3.5.84 \(\int (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}})^{3/2} \, dx\) [484]

Optimal. Leaf size=189 \[ \frac {9 a b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {9 a^2 b \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {3 b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log \left (\sqrt [3]{x}\right )}{a+\frac {b}{\sqrt [3]{x}}} \]

[Out]

9*a*b^2*x^(1/3)*(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))+9/2*a^2*b*x^(2/3)*(a^2+b^2/x^(2/3)+2*a*b/x
^(1/3))^(1/2)/(a+b/x^(1/3))+a^3*x*(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))+b^3*ln(x)*(a^2+b^2/x^(2/
3)+2*a*b/x^(1/3))^(1/2)/(a+b/x^(1/3))

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Rubi [A]
time = 0.07, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 1369, 269, 45} \begin {gather*} \frac {9 a^2 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {9 a b^2 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {3 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {a^3 x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9*a*b^2*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(1/3))/(a + b/x^(1/3)) + (9*a^2*b*Sqrt[a^2 + b^2/x^(2/3)
+ (2*a*b)/x^(1/3)]*x^(2/3))/(2*(a + b/x^(1/3))) + (a^3*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x)/(a + b/x^(
1/3)) + (3*b^3*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*Log[x^(1/3)])/(a + b/x^(1/3))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{3/2} \, dx &=3 \text {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{3/2} x^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \left (a b+\frac {b^2}{x}\right )^3 x^2 \, dx,x,\sqrt [3]{x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}\\ &=\frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \frac {\left (b^2+a b x\right )^3}{x} \, dx,x,\sqrt [3]{x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}\\ &=\frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \left (3 a b^5+\frac {b^6}{x}+3 a^2 b^4 x+a^3 b^3 x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{b^2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}\\ &=\frac {9 a b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a b+\frac {b^2}{\sqrt [3]{x}}}+\frac {9 a^2 b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}+\frac {a^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {b^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log (x)}{a b+\frac {b^2}{\sqrt [3]{x}}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 75, normalized size = 0.40 \begin {gather*} \frac {\left (b+a \sqrt [3]{x}\right ) \left (18 a b^2 \sqrt [3]{x}+9 a^2 b x^{2/3}+2 a^3 x+2 b^3 \log (x)\right )}{2 \sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*x^(1/3))*(18*a*b^2*x^(1/3) + 9*a^2*b*x^(2/3) + 2*a^3*x + 2*b^3*Log[x]))/(2*Sqrt[(b + a*x^(1/3))^2/x^(2
/3)]*x^(1/3))

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Maple [A]
time = 0.04, size = 69, normalized size = 0.37

method	result	size
derivativedivides	\(\frac {\left (\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} x \left (2 a^{3} x +9 a^{2} b \,x^{\frac {2}{3}}+2 b^{3} \ln \left (x \right )+18 a \,b^{2} x^{\frac {1}{3}}\right )}{2 \left (b +a \,x^{\frac {1}{3}}\right )^{3}}\)	\(69\)
default	\(\frac {\left (\frac {a^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {3}{2}} x \left (2 a^{3} x +9 a^{2} b \,x^{\frac {2}{3}}+2 b^{3} \ln \left (x \right )+18 a \,b^{2} x^{\frac {1}{3}}\right )}{2 \left (b +a \,x^{\frac {1}{3}}\right )^{3}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(3/2)*x*(2*a^3*x+9*a^2*b*x^(2/3)+2*b^3*ln(x)+18*a*b^2*x^(1/3))/(
b+a*x^(1/3))^3

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Maxima [A]
time = 0.29, size = 30, normalized size = 0.16 \begin {gather*} a^{3} x + b^{3} \log \left (x\right ) + \frac {9}{2} \, a^{2} b x^{\frac {2}{3}} + 9 \, a b^{2} x^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x + b^3*log(x) + 9/2*a^2*b*x^(2/3) + 9*a*b^2*x^(1/3)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.50, size = 79, normalized size = 0.42 \begin {gather*} a^{3} x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + \frac {9}{2} \, a^{2} b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 9 \, a b^{2} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x*sgn(a*x + b*x^(2/3))*sgn(x) + b^3*log(abs(x))*sgn(a*x + b*x^(2/3))*sgn(x) + 9/2*a^2*b*x^(2/3)*sgn(a*x +
b*x^(2/3))*sgn(x) + 9*a*b^2*x^(1/3)*sgn(a*x + b*x^(2/3))*sgn(x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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