\(\int \frac {(b x^{2/3}+a x)^{3/2}}{x^5} \, dx\) [183]

Optimal result

Integrand size = 19, antiderivative size = 291 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {429 a^9 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{15/2}} \]

[Out]

-1/3*(b*x^(2/3)+a*x)^(3/2)/x^4+429/32768*a^9*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(15/2)-1/16*a*(b
*x^(2/3)+a*x)^(1/2)/x^3-1/224*a^2*(b*x^(2/3)+a*x)^(1/2)/b/x^(8/3)+13/2688*a^3*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(7/3
)-143/26880*a^4*(b*x^(2/3)+a*x)^(1/2)/b^3/x^2+429/71680*a^5*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(5/3)-143/20480*a^6*(b
*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+143/16384*a^7*(b*x^(2/3)+a*x)^(1/2)/b^6/x-429/32768*a^8*(b*x^(2/3)+a*x)^(1/2)/
b^7/x^(2/3)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2045, 2050, 2054, 212} \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\frac {429 a^9 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac {429 a^8 \sqrt {a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac {143 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^6 x}-\frac {143 a^6 \sqrt {a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac {143 a^4 \sqrt {a x+b x^{2/3}}}{26880 b^3 x^2}+\frac {13 a^3 \sqrt {a x+b x^{2/3}}}{2688 b^2 x^{7/3}}-\frac {a^2 \sqrt {a x+b x^{2/3}}}{224 b x^{8/3}}-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}-\frac {a \sqrt {a x+b x^{2/3}}}{16 x^3} \]

[In]

Int[(b*x^(2/3) + a*x)^(3/2)/x^5,x]

[Out]

-1/16*(a*Sqrt[b*x^(2/3) + a*x])/x^3 - (a^2*Sqrt[b*x^(2/3) + a*x])/(224*b*x^(8/3)) + (13*a^3*Sqrt[b*x^(2/3) + a
*x])/(2688*b^2*x^(7/3)) - (143*a^4*Sqrt[b*x^(2/3) + a*x])/(26880*b^3*x^2) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(7
1680*b^4*x^(5/3)) - (143*a^6*Sqrt[b*x^(2/3) + a*x])/(20480*b^5*x^(4/3)) + (143*a^7*Sqrt[b*x^(2/3) + a*x])/(163
84*b^6*x) - (429*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^7*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(3*x^4) + (429*a^9*A
rcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(32768*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 2045

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b*
x^n)^p/(c*(m + j*p + 1))), x] - Dist[b*p*((n - j)/(c^n*(m + j*p + 1))), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 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 {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {1}{6} a \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {1}{96} a^2 \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (13 a^3\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{1344 b} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (143 a^4\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{16128 b^2} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (143 a^5\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{17920 b^3} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (143 a^6\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{20480 b^4} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (143 a^7\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{24576 b^5} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (143 a^8\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{32768 b^6} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac {\left (143 a^9\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{65536 b^7} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {\left (429 a^9\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 )}{32768 b^7} \\ & = -\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {429 a^9 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{15/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.21 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\frac {6 a^9 \left (b+a \sqrt [3]{x}\right )^2 \sqrt {b x^{2/3}+a x} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},10,\frac {7}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{5 b^{10} \sqrt [3]{x}} \]

[In]

Integrate[(b*x^(2/3) + a*x)^(3/2)/x^5,x]

[Out]

(6*a^9*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[5/2, 10, 7/2, 1 + (a*x^(1/3))/b])/(5*b^10*x^(
1/3))

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.62

method	result	size
derivativedivides	\(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 b^{\frac {31}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}-390390 b^{\frac {29}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}-2633274 b^{\frac {27}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+4349826 b^{\frac {25}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-4685824 b^{\frac {23}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+3317886 b^{\frac {21}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}}-1495494 b^{\frac {19}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}}+390390 b^{\frac {17}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}}-45045 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}}+45045 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}\right )}{3440640 x^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}}\)	\(181\)
default	\(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 b^{\frac {31}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}-390390 b^{\frac {29}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}-2633274 b^{\frac {27}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+4349826 b^{\frac {25}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-4685824 b^{\frac {23}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+3317886 b^{\frac {21}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}}-1495494 b^{\frac {19}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}}+390390 b^{\frac {17}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}}-45045 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}}+45045 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}\right )}{3440640 x^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}}\)	\(181\)

[In]

int((b*x^(2/3)+a*x)^(3/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

1/3440640*(b*x^(2/3)+a*x)^(3/2)*(45045*b^(31/2)*(b+a*x^(1/3))^(1/2)-390390*b^(29/2)*(b+a*x^(1/3))^(3/2)-263327
4*b^(27/2)*(b+a*x^(1/3))^(5/2)+4349826*b^(25/2)*(b+a*x^(1/3))^(7/2)-4685824*b^(23/2)*(b+a*x^(1/3))^(9/2)+33178
86*b^(21/2)*(b+a*x^(1/3))^(11/2)-1495494*b^(19/2)*(b+a*x^(1/3))^(13/2)+390390*b^(17/2)*(b+a*x^(1/3))^(15/2)-45
045*b^(15/2)*(b+a*x^(1/3))^(17/2)+45045*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*b^7*a^9*x^3)/x^4/(b+a*x^(1/3))^(3
/2)/b^(29/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\text {Timed out} \]

[In]

integrate((b*x^(2/3)+a*x)^(3/2)/x^5,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]

[In]

integrate((b*x**(2/3)+a*x)**(3/2)/x**5,x)

[Out]

Integral((a*x + b*x**(2/3))**(3/2)/x**5, x)

Maxima [F]

\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \]

[In]

integrate((b*x^(2/3)+a*x)^(3/2)/x^5,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(2/3))^(3/2)/x^5, x)

Giac [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.67 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=-\frac {\frac {45045 \, a^{10} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{10} - 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{10} b + 1495494 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{10} b^{2} - 3317886 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{10} b^{3} + 4685824 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{10} b^{4} - 4349826 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{10} b^{5} + 2633274 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{10} b^{6} + 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{10} b^{7} - 45045 \, \sqrt {a x^{\frac {1}{3}} + b} a^{10} b^{8}}{a^{9} b^{7} x^{3}}}{3440640 \, a} \]

[In]

integrate((b*x^(2/3)+a*x)^(3/2)/x^5,x, algorithm="giac")

[Out]

-1/3440640*(45045*a^10*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(17/2)*a^1
0 - 390390*(a*x^(1/3) + b)^(15/2)*a^10*b + 1495494*(a*x^(1/3) + b)^(13/2)*a^10*b^2 - 3317886*(a*x^(1/3) + b)^(
11/2)*a^10*b^3 + 4685824*(a*x^(1/3) + b)^(9/2)*a^10*b^4 - 4349826*(a*x^(1/3) + b)^(7/2)*a^10*b^5 + 2633274*(a*
x^(1/3) + b)^(5/2)*a^10*b^6 + 390390*(a*x^(1/3) + b)^(3/2)*a^10*b^7 - 45045*sqrt(a*x^(1/3) + b)*a^10*b^8)/(a^9
*b^7*x^3))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^5} \,d x \]

[In]

int((a*x + b*x^(2/3))^(3/2)/x^5,x)

[Out]

int((a*x + b*x^(2/3))^(3/2)/x^5, x)