\(\int \frac {x^3}{(b x^{2/3}+a x)^{3/2}} \, dx\) [174]

Optimal result

Integrand size = 19, antiderivative size = 248 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {32768 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^8}-\frac {65536 b^7 \sqrt {b x^{2/3}+a x}}{2145 a^9 \sqrt [3]{x}}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^6}-\frac {3584 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2} \] Output:

-6*x^3/a/(b*x^(2/3)+a*x)^(1/2)+32768/2145*b^6*(b*x^(2/3)+a*x)^(1/2)/a^8-65 
536/2145*b^7*(b*x^(2/3)+a*x)^(1/2)/a^9/x^(1/3)-8192/715*b^5*x^(1/3)*(b*x^( 
2/3)+a*x)^(1/2)/a^7+4096/429*b^4*x^(2/3)*(b*x^(2/3)+a*x)^(1/2)/a^6-3584/42 
9*b^3*x*(b*x^(2/3)+a*x)^(1/2)/a^5+5376/715*b^2*x^(4/3)*(b*x^(2/3)+a*x)^(1/ 
2)/a^4-448/65*b*x^(5/3)*(b*x^(2/3)+a*x)^(1/2)/a^3+32/5*x^2*(b*x^(2/3)+a*x) 
^(1/2)/a^2
 

Mathematica [A] (verified)

Time = 4.50 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.49 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {2 \left (-32768 b^8 \sqrt [3]{x}-16384 a b^7 x^{2/3}+4096 a^2 b^6 x-2048 a^3 b^5 x^{4/3}+1280 a^4 b^4 x^{5/3}-896 a^5 b^3 x^2+672 a^6 b^2 x^{7/3}-528 a^7 b x^{8/3}+429 a^8 x^3\right )}{2145 a^9 \sqrt {b x^{2/3}+a x}} \] Input:

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

Output:

(2*(-32768*b^8*x^(1/3) - 16384*a*b^7*x^(2/3) + 4096*a^2*b^6*x - 2048*a^3*b 
^5*x^(4/3) + 1280*a^4*b^4*x^(5/3) - 896*a^5*b^3*x^2 + 672*a^6*b^2*x^(7/3) 
- 528*a^7*b*x^(8/3) + 429*a^8*x^3))/(2145*a^9*Sqrt[b*x^(2/3) + a*x])
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1921, 1922, 1922, 1922, 1922, 1922, 1922, 1908, 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.

\(\displaystyle \int \frac {x^3}{\left (a x+b x^{2/3}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1921

\(\displaystyle \frac {16 \int \frac {x^2}{\sqrt {x^{2/3} b+a x}}dx}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \int \frac {x^{5/3}}{\sqrt {x^{2/3} b+a x}}dx}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \int \frac {x^{4/3}}{\sqrt {x^{2/3} b+a x}}dx}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \int \frac {x}{\sqrt {x^{2/3} b+a x}}dx}{11 a}\right )}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \int \frac {x^{2/3}}{\sqrt {x^{2/3} b+a x}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \int \frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \int \frac {1}{\sqrt {x^{2/3} b+a x}}dx}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1908

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {2 b \int \frac {1}{\sqrt [3]{x} \sqrt {x^{2/3} b+a x}}dx}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

\(\Big \downarrow \) 1920

\(\displaystyle \frac {16 \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {4 b \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{a}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}}\)

Input:

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

Output:

(-6*x^3)/(a*Sqrt[b*x^(2/3) + a*x]) + (16*((2*x^2*Sqrt[b*x^(2/3) + a*x])/(5 
*a) - (14*b*((6*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(13*a) - (12*b*((6*x^(4/3)* 
Sqrt[b*x^(2/3) + a*x])/(11*a) - (10*b*((2*x*Sqrt[b*x^(2/3) + a*x])/(3*a) - 
 (8*b*((6*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(7*a) - (6*b*((6*x^(1/3)*Sqrt[b*x 
^(2/3) + a*x])/(5*a) - (4*b*((2*Sqrt[b*x^(2/3) + a*x])/a - (4*b*Sqrt[b*x^( 
2/3) + a*x])/(a^2*x^(1/3))))/(5*a)))/(7*a)))/(9*a)))/(11*a)))/(13*a)))/(15 
*a)))/a
 

