displaystyle -\frac {10 a \int \frac {1}{x^3 \sqrt {\sqrt {x} b+a x}}dx}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\)

\(\Big \downarrow \) 1922

\(\displaystyle -\frac {10 a \left (-\frac {8 a \int \frac {1}{x^{5/2} \sqrt {\sqrt {x} b+a x}}dx}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\)

\(\Big \downarrow \) 1922

\(\displaystyle -\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \int \frac {1}{x^2 \sqrt {\sqrt {x} b+a x}}dx}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\)

\(\Big \downarrow \) 1922

\(\displaystyle -\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \int \frac {1}{x^{3/2} \sqrt {\sqrt {x} b+a x}}dx}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\)

\(\Big \downarrow \) 1922

\(\displaystyle -\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (-\frac {2 a \int \frac {1}{x \sqrt {\sqrt {x} b+a x}}dx}{3 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\)

\(\Big \downarrow \) 1920

\(\displaystyle -\frac {10 a \left (-\frac {8 a \left (-\frac {6 a \left (-\frac {4 a \left (\frac {8 a \sqrt {a x+b \sqrt {x}}}{3 b^2 \sqrt {x}}-\frac {4 \sqrt {a x+b \sqrt {x}}}{3 b x}\right )}{5 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{5 b x^{3/2}}\right )}{7 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{7 b x^2}\right )}{9 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{9 b x^{5/2}}\right )}{11 b}-\frac {4 \sqrt {a x+b \sqrt {x}}}{11 b x^3}\)

input

Int[1/(x^(7/2)*Sqrt[b*Sqrt[x] + a*x]),x]
 

output

(-4*Sqrt[b*Sqrt[x] + a*x])/(11*b*x^3) - (10*a*((-4*Sqrt[b*Sqrt[x] + a*x])/ 
(9*b*x^(5/2)) - (8*a*((-4*Sqrt[b*Sqrt[x] + a*x])/(7*b*x^2) - (6*a*((-4*Sqr 
t[b*Sqrt[x] + a*x])/(5*b*x^(3/2)) - (4*a*((-4*Sqrt[b*Sqrt[x] + a*x])/(3*b* 
x) + (8*a*Sqrt[b*Sqrt[x] + a*x])/(3*b^2*Sqrt[x])))/(5*b)))/(7*b)))/(9*b))) 
/(11*b)
 

3.2.23.3.1 Defintions of rubi rules used

rule 1920

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 1922

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] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(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])
 

3.2.23.4 Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(-\frac {4 \sqrt {b \sqrt {x}+a x}}{11 b \,x^{3}}-\frac {20 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{9 b \,x^{\frac {5}{2}}}-\frac {8 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{7 b \,x^{2}}-\frac {6 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{5 b \,x^{\frac {3}{2}}}-\frac {4 a \left (-\frac {2 \sqrt {b \sqrt {x}+a x}}{3 b x}+\frac {4 a \sqrt {b \sqrt {x}+a x}}{3 b^{2} \sqrt {x}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\)	\(145\)
default	\(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (1386 x^{\frac {13}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {13}{2}}+1386 x^{\frac {13}{2}} a^{\frac {13}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}-2772 x^{\frac {11}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {11}{2}}+693 x^{\frac {13}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{6} b -693 x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a^{6} b -1236 x^{\frac {9}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2}+1748 a^{\frac {9}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b \,x^{5}-532 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{4}+852 x^{4} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}+252 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{5} x^{3}\right )}{693 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{7} x^{\frac {13}{2}} \sqrt {a}}\)	\(284\)

input

int(1/x^(7/2)/(b*x^(1/2)+a*x)^(1/2),x,method=_RETURNVERBOSE)
 

output

-4/11*(b*x^(1/2)+a*x)^(1/2)/b/x^3-20/11*a/b*(-2/9*(b*x^(1/2)+a*x)^(1/2)/b/ 
x^(5/2)-8/9*a/b*(-2/7*(b*x^(1/2)+a*x)^(1/2)/b/x^2-6/7*a/b*(-2/5*(b*x^(1/2) 
+a*x)^(1/2)/b/x^(3/2)-4/5*a/b*(-2/3*(b*x^(1/2)+a*x)^(1/2)/b/x+4/3*a*(b*x^( 
1/2)+a*x)^(1/2)/b^2/x^(1/2)))))
 

3.2.23.5 Fricas [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.42 \[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=-\frac {4 \, {\left (128 \, a^{4} b x^{2} + 80 \, a^{2} b^{3} x + 63 \, b^{5} - 2 \, {\left (128 \, a^{5} x^{2} + 48 \, a^{3} b^{2} x + 35 \, a b^{4}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{693 \, b^{6} x^{3}} \]

input

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="fricas")
 

output

-4/693*(128*a^4*b*x^2 + 80*a^2*b^3*x + 63*b^5 - 2*(128*a^5*x^2 + 48*a^3*b^ 
2*x + 35*a*b^4)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(b^6*x^3)
 

3.2.23.6 Sympy [F]

\[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{\frac {7}{2}} \sqrt {a x + b \sqrt {x}}}\, dx \]

input

integrate(1/x**(7/2)/(b*x**(1/2)+a*x)**(1/2),x)
 

output

Integral(1/(x**(7/2)*sqrt(a*x + b*sqrt(x))), x)
 

3.2.23.7 Maxima [F]

\[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b \sqrt {x}} x^{\frac {7}{2}}} \,d x } \]

input

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="maxima")
 

output

integrate(1/(sqrt(a*x + b*sqrt(x))*x^(7/2)), x)
 

3.2.23.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=\frac {4 \, {\left (3696 \, a^{\frac {5}{2}} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{5} + 7920 \, a^{2} b {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{4} + 6930 \, a^{\frac {3}{2}} b^{2} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{3} + 3080 \, a b^{3} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{2} + 693 \, \sqrt {a} b^{4} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + 63 \, b^{5}\right )}}{693 \, {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )}^{11}} \]

input

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(1/2),x, algorithm="giac")
 

output

4/693*(3696*a^(5/2)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^5 + 7920*a^2 
*b*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^4 + 6930*a^(3/2)*b^2*(sqrt(a) 
*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3 + 3080*a*b^3*(sqrt(a)*sqrt(x) - sqrt(a 
*x + b*sqrt(x)))^2 + 693*sqrt(a)*b^4*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt( 
x))) + 63*b^5)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^11
 

3.2.23.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {1}{x^{7/2}\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]

input

int(1/(x^(7/2)*(a*x + b*x^(1/2))^(1/2)),x)
 

output

int(1/(x^(7/2)*(a*x + b*x^(1/2))^(1/2)), x)