\(\int x (b x^{2/3}+a x)^{3/2} \, dx\) [177]

Optimal result

Integrand size = 17, antiderivative size = 255 \[ \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a} \]

[Out]

-256/1615*b^3*(b*x^(2/3)+a*x)^(5/2)/a^4+65536/4849845*b^8*(b*x^(2/3)+a*x)^(5/2)/a^9/x^(5/3)-32768/969969*b^7*(
b*x^(2/3)+a*x)^(5/2)/a^8/x^(4/3)+8192/138567*b^6*(b*x^(2/3)+a*x)^(5/2)/a^7/x-4096/46189*b^5*(b*x^(2/3)+a*x)^(5
/2)/a^6/x^(2/3)+512/4199*b^4*(b*x^(2/3)+a*x)^(5/2)/a^5/x^(1/3)+64/323*b^2*x^(1/3)*(b*x^(2/3)+a*x)^(5/2)/a^3-32
/133*b*x^(2/3)*(b*x^(2/3)+a*x)^(5/2)/a^2+2/7*x*(b*x^(2/3)+a*x)^(5/2)/a

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2041, 2027, 2039} \[ \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac {256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]

[In]

Int[x*(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-256*b^3*(b*x^(2/3) + a*x)^(5/2))/(1615*a^4) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(4849845*a^9*x^(5/3)) - (3
2768*b^7*(b*x^(2/3) + a*x)^(5/2))/(969969*a^8*x^(4/3)) + (8192*b^6*(b*x^(2/3) + a*x)^(5/2))/(138567*a^7*x) - (
4096*b^5*(b*x^(2/3) + a*x)^(5/2))/(46189*a^6*x^(2/3)) + (512*b^4*(b*x^(2/3) + a*x)^(5/2))/(4199*a^5*x^(1/3)) +
(64*b^2*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(323*a^3) - (32*b*x^(2/3)*(b*x^(2/3) + a*x)^(5/2))/(133*a^2) + (2*x*
(b*x^(2/3) + a*x)^(5/2))/(7*a)

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 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 {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {(16 b) \int x^{2/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{21 a} \\ & = -\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (32 b^2\right ) \int \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{57 a^2} \\ & = \frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (128 b^3\right ) \int \left (b x^{2/3}+a x\right )^{3/2} \, dx}{323 a^3} \\ & = -\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (256 b^4\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{969 a^4} \\ & = -\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (2048 b^5\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{12597 a^5} \\ & = -\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (4096 b^6\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{46189 a^6} \\ & = -\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (16384 b^7\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{415701 a^7} \\ & = -\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (32768 b^8\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{2909907 a^8} \\ & = -\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.51 \[ \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \left (b+a \sqrt [3]{x}\right ) \left (b x^{2/3}+a x\right )^{3/2} \left (32768 b^8-81920 a b^7 \sqrt [3]{x}+143360 a^2 b^6 x^{2/3}-215040 a^3 b^5 x+295680 a^4 b^4 x^{4/3}-384384 a^5 b^3 x^{5/3}+480480 a^6 b^2 x^2-583440 a^7 b x^{7/3}+692835 a^8 x^{8/3}\right )}{4849845 a^9 x} \]

[In]

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

[Out]

(2*(b + a*x^(1/3))*(b*x^(2/3) + a*x)^(3/2)*(32768*b^8 - 81920*a*b^7*x^(1/3) + 143360*a^2*b^6*x^(2/3) - 215040*
a^3*b^5*x + 295680*a^4*b^4*x^(4/3) - 384384*a^5*b^3*x^(5/3) + 480480*a^6*b^2*x^2 - 583440*a^7*b*x^(7/3) + 6928
35*a^8*x^(8/3)))/(4849845*a^9*x)

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.44

method	result	size
derivativedivides	\(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (692835 a^{8} x^{\frac {8}{3}}-583440 a^{7} b \,x^{\frac {7}{3}}+480480 a^{6} x^{2} b^{2}-384384 a^{5} b^{3} x^{\frac {5}{3}}+295680 x^{\frac {4}{3}} a^{4} b^{4}-215040 a^{3} b^{5} x +143360 a^{2} b^{6} x^{\frac {2}{3}}-81920 x^{\frac {1}{3}} a \,b^{7}+32768 b^{8}\right )}{4849845 x \,a^{9}}\)	\(112\)
default	\(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (692835 a^{8} x^{\frac {8}{3}}-583440 a^{7} b \,x^{\frac {7}{3}}+480480 a^{6} x^{2} b^{2}-384384 a^{5} b^{3} x^{\frac {5}{3}}+295680 x^{\frac {4}{3}} a^{4} b^{4}-215040 a^{3} b^{5} x +143360 a^{2} b^{6} x^{\frac {2}{3}}-81920 x^{\frac {1}{3}} a \,b^{7}+32768 b^{8}\right )}{4849845 x \,a^{9}}\)	\(112\)

