3.2.41 \(\int x (b \sqrt [3]{x}+a x)^{3/2} \, dx\) [141]

Optimal. Leaf size=408 \[ -\frac {88 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {88 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {44 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

2/7*x^2*(b*x^(1/3)+a*x)^(3/2)-88/1105*b^5*(b+a*x^(2/3))*x^(1/3)/a^(7/2)/(x^(1/3)*a^(1/2)+b^(1/2))/(b*x^(1/3)+a
*x)^(1/2)+88/3315*b^4*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)/a^3-88/4641*b^3*x*(b*x^(1/3)+a*x)^(1/2)/a^2+24/1547*b^2*x^
(5/3)*(b*x^(1/3)+a*x)^(1/2)/a+12/119*b*x^(7/3)*(b*x^(1/3)+a*x)^(1/2)+88/1105*b^(21/4)*x^(1/6)*(cos(2*arctan(a^
(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))*EllipticE(sin(2*arctan(a^(1/4)*x^(1/6)
/b^(1/4))),1/2*2^(1/2))*(x^(1/3)*a^(1/2)+b^(1/2))*((b+a*x^(2/3))/(x^(1/3)*a^(1/2)+b^(1/2))^2)^(1/2)/a^(15/4)/(
b*x^(1/3)+a*x)^(1/2)-44/1105*b^(21/4)*x^(1/6)*(cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/cos(2*arctan(a^
(1/4)*x^(1/6)/b^(1/4)))*EllipticF(sin(2*arctan(a^(1/4)*x^(1/6)/b^(1/4))),1/2*2^(1/2))*(x^(1/3)*a^(1/2)+b^(1/2)
)*((b+a*x^(2/3))/(x^(1/3)*a^(1/2)+b^(1/2))^2)^(1/2)/a^(15/4)/(b*x^(1/3)+a*x)^(1/2)

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Rubi [A]
time = 0.77, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2043, 2046, 2049, 2057, 335, 311, 226, 1210} \begin {gather*} -\frac {44 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {88 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 a^{7/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{3315 a^3}-\frac {88 b^3 x \sqrt {a x+b \sqrt [3]{x}}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}+\frac {2}{7} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(b*x^(1/3) + a*x)^(3/2),x]

[Out]

(-88*b^5*(b + a*x^(2/3))*x^(1/3))/(1105*a^(7/2)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[b*x^(1/3) + a*x]) + (88*b^4*x
^(1/3)*Sqrt[b*x^(1/3) + a*x])/(3315*a^3) - (88*b^3*x*Sqrt[b*x^(1/3) + a*x])/(4641*a^2) + (24*b^2*x^(5/3)*Sqrt[
b*x^(1/3) + a*x])/(1547*a) + (12*b*x^(7/3)*Sqrt[b*x^(1/3) + a*x])/119 + (2*x^2*(b*x^(1/3) + a*x)^(3/2))/7 + (8
8*b^(21/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticE[2
*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(1105*a^(15/4)*Sqrt[b*x^(1/3) + a*x]) - (44*b^(21/4)*(Sqrt[b] + Sqrt
[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/
b^(1/4)], 1/2])/(1105*a^(15/4)*Sqrt[b*x^(1/3) + a*x])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 2043

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

Rule 2046

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 + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n +
1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*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]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps

\begin {align*} \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx &=3 \text {Subst}\left (\int x^5 \left (b x+a x^3\right )^{3/2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{7} (6 b) \text {Subst}\left (\int x^6 \sqrt {b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{119} \left (12 b^2\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (132 b^3\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1547 a}\\ &=-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (44 b^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 a^2}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (44 b^5\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 a^3}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (44 b^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{1105 a^3 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (88 b^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1105 a^3 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (88 b^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1105 a^{7/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (88 b^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1105 a^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {88 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {88 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {44 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.07, size = 123, normalized size = 0.30 \begin {gather*} \frac {2 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x} \left (\left (b+a x^{2/3}\right )^2 \sqrt {1+\frac {a x^{2/3}}{b}} \left (77 b^2-143 a b x^{2/3}+221 a^2 x^{4/3}\right )-77 b^4 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )\right )}{1547 a^3 \sqrt {1+\frac {a x^{2/3}}{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(1/3)*Sqrt[b*x^(1/3) + a*x]*((b + a*x^(2/3))^2*Sqrt[1 + (a*x^(2/3))/b]*(77*b^2 - 143*a*b*x^(2/3) + 221*a^
2*x^(4/3)) - 77*b^4*Hypergeometric2F1[-3/2, 3/4, 7/4, -((a*x^(2/3))/b)]))/(1547*a^3*Sqrt[1 + (a*x^(2/3))/b])

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Maple [A]
time = 0.36, size = 261, normalized size = 0.64

method	result	size
default	\(\frac {\frac {622 x^{\frac {8}{3}} a^{4} b^{2}}{1547}+\frac {80 x^{\frac {10}{3}} a^{5} b}{119}-\frac {16 a^{3} b^{3} x^{2}}{4641}-\frac {88 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {44 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {2 a^{6} x^{4}}{7}+\frac {88 x^{\frac {2}{3}} a \,b^{5}}{3315}+\frac {176 x^{\frac {4}{3}} a^{2} b^{4}}{23205}}{a^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\)	\(261\)
derivativedivides	\(\frac {2 a \,x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7}+\frac {46 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119}+\frac {24 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a}-\frac {88 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{2}}+\frac {88 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3315 a^{3}}-\frac {44 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{1105 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23205/a^4*(4665*x^(8/3)*a^4*b^2+7800*x^(10/3)*a^5*b-40*a^3*b^3*x^2-924*b^6*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^
(1/2))^(1/2)*(-2*(a*x^(1/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)*a/(-a*b)^(1/2))^(1/2)*EllipticE(((a*x^
(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+462*b^6*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2
*(a*x^(1/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)*a/(-a*b)^(1/2))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/
2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+3315*a^6*x^4+308*x^(2/3)*a*b^5+88*x^(4/3)*a^2*b^4)/(x^(1/3)*(b+a*x^(2/3))
)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*x^2 + b*x^(4/3))*sqrt(a*x + b*x^(1/3)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a*x + b*x^(1/3))^(3/2),x)

[Out]

int(x*(a*x + b*x^(1/3))^(3/2), x)

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