3.463 \(\int (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3})^{5/2} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]

[Out]

(a^2*(a + b*x^(1/3))^5*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(2*b^3) - (6*a*(a + b*x^(1/3))^6*Sqrt[a^2 + 2*
a*b*x^(1/3) + b^2*x^(2/3)])/(7*b^3) + (3*(a + b*x^(1/3))^7*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(8*b^3)

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Rubi [A]  time = 0.0706979, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1341, 646, 43} \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]

[Out]

(a^2*(a + b*x^(1/3))^5*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(2*b^3) - (6*a*(a + b*x^(1/3))^6*Sqrt[a^2 + 2*
a*b*x^(1/3) + b^2*x^(2/3)])/(7*b^3) + (3*(a + b*x^(1/3))^7*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])/(8*b^3)

Rule 1341

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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 43

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])

Rubi steps

\begin{align*} \int \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{5/2} \, dx &=3 \operatorname{Subst}\left (\int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right )^5 \, dx,x,\sqrt [3]{x}\right )}{b^5 \left (a+b \sqrt [3]{x}\right )}\\ &=\frac{\left (3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}\right ) \operatorname{Subst}\left (\int \left (\frac{a^2 \left (a b+b^2 x\right )^5}{b^2}-\frac{2 a \left (a b+b^2 x\right )^6}{b^3}+\frac{\left (a b+b^2 x\right )^7}{b^4}\right ) \, dx,x,\sqrt [3]{x}\right )}{b^5 \left (a+b \sqrt [3]{x}\right )}\\ &=\frac{a^2 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{2 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^6 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{7 b^3}+\frac{3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{8 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0365837, size = 56, normalized size = 0.41 \[ \frac{\left (a+b \sqrt [3]{x}\right )^5 \sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (a^2-6 a b \sqrt [3]{x}+21 b^2 x^{2/3}\right )}{56 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]

[Out]

((a + b*x^(1/3))^5*Sqrt[(a + b*x^(1/3))^2]*(a^2 - 6*a*b*x^(1/3) + 21*b^2*x^(2/3)))/(56*b^3)

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Maple [A]  time = 0.002, size = 87, normalized size = 0.6 \begin{align*}{\frac{1}{56}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 21\,{b}^{5}{x}^{8/3}+120\,a{b}^{4}{x}^{7/3}+336\,{a}^{3}{b}^{2}{x}^{5/3}+210\,{a}^{4}b{x}^{4/3}+280\,{a}^{2}{b}^{3}{x}^{2}+56\,{a}^{5}x \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x)

[Out]

1/56*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(21*b^5*x^(8/3)+120*a*b^4*x^(7/3)+336*a^3*b^2*x^(5/3)+210*a^4*b*x^(
4/3)+280*a^2*b^3*x^2+56*a^5*x)/(a+b*x^(1/3))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.03009, size = 140, normalized size = 1.02 \begin{align*} 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac{3}{8} \,{\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac{2}{3}} + \frac{15}{28} \,{\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x, algorithm="fricas")

[Out]

5*a^2*b^3*x^2 + a^5*x + 3/8*(b^5*x^2 + 16*a^3*b^2*x)*x^(2/3) + 15/28*(4*a*b^4*x^2 + 7*a^4*b*x)*x^(1/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(5/2),x)

[Out]

Integral((a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2), x)

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Giac [A]  time = 1.13114, size = 138, normalized size = 1.01 \begin{align*} \frac{3}{8} \, b^{5} x^{\frac{8}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) + 5 \, a^{2} b^{3} x^{2} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) + 6 \, a^{3} b^{2} x^{\frac{5}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}} \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) + a^{5} x \mathrm{sgn}\left (b x^{\frac{1}{3}} + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x, algorithm="giac")

[Out]

3/8*b^5*x^(8/3)*sgn(b*x^(1/3) + a) + 15/7*a*b^4*x^(7/3)*sgn(b*x^(1/3) + a) + 5*a^2*b^3*x^2*sgn(b*x^(1/3) + a)
+ 6*a^3*b^2*x^(5/3)*sgn(b*x^(1/3) + a) + 15/4*a^4*b*x^(4/3)*sgn(b*x^(1/3) + a) + a^5*x*sgn(b*x^(1/3) + a)