[In]

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

[Out]

2/4849845*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(692835*a^8*x^(8/3)-583440*a^7*b*x^(7/3)+480480*a^6*x^2*b^2-3843
84*a^5*b^3*x^(5/3)+295680*x^(4/3)*a^4*b^4-215040*a^3*b^5*x+143360*a^2*b^6*x^(2/3)-81920*x^(1/3)*a*b^7+32768*b^
8)/x/a^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (189) = 378\).

Time = 144.93 (sec) , antiderivative size = 1031, normalized size of antiderivative = 4.04 \[ \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4849845*((824633720832*b^16 + 171798691840*b^15 + 805306368*(64*a^3 - 3)*b^13 - 16106127360*b^14 - 4194304*
(11264*a^3 - 53)*b^12 - 8070142080*a^12 - 1572864*(5504*a^3 + 1)*b^11 + 49152*(3194880*a^6 - 114688*a^3 - 3)*b
^10 + 1835008*(18816*a^6 + 103*a^3)*b^9 - 196608*(48816*a^6 + 23*a^3)*b^8 + 3072*(6575923200*a^9 + 495872*a^6
+ 15*a^3)*b^7 + 28672*(146455680*a^9 - 3439*a^6)*b^6 - 419328*(934400*a^9 - 7*a^6)*b^5 + 1584*(12166103040*a^1
2 - 38275840*a^9 - 21*a^6)*b^4 + 164736*(43008000*a^12 + 33737*a^9)*b^3 + 51480*(10838016*a^12 - 799*a^9)*b^2
- 109395*(401408*a^12 + 33*a^9)*b)*x - 2*(692835*(16777216*a^10*b^6 + 6291456*a^10*b^5 + 196608*a^10*b^4 - 262
144*a^13 - 114688*a^10*b^3 - 2304*a^10*b^2 + 864*a^10*b - 27*a^10)*x^4 - 6864*(16777216*a^7*b^9 + 6291456*a^7*
b^8 + 196608*a^7*b^7 - 114688*a^7*b^6 - 2304*a^7*b^5 + 864*a^7*b^4 - (262144*a^10 + 27*a^7)*b^3)*x^3 + 8960*(1
6777216*a^4*b^12 + 6291456*a^4*b^11 + 196608*a^4*b^10 - 114688*a^4*b^9 - 2304*a^4*b^8 + 864*a^4*b^7 - (262144*
a^7 + 27*a^4)*b^6)*x^2 - 16384*(16777216*a*b^15 + 6291456*a*b^14 + 196608*a*b^13 - 114688*a*b^12 - 2304*a*b^11
+ 864*a*b^10 - (262144*a^4 + 27*a)*b^9)*x + 2*(274877906944*b^16 + 103079215104*b^15 + 3221225472*b^14 - 1879
048192*b^13 - 37748736*b^12 - 16384*(262144*a^3 + 27)*b^10 + 14155776*b^11 + 401115*(16777216*a^9*b^7 + 629145
6*a^9*b^6 + 196608*a^9*b^5 - 114688*a^9*b^4 - 2304*a^9*b^3 + 864*a^9*b^2 - (262144*a^12 + 27*a^9)*b)*x^3 + 369
6*(16777216*a^6*b^10 + 6291456*a^6*b^9 + 196608*a^6*b^8 - 114688*a^6*b^7 - 2304*a^6*b^6 + 864*a^6*b^5 - (26214
4*a^9 + 27*a^6)*b^4)*x^2 - 5120*(16777216*a^3*b^13 + 6291456*a^3*b^12 + 196608*a^3*b^11 - 114688*a^3*b^10 - 23
04*a^3*b^9 + 864*a^3*b^8 - (262144*a^6 + 27*a^3)*b^7)*x)*x^(2/3) + 3*(2145*(16777216*a^8*b^8 + 6291456*a^8*b^7
+ 196608*a^8*b^6 - 114688*a^8*b^5 - 2304*a^8*b^4 + 864*a^8*b^3 - (262144*a^11 + 27*a^8)*b^2)*x^3 - 2688*(1677
7216*a^5*b^11 + 6291456*a^5*b^10 + 196608*a^5*b^9 - 114688*a^5*b^8 - 2304*a^5*b^7 + 864*a^5*b^6 - (262144*a^8
+ 27*a^5)*b^5)*x^2 + 4096*(16777216*a^2*b^14 + 6291456*a^2*b^13 + 196608*a^2*b^12 - 114688*a^2*b^11 - 2304*a^2
*b^10 + 864*a^2*b^9 - (262144*a^5 + 27*a^2)*b^8)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((16777216*a^9*b^6 + 62914
56*a^9*b^5 + 196608*a^9*b^4 - 262144*a^12 - 114688*a^9*b^3 - 2304*a^9*b^2 + 864*a^9*b - 27*a^9)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (189) = 378\).

