\(\int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx\) [192]

Optimal result

Integrand size = 19, antiderivative size = 241 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac {1287 a^7 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^{15/2}} \]

[Out]

1287/1024*a^7*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(15/2)-3/7*(b*x^(2/3)+a*x)^(1/2)/b/x^(8/3)+13/2
8*a*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(7/3)-143/280*a^2*(b*x^(2/3)+a*x)^(1/2)/b^3/x^2+1287/2240*a^3*(b*x^(2/3)+a*x)^
(1/2)/b^4/x^(5/3)-429/640*a^4*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+429/512*a^5*(b*x^(2/3)+a*x)^(1/2)/b^6/x-1287/1
024*a^6*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(2/3)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2050, 2054, 212} \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\frac {1287 a^7 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{1024 b^{15/2}}-\frac {1287 a^6 \sqrt {a x+b x^{2/3}}}{1024 b^7 x^{2/3}}+\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{512 b^6 x}-\frac {429 a^4 \sqrt {a x+b x^{2/3}}}{640 b^5 x^{4/3}}+\frac {1287 a^3 \sqrt {a x+b x^{2/3}}}{2240 b^4 x^{5/3}}-\frac {143 a^2 \sqrt {a x+b x^{2/3}}}{280 b^3 x^2}+\frac {13 a \sqrt {a x+b x^{2/3}}}{28 b^2 x^{7/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}} \]

[In]

Int[1/(x^3*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(7*b*x^(8/3)) + (13*a*Sqrt[b*x^(2/3) + a*x])/(28*b^2*x^(7/3)) - (143*a^2*Sqrt[b*x^(
2/3) + a*x])/(280*b^3*x^2) + (1287*a^3*Sqrt[b*x^(2/3) + a*x])/(2240*b^4*x^(5/3)) - (429*a^4*Sqrt[b*x^(2/3) + a
*x])/(640*b^5*x^(4/3)) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^6*x) - (1287*a^6*Sqrt[b*x^(2/3) + a*x])/(1024*
b^7*x^(2/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2050

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, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}-\frac {(13 a) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{14 b} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}+\frac {\left (143 a^2\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{168 b^2} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}-\frac {\left (429 a^3\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{560 b^3} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}+\frac {\left (429 a^4\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{640 b^4} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}-\frac {\left (143 a^5\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{256 b^5} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}+\frac {\left (429 a^6\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{1024 b^6} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}-\frac {\left (429 a^7\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{2048 b^7} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac {\left (1287 a^7\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^7} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac {1287 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-15360 b^6+16640 a b^5 \sqrt [3]{x}-18304 a^2 b^4 x^{2/3}+20592 a^3 b^3 x-24024 a^4 b^2 x^{4/3}+30030 a^5 b x^{5/3}-45045 a^6 x^2\right )}{35840 b^7 x^{8/3}}+\frac {1287 a^7 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^{15/2}} \]

[In]

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

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-15360*b^6 + 16640*a*b^5*x^(1/3) - 18304*a^2*b^4*x^(2/3) + 20592*a^3*b^3*x - 24024*a^4
*b^2*x^(4/3) + 30030*a^5*b*x^(5/3) - 45045*a^6*x^2))/(35840*b^7*x^(8/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))
/Sqrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (45045 b^{\frac {3}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{6} x^{2}-45045 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{7} b \,x^{\frac {7}{3}}-30030 b^{\frac {5}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{5} x^{\frac {5}{3}}+24024 b^{\frac {7}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{4} x^{\frac {4}{3}}-20592 b^{\frac {9}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{3} x +18304 b^{\frac {11}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{2} x^{\frac {2}{3}}-16640 b^{\frac {13}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a \,x^{\frac {1}{3}}+15360 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {15}{2}}\right )}{35840 x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {17}{2}}}\)	\(183\)
default	\(\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (45045 x^{\frac {13}{3}} \operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{7} b +30030 x^{\frac {11}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {5}{2}} a^{5}-24024 x^{\frac {10}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {7}{2}} a^{4}-18304 x^{\frac {8}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {11}{2}} a^{2}+16640 x^{\frac {7}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {13}{2}} a -15360 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {15}{2}} x^{2}+20592 x^{3} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {9}{2}} a^{3}-45045 x^{4} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}} a^{6}\right )}{35840 x^{4} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {17}{2}}}\)	\(188\)

[In]

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

[Out]

-1/35840/x^2*(b+a*x^(1/3))^(1/2)*(45045*b^(3/2)*(b+a*x^(1/3))^(1/2)*a^6*x^2-45045*arctanh((b+a*x^(1/3))^(1/2)/
b^(1/2))*a^7*b*x^(7/3)-30030*b^(5/2)*(b+a*x^(1/3))^(1/2)*a^5*x^(5/3)+24024*b^(7/2)*(b+a*x^(1/3))^(1/2)*a^4*x^(
4/3)-20592*b^(9/2)*(b+a*x^(1/3))^(1/2)*a^3*x+18304*b^(11/2)*(b+a*x^(1/3))^(1/2)*a^2*x^(2/3)-16640*b^(13/2)*(b+
a*x^(1/3))^(1/2)*a*x^(1/3)+15360*(b+a*x^(1/3))^(1/2)*b^(15/2))/(b*x^(2/3)+a*x)^(1/2)/b^(17/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {\frac {45045 \, a^{8} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{8} - 300300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{8} b + 849849 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{8} b^{2} - 1317888 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{8} b^{3} + 1200199 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{8} b^{4} - 631540 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{8} b^{5} + 169995 \, \sqrt {a x^{\frac {1}{3}} + b} a^{8} b^{6}}{a^{7} b^{7} x^{\frac {7}{3}}}}{35840 \, a} \]

[In]

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

[Out]

-1/35840*(45045*a^8*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(13/2)*a^8 -
300300*(a*x^(1/3) + b)^(11/2)*a^8*b + 849849*(a*x^(1/3) + b)^(9/2)*a^8*b^2 - 1317888*(a*x^(1/3) + b)^(7/2)*a^8
*b^3 + 1200199*(a*x^(1/3) + b)^(5/2)*a^8*b^4 - 631540*(a*x^(1/3) + b)^(3/2)*a^8*b^5 + 169995*sqrt(a*x^(1/3) +
b)*a^8*b^6)/(a^7*b^7*x^(7/3)))/a

Mupad [F(-1)]

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

[In]

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

[Out]

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