3.2.59 \(\int \frac {x^3}{(b \sqrt [3]{x}+a x)^{3/2}} \, dx\) [159]

Optimal. Leaf size=239 \[ -\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {663 b^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

-3*x^3/a/(b*x^(1/3)+a*x)^(1/2)-663/77*b^3*(b*x^(1/3)+a*x)^(1/2)/a^5+1989/385*b^2*x^(2/3)*(b*x^(1/3)+a*x)^(1/2)
/a^4-221/55*b*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^3+17/5*x^2*(b*x^(1/3)+a*x)^(1/2)/a^2+663/154*b^(15/4)*x^(1/6)*(c
os(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^(21/4)/(b*x^(1/3)+a*x)^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2047, 2049, 2036, 335, 226} \begin {gather*} \frac {663 b^{15/4} \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 )}{154 a^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {663 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{55 a^3}+\frac {17 x^2 \sqrt {a x+b \sqrt [3]{x}}}{5 a^2}-\frac {3 x^3}{a \sqrt {a x+b \sqrt [3]{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x^(1/3) + a*x)^(3/2),x]

[Out]

(-3*x^3)/(a*Sqrt[b*x^(1/3) + a*x]) - (663*b^3*Sqrt[b*x^(1/3) + a*x])/(77*a^5) + (1989*b^2*x^(2/3)*Sqrt[b*x^(1/
3) + a*x])/(385*a^4) - (221*b*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(55*a^3) + (17*x^2*Sqrt[b*x^(1/3) + a*x])/(5*a^2)
+ (663*b^(15/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*Ellip
ticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(154*a^(21/4)*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 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 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

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 2047

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*(n - j)*(p + 1))), x] - Dist[c^n*((m + j*p - n + j + 1)/(b*(n - j)*(p + 1))), I
nt[(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] && (I
ntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1] && GtQ[m + j*p + 1, n - j]

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]

Rubi steps

\begin {align*} \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \text {Subst}\left (\int \frac {x^{11}}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {51 \text {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}-\frac {(221 b) \text {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 a^2}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (1989 b^2\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 a^3}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}-\frac {\left (1989 b^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^4}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^5}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{154 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4 \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 )}{77 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {663 b^{15/4} \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 )}{154 a^{21/4} \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 = 10.08, size = 124, normalized size = 0.52 \begin {gather*} \frac {\sqrt {b \sqrt [3]{x}+a x} \left (-3315 b^4-1326 a b^3 x^{2/3}+442 a^2 b^2 x^{4/3}-238 a^3 b x^2+154 a^4 x^{8/3}+3315 b^4 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )\right )}{385 a^5 \left (b+a x^{2/3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x^(1/3) + a*x]*(-3315*b^4 - 1326*a*b^3*x^(2/3) + 442*a^2*b^2*x^(4/3) - 238*a^3*b*x^2 + 154*a^4*x^(8/3)
+ 3315*b^4*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((a*x^(2/3))/b)]))/(385*a^5*(b + a*x^(2/
3)))

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Maple [A]
time = 0.35, size = 260, normalized size = 1.09

method	result	size
derivativedivides	\(-\frac {3 x^{\frac {1}{3}} b^{4}}{a^{5} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {2 x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{5 a^{2}}-\frac {56 b \,x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{55 a^{3}}+\frac {834 b^{2} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 a^{4}}-\frac {432 b^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 a^{5}}+\frac {663 b^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{154 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(228\)
default	\(-\frac {-884 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b^{2}+476 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{4} b -3315 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{4}+2652 x \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{3}-308 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5} x^{3}+2310 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {1}{3}} a \,b^{4}+4320 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{4}}{770 x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right ) a^{6}}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/770*(-884*x^(5/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^3*b^2+476*x^(7/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^4*b-331
5*(-a*b)^(1/2)*((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))*(x^(1/3)*
(b+a*x^(2/3)))^(1/2)*b^4+2652*x*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^2*b^3-308*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a^5*x^
3+2310*(b*x^(1/3)+a*x)^(1/2)*x^(1/3)*a*b^4+4320*x^(1/3)*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*a*b^4)/x^(1/3)/(b+a*x^(2
/3))/a^6

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

[Out]

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

[Out]

integral((a^4*x^5 + 3*a^2*b^2*x^(11/3) - 2*a*b^3*x^3 - (2*a^3*b*x^4 - b^4*x^2)*x^(1/3))*sqrt(a*x + b*x^(1/3))/
(a^6*x^4 + 2*a^3*b^3*x^2 + b^6), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\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**3/(b*x**(1/3)+a*x)**(3/2),x)

[Out]

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

[Out]

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

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

[Out]

int(x^3/(a*x + b*x^(1/3))^(3/2), x)

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