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

Optimal. Leaf size=291 \[ \frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}}+\frac{25 a^4 b x^{4/5} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{4 \left (a+\frac{b}{\sqrt [5]{x}}\right )}+\frac{50 a^3 b^2 x^{3/5} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{3 \left (a+\frac{b}{\sqrt [5]{x}}\right )}+\frac{25 a^2 b^3 x^{2/5} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}}+\frac{25 a b^4 \sqrt [5]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}}+\frac{5 b^5 \log \left (\sqrt [5]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}} \]

[Out]

(25*a*b^4*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x^(1/5))/(a + b/x^(1/5)) + (25*a^2*b^3*Sqrt[a^2 + b^2/x^(2
/5) + (2*a*b)/x^(1/5)]*x^(2/5))/(a + b/x^(1/5)) + (50*a^3*b^2*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x^(3/5
))/(3*(a + b/x^(1/5))) + (25*a^4*b*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x^(4/5))/(4*(a + b/x^(1/5))) + (a
^5*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x)/(a + b/x^(1/5)) + (5*b^5*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1
/5)]*Log[x^(1/5)])/(a + b/x^(1/5))

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Rubi [A]  time = 0.136913, antiderivative size = 291, normalized size of antiderivative = 1., 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 = {1341, 1355, 263, 43} \[ \frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}}+\frac{25 a^4 b x^{4/5} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{4 \left (a+\frac{b}{\sqrt [5]{x}}\right )}+\frac{50 a^3 b^2 x^{3/5} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{3 \left (a+\frac{b}{\sqrt [5]{x}}\right )}+\frac{25 a^2 b^3 x^{2/5} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}}+\frac{25 a b^4 \sqrt [5]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}}+\frac{5 b^5 \log \left (\sqrt [5]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [5]{x}}+\frac{b^2}{x^{2/5}}}}{a+\frac{b}{\sqrt [5]{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(25*a*b^4*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x^(1/5))/(a + b/x^(1/5)) + (25*a^2*b^3*Sqrt[a^2 + b^2/x^(2
/5) + (2*a*b)/x^(1/5)]*x^(2/5))/(a + b/x^(1/5)) + (50*a^3*b^2*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x^(3/5
))/(3*(a + b/x^(1/5))) + (25*a^4*b*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x^(4/5))/(4*(a + b/x^(1/5))) + (a
^5*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1/5)]*x)/(a + b/x^(1/5)) + (5*b^5*Sqrt[a^2 + b^2/x^(2/5) + (2*a*b)/x^(1
/5)]*Log[x^(1/5)])/(a + b/x^(1/5))

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 1355

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]

Rule 263

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 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+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}\right )^{5/2} \, dx &=5 \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{5/2} x^4 \, dx,x,\sqrt [5]{x}\right )\\ &=\frac{\left (5 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}}\right ) \operatorname{Subst}\left (\int \left (a b+\frac{b^2}{x}\right )^5 x^4 \, dx,x,\sqrt [5]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [5]{x}}\right )}\\ &=\frac{\left (5 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}}\right ) \operatorname{Subst}\left (\int \frac{\left (b^2+a b x\right )^5}{x} \, dx,x,\sqrt [5]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [5]{x}}\right )}\\ &=\frac{\left (5 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}}\right ) \operatorname{Subst}\left (\int \left (5 a b^9+\frac{b^{10}}{x}+10 a^2 b^8 x+10 a^3 b^7 x^2+5 a^4 b^6 x^3+a^5 b^5 x^4\right ) \, dx,x,\sqrt [5]{x}\right )}{b^4 \left (a b+\frac{b^2}{\sqrt [5]{x}}\right )}\\ &=\frac{25 a b^5 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}} \sqrt [5]{x}}{a b+\frac{b^2}{\sqrt [5]{x}}}+\frac{25 a^2 b^4 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}} x^{2/5}}{a b+\frac{b^2}{\sqrt [5]{x}}}+\frac{50 a^3 b^3 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}} x^{3/5}}{3 \left (a b+\frac{b^2}{\sqrt [5]{x}}\right )}+\frac{25 a^4 b^2 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}} x^{4/5}}{4 \left (a b+\frac{b^2}{\sqrt [5]{x}}\right )}+\frac{a^5 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}} x}{a+\frac{b}{\sqrt [5]{x}}}+\frac{b^6 \sqrt{a^2+\frac{b^2}{x^{2/5}}+\frac{2 a b}{\sqrt [5]{x}}} \log (x)}{a b+\frac{b^2}{\sqrt [5]{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0508853, size = 103, normalized size = 0.35 \[ \frac{\sqrt{\frac{\left (a \sqrt [5]{x}+b\right )^2}{x^{2/5}}} \left (200 a^3 b^2 x^{4/5}+300 a^2 b^3 x^{3/5}+75 a^4 b x+12 a^5 x^{6/5}+300 a b^4 x^{2/5}+12 b^5 \sqrt [5]{x} \log (x)\right )}{12 \left (a \sqrt [5]{x}+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(b + a*x^(1/5))^2/x^(2/5)]*(300*a*b^4*x^(2/5) + 300*a^2*b^3*x^(3/5) + 200*a^3*b^2*x^(4/5) + 75*a^4*b*x +
12*a^5*x^(6/5) + 12*b^5*x^(1/5)*Log[x]))/(12*(b + a*x^(1/5)))

