\(\int \frac {x^3}{(b x^{2/3}+a x)^{3/2}} \, dx\) [195]

Optimal result

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

[Out]

-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^8-65536/2145*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/
429*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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2040, 2041, 2027, 2039} \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {65536 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^9 \sqrt [3]{x}}+\frac {32768 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^8}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^6}-\frac {3584 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^3}+\frac {32 x^2 \sqrt {a x+b x^{2/3}}}{5 a^2}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}} \]

[In]

Int[x^3/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-6*x^3)/(a*Sqrt[b*x^(2/3) + a*x]) + (32768*b^6*Sqrt[b*x^(2/3) + a*x])/(2145*a^8) - (65536*b^7*Sqrt[b*x^(2/3)
+ a*x])/(2145*a^9*x^(1/3)) - (8192*b^5*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^7) + (4096*b^4*x^(2/3)*Sqrt[b*x^(
2/3) + a*x])/(429*a^6) - (3584*b^3*x*Sqrt[b*x^(2/3) + a*x])/(429*a^5) + (5376*b^2*x^(4/3)*Sqrt[b*x^(2/3) + a*x
])/(715*a^4) - (448*b*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(65*a^3) + (32*x^2*Sqrt[b*x^(2/3) + a*x])/(5*a^2)

Rule 2027

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] - Dist[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 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 {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {16 \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx}{a} \\ & = -\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}-\frac {(224 b) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{15 a^2} \\ & = -\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}-\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}+\frac {\left (896 b^2\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{65 a^3} \\ & = -\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\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}-\frac {\left (1792 b^3\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{143 a^4} \\ & = -\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}-\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}+\frac {\left (14336 b^4\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{1287 a^5} \\ & = -\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\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}-\frac {\left (4096 b^5\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{429 a^6} \\ & = -\frac {6 x^3}{a \sqrt {b x^{2/3}+a 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}+\frac {\left (16384 b^6\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{2145 a^7} \\ & = -\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 {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}-\frac {\left (32768 b^7\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{6435 a^8} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.49 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {2 \left (-32768 b^8 \sqrt [3]{x}-16384 a b^7 x^{2/3}+4096 a^2 b^6 x-2048 a^3 b^5 x^{4/3}+1280 a^4 b^4 x^{5/3}-896 a^5 b^3 x^2+672 a^6 b^2 x^{7/3}-528 a^7 b x^{8/3}+429 a^8 x^3\right )}{2145 a^9 \sqrt {b x^{2/3}+a x}} \]

[In]

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

[Out]

(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])

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.44

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

[In]

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

[Out]

