\(\int \frac {1}{x^{7/2} (b \sqrt {x}+a x)^{3/2}} \, dx\) [130]

Optimal result

Integrand size = 21, antiderivative size = 223 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}-\frac {8192 a^6 \sqrt {b \sqrt {x}+a x}}{429 b^8 \sqrt {x}} \]

[Out]

4/b/x^3/(b*x^(1/2)+a*x)^(1/2)-56/13*(b*x^(1/2)+a*x)^(1/2)/b^2/x^(7/2)+672/143*a*(b*x^(1/2)+a*x)^(1/2)/b^3/x^3-
2240/429*a^2*(b*x^(1/2)+a*x)^(1/2)/b^4/x^(5/2)+2560/429*a^3*(b*x^(1/2)+a*x)^(1/2)/b^5/x^2-1024/143*a^4*(b*x^(1
/2)+a*x)^(1/2)/b^6/x^(3/2)+4096/429*a^5*(b*x^(1/2)+a*x)^(1/2)/b^7/x-8192/429*a^6*(b*x^(1/2)+a*x)^(1/2)/b^8/x^(
1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2040, 2041, 2039} \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {8192 a^6 \sqrt {a x+b \sqrt {x}}}{429 b^8 \sqrt {x}}+\frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{429 b^7 x}-\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{143 b^6 x^{3/2}}+\frac {2560 a^3 \sqrt {a x+b \sqrt {x}}}{429 b^5 x^2}-\frac {2240 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^4 x^{5/2}}+\frac {672 a \sqrt {a x+b \sqrt {x}}}{143 b^3 x^3}-\frac {56 \sqrt {a x+b \sqrt {x}}}{13 b^2 x^{7/2}}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}} \]

[In]

Int[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

4/(b*x^3*Sqrt[b*Sqrt[x] + a*x]) - (56*Sqrt[b*Sqrt[x] + a*x])/(13*b^2*x^(7/2)) + (672*a*Sqrt[b*Sqrt[x] + a*x])/
(143*b^3*x^3) - (2240*a^2*Sqrt[b*Sqrt[x] + a*x])/(429*b^4*x^(5/2)) + (2560*a^3*Sqrt[b*Sqrt[x] + a*x])/(429*b^5
*x^2) - (1024*a^4*Sqrt[b*Sqrt[x] + a*x])/(143*b^6*x^(3/2)) + (4096*a^5*Sqrt[b*Sqrt[x] + a*x])/(429*b^7*x) - (8
192*a^6*Sqrt[b*Sqrt[x] + a*x])/(429*b^8*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 2040

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] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n,
j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (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*} \text {integral}& = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}+\frac {14 \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx}{b} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}-\frac {(168 a) \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx}{13 b^2} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}+\frac {\left (1680 a^2\right ) \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^3} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}-\frac {\left (4480 a^3\right ) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^4} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}+\frac {\left (1280 a^4\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^5} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}-\frac {\left (1024 a^5\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^6} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}+\frac {\left (2048 a^6\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^7} \\ & = \frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}-\frac {8192 a^6 \sqrt {b \sqrt {x}+a x}}{429 b^8 \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \sqrt {b \sqrt {x}+a x} \left (33 b^7-42 a b^6 \sqrt {x}+56 a^2 b^5 x-80 a^3 b^4 x^{3/2}+128 a^4 b^3 x^2-256 a^5 b^2 x^{5/2}+1024 a^6 b x^3+2048 a^7 x^{7/2}\right )}{429 b^8 \left (b+a \sqrt {x}\right ) x^{7/2}} \]

[In]

Integrate[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x]*(33*b^7 - 42*a*b^6*Sqrt[x] + 56*a^2*b^5*x - 80*a^3*b^4*x^(3/2) + 128*a^4*b^3*x^2 - 2
56*a^5*b^2*x^(5/2) + 1024*a^6*b*x^3 + 2048*a^7*x^(7/2)))/(429*b^8*(b + a*Sqrt[x])*x^(7/2))