Defintions of rubi rules used

rule 1908

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Simp[b*((n*p + n - j + 1)/(a*( 
j*p + 1)))   Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, 
p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n 
- j)], 0] && NeQ[j*p + 1, 0]
 

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 1921

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] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))   In 
t[(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 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])
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.44

method	result	size
derivativedivides	\(\frac {2 x \left (x^{\frac {1}{3}} a +b \right ) \left (429 a^{8} x^{\frac {8}{3}}-528 a^{7} b \,x^{\frac {7}{3}}+672 a^{6} b^{2} x^{2}-896 a^{5} b^{3} x^{\frac {5}{3}}+1280 x^{\frac {4}{3}} a^{4} b^{4}-2048 a^{3} b^{5} x +4096 a^{2} b^{6} x^{\frac {2}{3}}-16384 x^{\frac {1}{3}} a \,b^{7}-32768 b^{8}\right )}{2145 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{9}}\)	\(110\)
default	\(\frac {2 x \left (x^{\frac {1}{3}} a +b \right ) \left (429 a^{8} x^{\frac {8}{3}}-528 a^{7} b \,x^{\frac {7}{3}}+672 a^{6} b^{2} x^{2}-896 a^{5} b^{3} x^{\frac {5}{3}}+1280 x^{\frac {4}{3}} a^{4} b^{4}-2048 a^{3} b^{5} x +4096 a^{2} b^{6} x^{\frac {2}{3}}-16384 x^{\frac {1}{3}} a \,b^{7}-32768 b^{8}\right )}{2145 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{9}}\)	\(110\)

Input:

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

Output:

2/2145*x*(x^(1/3)*a+b)*(429*a^8*x^(8/3)-528*a^7*b*x^(7/3)+672*a^6*b^2*x^2- 
896*a^5*b^3*x^(5/3)+1280*x^(4/3)*a^4*b^4-2048*a^3*b^5*x+4096*a^2*b^6*x^(2/ 
3)-16384*x^(1/3)*a*b^7-32768*b^8)/(b*x^(2/3)+a*x)^(3/2)/a^9
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2083 vs. \(2 (186) = 372\).