2/2145*x*(b+a*x^(1/3))*(429*a^8*x^(8/3)-528*a^7*b*x^(7/3)+672*a^6*x^2*b^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 = 123.02 (sec) , antiderivative size = 2083, normalized size of antiderivative = 8.40 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2145*((402653184*a^3*b^16 + 335544320*a^3*b^15 - 125829120*a^3*b^14 + 624624*a^15 + 25165824*(17*a^6 - 3*a^3
)*b^13 + 524288*(464*a^6 + 53*a^3)*b^12 - 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 - 768*(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 + 25165824*(17*a^3 - 3)*b^16 - 125829120*b^17 + 524288*(464*a^3 + 53)*b^1
5 + 624624*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 - 735
7*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 - 1
05031*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 + 61
44*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^11 + 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 + 27*a^7)*b^6 - (4096*a^13 + 27*a^10)*b^3)*x^3 + 7424*(4096*a^4*b
^15 + 6144*a^4*b^14 + 768*a^4*b^13 - 144*a^7*b^8 + 216*a^7*b^7 + 256*(16*a^7 - 7*a^4)*b^12 + 48*(128*a^7 - 3*a
^4)*b^11 + 24*(32*a^7 + 9*a^4)*b^10 - (5888*a^7 + 27*a^4)*b^9 - (4096*a^10 + 27*a^7)*b^6)*x^2 + 16384*(4096*a*
b^18 + 6144*a*b^17 + 768*a*b^16 + 256*(16*a^4 - 7*a)*b^15 - 144*a^4*b^11 + 48*(128*a^4 - 3*a)*b^14 + 216*a^4*b
^10 + 24*(32*a^4 + 9*a)*b^13 - (5888*a^4 + 27*a)*b^12 - (4096*a^7 + 27*a^4)*b^9)*x - (134217728*b^19 + 2013265
92*b^18 + 8388608*(16*a^3 - 7)*b^16 + 25165824*b^17 + 1572864*(128*a^3 - 3)*b^15 - 4718592*a^3*b^12 + 786432*(
32*a^3 + 9)*b^14 + 7077888*a^3*b^11 - 32768*(5888*a^3 + 27)*b^13 - 32768*(4096*a^6 + 27*a^3)*b^10 + 957*(4096*
a^9*b^10 + 6144*a^9*b^9 + 768*a^9*b^8 - 144*a^12*b^3 + 216*a^12*b^2 + 256*(16*a^12 - 7*a^9)*b^7 + 48*(128*a^12
- 3*a^9)*b^6 + 24*(32*a^12 + 9*a^9)*b^5 - (5888*a^12 + 27*a^9)*b^4 - (4096*a^15 + 27*a^12)*b)*x^3 - 2848*(409
6*a^6*b^13 + 6144*a^6*b^12 + 768*a^6*b^11 - 144*a^9*b^6 + 216*a^9*b^5 + 256*(16*a^9 - 7*a^6)*b^10 + 48*(128*a^
9 - 3*a^6)*b^9 + 24*(32*a^9 + 9*a^6)*b^8 - (5888*a^9 + 27*a^6)*b^7 - (4096*a^12 + 27*a^9)*b^4)*x^2 + 22528*(40
96*a^3*b^16 + 6144*a^3*b^15 + 768*a^3*b^14 - 144*a^6*b^9 + 216*a^6*b^8 + 256*(16*a^6 - 7*a^3)*b^13 + 48*(128*a
^6 - 3*a^3)*b^12 + 24*(32*a^6 + 9*a^3)*b^11 - (5888*a^6 + 27*a^3)*b^10 - (4096*a^9 + 27*a^6)*b^7)*x)*x^(2/3) +
3*(543*(4096*a^8*b^11 + 6144*a^8*b^10 + 768*a^8*b^9 - 144*a^11*b^4 + 216*a^11*b^3 + 256*(16*a^11 - 7*a^8)*b^8
+ 48*(128*a^11 - 3*a^8)*b^7 + 24*(32*a^11 + 9*a^8)*b^6 - (5888*a^11 + 27*a^8)*b^5 - (4096*a^14 + 27*a^11)*b^2
)*x^3 - 1408*(4096*a^5*b^14 + 6144*a^5*b^13 + 768*a^5*b^12 - 144*a^8*b^7 + 216*a^8*b^6 + 256*(16*a^8 - 7*a^5)*
b^11 + 48*(128*a^8 - 3*a^5)*b^10 + 24*(32*a^8 + 9*a^5)*b^9 - (5888*a^8 + 27*a^5)*b^8 - (4096*a^11 + 27*a^8)*b^
5)*x^2 - 4096*(4096*a^2*b^17 + 6144*a^2*b^16 + 768*a^2*b^15 - 144*a^5*b^10 + 256*(16*a^5 - 7*a^2)*b^14 + 216*a
^5*b^9 + 48*(128*a^5 - 3*a^2)*b^13 + 24*(32*a^5 + 9*a^2)*b^12 - (5888*a^5 + 27*a^2)*b^11 - (4096*a^8 + 27*a^5)
*b^8)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((4096*a^12*b^9 + 6144*a^12*b^8 + 768*a^12*b^7 - 4096*a^18 - 144*a^15
*b^2 + 216*a^15*b - 27*a^15 + 256*(16*a^15 - 7*a^12)*b^6 + 48*(128*a^15 - 3*a^12)*b^5 + 24*(32*a^15 + 9*a^12)*
b^4 - (5888*a^15 + 27*a^12)*b^3)*x^2 + (4096*a^9*b^12 + 6144*a^9*b^11 + 768*a^9*b^10 - 144*a^12*b^5 + 216*a^12
*b^4 + 256*(16*a^12 - 7*a^9)*b^9 + 48*(128*a^12 - 3*a^9)*b^8 + 24*(32*a^12 + 9*a^9)*b^7 - (5888*a^12 + 27*a^9)
*b^6 - (4096*a^15 + 27*a^12)*b^3)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.66 \[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {65536 \, b^{\frac {15}{2}}}{2145 \, a^{9}} - \frac {6 \, b^{8}}{\sqrt {a x^{\frac {1}{3}} + b} a^{9}} + \frac {2 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{126} - 3960 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{126} b + 16380 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{126} b^{2} - 40040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{126} b^{3} + 64350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{126} b^{4} - 72072 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{126} b^{5} + 60060 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{126} b^{6} - 51480 \, \sqrt {a x^{\frac {1}{3}} + b} a^{126} b^{7}\right )}}{2145 \, a^{135}} \]

[In]

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

[Out]

65536/2145*b^(15/2)/a^9 - 6*b^8/(sqrt(a*x^(1/3) + b)*a^9) + 2/2145*(429*(a*x^(1/3) + b)^(15/2)*a^126 - 3960*(a
*x^(1/3) + b)^(13/2)*a^126*b + 16380*(a*x^(1/3) + b)^(11/2)*a^126*b^2 - 40040*(a*x^(1/3) + b)^(9/2)*a^126*b^3
+ 64350*(a*x^(1/3) + b)^(7/2)*a^126*b^4 - 72072*(a*x^(1/3) + b)^(5/2)*a^126*b^5 + 60060*(a*x^(1/3) + b)^(3/2)*
a^126*b^6 - 51480*sqrt(a*x^(1/3) + b)*a^126*b^7)/a^135

Mupad [F(-1)]

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

[In]

int(x^3/(a*x + b*x^(2/3))^(3/2),x)

[Out]

int(x^3/(a*x + b*x^(2/3))^(3/2), x)