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {4}{13 b \,x^{3} \sqrt {b \sqrt {x}+a x}}-\frac {28 a \left (-\frac {2}{11 b \,x^{\frac {5}{2}} \sqrt {b \sqrt {x}+a x}}-\frac {12 a \left (-\frac {2}{9 b \,x^{2} \sqrt {b \sqrt {x}+a x}}-\frac {10 a \left (-\frac {2}{7 b \,x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}-\frac {8 a \left (-\frac {2}{5 b x \sqrt {b \sqrt {x}+a x}}-\frac {6 a \left (-\frac {2}{3 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}+\frac {8 a \left (b +2 a \sqrt {x}\right )}{3 b^{3} \sqrt {b \sqrt {x}+a x}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )}{13 b}\)	\(176\)
default	\(\frac {2 \sqrt {b \sqrt {x}+a x}\, \left (2574 x^{\frac {17}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {19}{2}}+2574 x^{\frac {17}{2}} a^{\frac {19}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}-6006 x^{\frac {15}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {17}{2}}+858 x^{\frac {15}{2}} a^{\frac {17}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}}+2574 x^{\frac {15}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {15}{2}} b^{2}+2574 x^{\frac {15}{2}} a^{\frac {15}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{2}-2048 x^{\frac {13}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {13}{2}} b^{2}+1287 x^{\frac {17}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{9} b -1287 x^{\frac {17}{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^{9} b -256 x^{\frac {11}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{4}+5148 x^{8} \sqrt {b \sqrt {x}+a x}\, a^{\frac {17}{2}} b +5148 x^{8} a^{\frac {17}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b +1287 x^{\frac {15}{2}} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{7} b^{3}-1287 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^{3}-9244 x^{7} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {15}{2}} b -112 x^{\frac {9}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{6}+512 x^{6} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {11}{2}} b^{3}+160 x^{5} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{5}-66 x^{\frac {7}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {a}\, b^{8}+2574 x^{8} \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{8} b^{2}-2574 x^{8} \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^{8} b^{2}+84 x^{4} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{7}\right )}{429 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{9} x^{\frac {15}{2}} \sqrt {a}\, \left (a \sqrt {x}+b \right )^{2}}\)	\(636\)

[In]

int(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-4/13/b/x^3/(b*x^(1/2)+a*x)^(1/2)-28/13*a/b*(-2/11/b/x^(5/2)/(b*x^(1/2)+a*x)^(1/2)-12/11*a/b*(-2/9/b/x^2/(b*x^
(1/2)+a*x)^(1/2)-10/9*a/b*(-2/7/b/x^(3/2)/(b*x^(1/2)+a*x)^(1/2)-8/7*a/b*(-2/5/b/x/(b*x^(1/2)+a*x)^(1/2)-6/5*a/
b*(-2/3/b/x^(1/2)/(b*x^(1/2)+a*x)^(1/2)+8/3*a/b^3*(b+2*a*x^(1/2))/(b*x^(1/2)+a*x)^(1/2))))))

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4 \, {\left (1024 \, a^{7} b x^{4} - 384 \, a^{5} b^{3} x^{3} - 136 \, a^{3} b^{5} x^{2} - 75 \, a b^{7} x - {\left (2048 \, a^{8} x^{4} - 1280 \, a^{6} b^{2} x^{3} - 208 \, a^{4} b^{4} x^{2} - 98 \, a^{2} b^{6} x - 33 \, b^{8}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{429 \, {\left (a^{2} b^{8} x^{5} - b^{10} x^{4}\right )}} \]

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")

[Out]

4/429*(1024*a^7*b*x^4 - 384*a^5*b^3*x^3 - 136*a^3*b^5*x^2 - 75*a*b^7*x - (2048*a^8*x^4 - 1280*a^6*b^2*x^3 - 20
8*a^4*b^4*x^2 - 98*a^2*b^6*x - 33*b^8)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^8*x^5 - b^10*x^4)

Sympy [F]

\[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{\frac {7}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/x**(7/2)/(b*x**(1/2)+a*x)**(3/2),x)

[Out]

Integral(1/(x**(7/2)*(a*x + b*sqrt(x))**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)

Giac [F]

\[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{7/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]

[In]

int(1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)),x)

[Out]

int(1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)), x)