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Maple [A]  time = 0.02, size = 91, normalized size = 0.3 \begin{align*}{\frac{x}{12} \left ({ \left ({a}^{2}{x}^{{\frac{2}{5}}}+2\,ab\sqrt [5]{x}+{b}^{2} \right ){x}^{-{\frac{2}{5}}}} \right ) ^{{\frac{5}{2}}} \left ( 75\,{a}^{4}b{x}^{4/5}+200\,{a}^{3}{b}^{2}{x}^{3/5}+300\,{a}^{2}{b}^{3}{x}^{2/5}+300\,a{b}^{4}\sqrt [5]{x}+12\,{b}^{5}\ln \left ( x \right ) +12\,{a}^{5}x \right ) \left ( a\sqrt [5]{x}+b \right ) ^{-5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((a^2*x^(2/5)+2*a*b*x^(1/5)+b^2)/x^(2/5))^(5/2)*x*(75*a^4*b*x^(4/5)+200*a^3*b^2*x^(3/5)+300*a^2*b^3*x^(2/
5)+300*a*b^4*x^(1/5)+12*b^5*ln(x)+12*a^5*x)/(a*x^(1/5)+b)^5

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Maxima [A]  time = 0.989778, size = 70, normalized size = 0.24 \begin{align*} a^{5} x + b^{5} \log \left (x\right ) + \frac{25}{4} \, a^{4} b x^{\frac{4}{5}} + \frac{50}{3} \, a^{3} b^{2} x^{\frac{3}{5}} + 25 \, a^{2} b^{3} x^{\frac{2}{5}} + 25 \, a b^{4} x^{\frac{1}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^5*x + b^5*log(x) + 25/4*a^4*b*x^(4/5) + 50/3*a^3*b^2*x^(3/5) + 25*a^2*b^3*x^(2/5) + 25*a*b^4*x^(1/5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18118, size = 169, normalized size = 0.58 \begin{align*} a^{5} x \mathrm{sgn}\left (a x + b x^{\frac{4}{5}}\right ) \mathrm{sgn}\left (x\right ) + b^{5} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (a x + b x^{\frac{4}{5}}\right ) \mathrm{sgn}\left (x\right ) + \frac{25}{4} \, a^{4} b x^{\frac{4}{5}} \mathrm{sgn}\left (a x + b x^{\frac{4}{5}}\right ) \mathrm{sgn}\left (x\right ) + \frac{50}{3} \, a^{3} b^{2} x^{\frac{3}{5}} \mathrm{sgn}\left (a x + b x^{\frac{4}{5}}\right ) \mathrm{sgn}\left (x\right ) + 25 \, a^{2} b^{3} x^{\frac{2}{5}} \mathrm{sgn}\left (a x + b x^{\frac{4}{5}}\right ) \mathrm{sgn}\left (x\right ) + 25 \, a b^{4} x^{\frac{1}{5}} \mathrm{sgn}\left (a x + b x^{\frac{4}{5}}\right ) \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^5*x*sgn(a*x + b*x^(4/5))*sgn(x) + b^5*log(abs(x))*sgn(a*x + b*x^(4/5))*sgn(x) + 25/4*a^4*b*x^(4/5)*sgn(a*x +
b*x^(4/5))*sgn(x) + 50/3*a^3*b^2*x^(3/5)*sgn(a*x + b*x^(4/5))*sgn(x) + 25*a^2*b^3*x^(2/5)*sgn(a*x + b*x^(4/5)
)*sgn(x) + 25*a*b^4*x^(1/5)*sgn(a*x + b*x^(4/5))*sgn(x)