3.148 \(\int \frac {(b \sqrt [3]{x}+a x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=301 \[ -\frac {884 a^{27/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{100947 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {1768 a^6 \sqrt {a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac {1768 a^5 \sqrt {a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac {1768 a^4 \sqrt {a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac {136 a^3 \sqrt {a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac {8 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5}-\frac {4 a \sqrt {a x+b \sqrt [3]{x}}}{69 x^4} \]

[Out]

-2/9*(b*x^(1/3)+a*x)^(3/2)/x^5-4/69*a*(b*x^(1/3)+a*x)^(1/2)/x^4-8/1311*a^2*(b*x^(1/3)+a*x)^(1/2)/b/x^(10/3)+13
6/19665*a^3*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(8/3)-1768/216315*a^4*(b*x^(1/3)+a*x)^(1/2)/b^3/x^2+1768/168245*a^5*(b
*x^(1/3)+a*x)^(1/2)/b^4/x^(4/3)-1768/100947*a^6*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)-884/100947*a^(27/4)*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)/b^(21/4)/(b*x^(1/3)+a*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2018, 2020, 2025, 2011, 329, 220} \[ -\frac {1768 a^6 \sqrt {a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac {1768 a^5 \sqrt {a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac {1768 a^4 \sqrt {a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac {136 a^3 \sqrt {a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac {884 a^{27/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{100947 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac {4 a \sqrt {a x+b \sqrt [3]{x}}}{69 x^4}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^(1/3) + a*x)^(3/2)/x^6,x]

[Out]

(-4*a*Sqrt[b*x^(1/3) + a*x])/(69*x^4) - (8*a^2*Sqrt[b*x^(1/3) + a*x])/(1311*b*x^(10/3)) + (136*a^3*Sqrt[b*x^(1
/3) + a*x])/(19665*b^2*x^(8/3)) - (1768*a^4*Sqrt[b*x^(1/3) + a*x])/(216315*b^3*x^2) + (1768*a^5*Sqrt[b*x^(1/3)
+ a*x])/(168245*b^4*x^(4/3)) - (1768*a^6*Sqrt[b*x^(1/3) + a*x])/(100947*b^5*x^(2/3)) - (2*(b*x^(1/3) + a*x)^(
3/2))/(9*x^5) - (884*a^(27/4)*(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])/(100947*b^(21/4)*Sqrt[b*x^(1/3) + a*x])

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 329

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 2011

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 2018

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 2020

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

Rule 2025

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, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx &=3 \operatorname {Subst}\left (\int \frac {\left (b x+a x^3\right )^{3/2}}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {1}{3} (2 a) \operatorname {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {1}{69} \left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (68 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 b}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {\left (884 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 b^2}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (884 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 b^3}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {\left (884 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^4}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (884 a^7\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{100947 b^5}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (884 a^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{100947 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (1768 a^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{100947 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {884 a^{27/4} \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 )}{100947 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.07, size = 62, normalized size = 0.21 \[ -\frac {2 b \sqrt {a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac {27}{4},-\frac {3}{2};-\frac {23}{4};-\frac {a x^{2/3}}{b}\right )}{9 x^{14/3} \sqrt {\frac {a x^{2/3}}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^(1/3) + a*x)^(3/2)/x^6,x]

[Out]

(-2*b*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-27/4, -3/2, -23/4, -((a*x^(2/3))/b)])/(9*Sqrt[1 + (a*x^(2/3))/b
]*x^(14/3))

________________________________________________________________________________________

fricas [F]  time = 1.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(3/2)/x^6,x, algorithm="fricas")

[Out]

integral((a*x + b*x^(1/3))^(3/2)/x^6, x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(3/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(x)]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time: 3.242*((((
((-1264936572900*b^12/11384429156100/b^11/x^(1/3)/x^(1/3)-1594920026700*b^11*a/11384429156100/b^11)/x^(1/3)/x^
(1/3)-34735100400*b^10*a^2/11384429156100/b^11)/x^(1/3)/x^(1/3)[phone]*b^9*a^3/11384429156100/b^11)/x^(1/
3)/x^(1/3)-46523982960*b^8*a^4/11384429156100/b^11)/x^(1/3)/x^(1/3)[phone]*b^7*a^5/11384429156100/b^11)/x
^(1/3)/x^(1/3)-99694249200*b^6*a^6/11384429156100/b^11)*sqrt(a/x^(1/3)+b/x)+integrate(-99694249200*b^6*a^7/113
84429156100/b^11/3/((x^(1/6))^5*sqrt(a*(x^(1/3))^2+b)*sign(x)),x)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 201, normalized size = 0.67 \[ -\frac {2 \left (13260 a^{7} x^{9}+6630 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, a^{6} x^{\frac {26}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+5304 a^{6} b \,x^{\frac {25}{3}}-1768 a^{5} b^{2} x^{\frac {23}{3}}+952 a^{4} b^{3} x^{7}-616 a^{3} b^{4} x^{\frac {19}{3}}+216755 a^{2} b^{5} x^{\frac {17}{3}}+380380 a \,b^{6} x^{5}+168245 b^{7} x^{\frac {13}{3}}\right )}{1514205 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{5} x^{\frac {26}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b*x^(1/3))^(3/2)/x^6,x)

[Out]

-2/1514205*(6630*a^6*(-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)*(-1/(-a*b)^(1/2)*a*x^(1/3))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1
/2*2^(1/2))*x^(26/3)-1768*x^(23/3)*a^5*b^2+5304*x^(25/3)*a^6*b+952*x^7*a^4*b^3+216755*x^(17/3)*a^2*b^5-616*x^(
19/3)*a^3*b^4+380380*x^5*a*b^6+13260*x^9*a^7+168245*x^(13/3)*b^7)/b^5/((a*x^(2/3)+b)*x^(1/3))^(1/2)/x^(26/3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(1/3)+a*x)^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)/x^6, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^6} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^(1/3))^(3/2)/x^6,x)

[Out]

int((a*x + b*x^(1/3))^(3/2)/x^6, x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**(1/3)+a*x)**(3/2)/x**6,x)

[Out]

Timed out

________________________________________________________________________________________