Time = 0.32 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.36 \[ \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx=-\frac {2}{692835} \, b {\left (\frac {32768 \, b^{\frac {19}{2}}}{a^{9}} - \frac {\frac {19 \, {\left (6435 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} - 58344 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b + 235620 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{2} - 556920 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{3} + 850850 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{4} - 875160 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{5} + 612612 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{6} - 291720 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{7} + 109395 \, \sqrt {a x^{\frac {1}{3}} + b} b^{8}\right )} b}{a^{8}} + \frac {9 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )}}{a^{8}}}{a}\right )} + \frac {2}{1616615} \, a {\left (\frac {65536 \, b^{\frac {21}{2}}}{a^{10}} + \frac {\frac {21 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )} b}{a^{9}} + \frac {5 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{a^{9}}}{a}\right )} \]

[In]

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

[Out]

-2/692835*b*(32768*b^(19/2)/a^9 - (19*(6435*(a*x^(1/3) + b)^(17/2) - 58344*(a*x^(1/3) + b)^(15/2)*b + 235620*(
a*x^(1/3) + b)^(13/2)*b^2 - 556920*(a*x^(1/3) + b)^(11/2)*b^3 + 850850*(a*x^(1/3) + b)^(9/2)*b^4 - 875160*(a*x
^(1/3) + b)^(7/2)*b^5 + 612612*(a*x^(1/3) + b)^(5/2)*b^6 - 291720*(a*x^(1/3) + b)^(3/2)*b^7 + 109395*sqrt(a*x^
(1/3) + b)*b^8)*b/a^8 + 9*(12155*(a*x^(1/3) + b)^(19/2) - 122265*(a*x^(1/3) + b)^(17/2)*b + 554268*(a*x^(1/3)
+ b)^(15/2)*b^2 - 1492260*(a*x^(1/3) + b)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 - 3233230*(a*x^(1/3)
+ b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b)^(7/2)*b^6 - 1662804*(a*x^(1/3) + b)^(5/2)*b^7 + 692835*(a*x^(1/3) +
b)^(3/2)*b^8 - 230945*sqrt(a*x^(1/3) + b)*b^9)/a^8)/a) + 2/1616615*a*(65536*b^(21/2)/a^10 + (21*(12155*(a*x^(1
/3) + b)^(19/2) - 122265*(a*x^(1/3) + b)^(17/2)*b + 554268*(a*x^(1/3) + b)^(15/2)*b^2 - 1492260*(a*x^(1/3) + b
)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 - 3233230*(a*x^(1/3) + b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b
)^(7/2)*b^6 - 1662804*(a*x^(1/3) + b)^(5/2)*b^7 + 692835*(a*x^(1/3) + b)^(3/2)*b^8 - 230945*sqrt(a*x^(1/3) + b
)*b^9)*b/a^9 + 5*(46189*(a*x^(1/3) + b)^(21/2) - 510510*(a*x^(1/3) + b)^(19/2)*b + 2567565*(a*x^(1/3) + b)^(17
/2)*b^2 - 7759752*(a*x^(1/3) + b)^(15/2)*b^3 + 15668730*(a*x^(1/3) + b)^(13/2)*b^4 - 22221108*(a*x^(1/3) + b)^
(11/2)*b^5 + 22632610*(a*x^(1/3) + b)^(9/2)*b^6 - 16628040*(a*x^(1/3) + b)^(7/2)*b^7 + 8729721*(a*x^(1/3) + b)
^(5/2)*b^8 - 3233230*(a*x^(1/3) + b)^(3/2)*b^9 + 969969*sqrt(a*x^(1/3) + b)*b^10)/a^9)/a)

Mupad [F(-1)]

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

[In]

int(x*(a*x + b*x^(2/3))^(3/2),x)

[Out]

int(x*(a*x + b*x^(2/3))^(3/2), x)