Time = 124.49 (sec) , antiderivative size = 2083, normalized size of antiderivative = 8.40 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2145*((402653184*a^3*b^16 + 335544320*a^3*b^15 - 125829120*a^3*b^14 + 62 
4624*a^15 + 25165824*(17*a^6 - 3*a^3)*b^13 + 524288*(464*a^6 + 53*a^3)*b^1 
2 - 786432*(246*a^6 + a^3)*b^11 + 98304*(1036*a^9 - 2560*a^6 - 3*a^3)*b^10 
 - 32768*(758*a^9 - 1569*a^6)*b^9 - 24576*(5803*a^9 + 124*a^6)*b^8 + 6144* 
(600*a^12 - 20924*a^9 - 33*a^6)*b^7 - 1536*(7666*a^12 - 7357*a^9)*b^6 - 76 
8*(40107*a^12 + 1033*a^9)*b^5 + 96*(63360*a^15 + 167852*a^12 + 267*a^9)*b^ 
4 + 32*(613440*a^15 - 105031*a^12)*b^3 + 468*(34560*a^15 + 661*a^12)*b^2 - 
 99*(68480*a^15 + 87*a^12)*b)*x^2 + (402653184*b^19 + 335544320*b^18 + 251 
65824*(17*a^3 - 3)*b^16 - 125829120*b^17 + 524288*(464*a^3 + 53)*b^15 + 62 
4624*a^12*b^3 - 786432*(246*a^3 + 1)*b^14 + 98304*(1036*a^6 - 2560*a^3 - 3 
)*b^13 - 32768*(758*a^6 - 1569*a^3)*b^12 - 24576*(5803*a^6 + 124*a^3)*b^11 
 + 6144*(600*a^9 - 20924*a^6 - 33*a^3)*b^10 - 1536*(7666*a^9 - 7357*a^6)*b 
^9 - 768*(40107*a^9 + 1033*a^6)*b^8 + 96*(63360*a^12 + 167852*a^9 + 267*a^ 
6)*b^7 + 32*(613440*a^12 - 105031*a^9)*b^6 + 468*(34560*a^12 + 661*a^9)*b^ 
5 - 99*(68480*a^12 + 87*a^9)*b^4)*x + 2*(429*(4096*a^10*b^9 + 6144*a^10*b^ 
8 + 768*a^10*b^7 - 4096*a^16 - 144*a^13*b^2 + 216*a^13*b - 27*a^13 + 256*( 
16*a^13 - 7*a^10)*b^6 + 48*(128*a^13 - 3*a^10)*b^5 + 24*(32*a^13 + 9*a^10) 
*b^4 - (5888*a^13 + 27*a^10)*b^3)*x^4 - 2096*(4096*a^7*b^12 + 6144*a^7*b^1 
1 + 768*a^7*b^10 - 144*a^10*b^5 + 216*a^10*b^4 + 256*(16*a^10 - 7*a^7)*b^9 
 + 48*(128*a^10 - 3*a^7)*b^8 + 24*(32*a^10 + 9*a^7)*b^7 - (5888*a^10 + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.68 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {65536 \, b^{\frac {15}{2}}}{2145 \, a^{9}} - \frac {2 \, {\left (\frac {6435 \, b^{8}}{\sqrt {a x^{\frac {1}{3}} + b} a} - \frac {429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{14} - 3960 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{14} b + 16380 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{14} b^{2} - 40040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{14} b^{3} + 64350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{14} b^{4} - 72072 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{14} b^{5} + 60060 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{14} b^{6} - 51480 \, \sqrt {a x^{\frac {1}{3}} + b} a^{14} b^{7}}{a^{15}}\right )}}{2145 \, a^{8}} \] Input:

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

Output:

65536/2145*b^(15/2)/a^9 - 2/2145*(6435*b^8/(sqrt(a*x^(1/3) + b)*a) - (429* 
(a*x^(1/3) + b)^(15/2)*a^14 - 3960*(a*x^(1/3) + b)^(13/2)*a^14*b + 16380*( 
a*x^(1/3) + b)^(11/2)*a^14*b^2 - 40040*(a*x^(1/3) + b)^(9/2)*a^14*b^3 + 64 
350*(a*x^(1/3) + b)^(7/2)*a^14*b^4 - 72072*(a*x^(1/3) + b)^(5/2)*a^14*b^5 
+ 60060*(a*x^(1/3) + b)^(3/2)*a^14*b^6 - 51480*sqrt(a*x^(1/3) + b)*a^14*b^ 
7)/a^15)/a^8
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.40 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {\frac {2 x^{\frac {8}{3}} a^{8}}{5}-\frac {1792 x^{\frac {5}{3}} a^{5} b^{3}}{2145}+\frac {8192 x^{\frac {2}{3}} a^{2} b^{6}}{2145}-\frac {32 x^{\frac {7}{3}} a^{7} b}{65}+\frac {512 x^{\frac {4}{3}} a^{4} b^{4}}{429}-\frac {32768 x^{\frac {1}{3}} a \,b^{7}}{2145}+\frac {448 a^{6} b^{2} x^{2}}{715}-\frac {4096 a^{3} b^{5} x}{2145}-\frac {65536 b^{8}}{2145}}{\sqrt {x^{\frac {1}{3}} a +b}\, a^{9}} \] Input:

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

Output:

(2*(429*x**(2/3)*a**8*x**2 - 896*x**(2/3)*a**5*b**3*x + 4096*x**(2/3)*a**2 
*b**6 - 528*x**(1/3)*a**7*b*x**2 + 1280*x**(1/3)*a**4*b**4*x - 16384*x**(1 
/3)*a*b**7 + 672*a**6*b**2*x**2 - 2048*a**3*b**5*x - 32768*b**8))/(2145*sq 
rt(x**(1/3)*a + b)*a**9)