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

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 291, 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 = {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 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])

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 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]

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*}

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Mathematica [A]  time = 0.05, size = 103, normalized size = 0.35 \[ \frac {\sqrt {\frac {\left (a \sqrt [5]{x}+b\right )^2}{x^{2/5}}} \left (12 a^5 x^{6/5}+75 a^4 b x+200 a^3 b^2 x^{4/5}+300 a^2 b^3 x^{3/5}+300 a b^4 x^{2/5}+12 b^5 \sqrt [5]{x} \log (x)\right )}{12 \left (a \sqrt [5]{x}+b\right )} \]

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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fricas [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((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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giac [A]  time = 0.48, size = 125, normalized size = 0.43 \[ a^{5} x \mathrm {sgn}\left (a x + b x^{\frac {4}{5}}\right ) \mathrm {sgn}\relax (x) + b^{5} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {4}{5}}\right ) \mathrm {sgn}\relax (x) + \frac {25}{4} \, a^{4} b x^{\frac {4}{5}} \mathrm {sgn}\left (a x + b x^{\frac {4}{5}}\right ) \mathrm {sgn}\relax (x) + \frac {50}{3} \, a^{3} b^{2} x^{\frac {3}{5}} \mathrm {sgn}\left (a x + b x^{\frac {4}{5}}\right ) \mathrm {sgn}\relax (x) + 25 \, a^{2} b^{3} x^{\frac {2}{5}} \mathrm {sgn}\left (a x + b x^{\frac {4}{5}}\right ) \mathrm {sgn}\relax (x) + 25 \, a b^{4} x^{\frac {1}{5}} \mathrm {sgn}\left (a x + b x^{\frac {4}{5}}\right ) \mathrm {sgn}\relax (x) \]

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)

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maple [A]  time = 0.03, size = 91, normalized size = 0.31 \[ \frac {\left (\frac {a^{2} x^{\frac {2}{5}}+2 a b \,x^{\frac {1}{5}}+b^{2}}{x^{\frac {2}{5}}}\right )^{\frac {5}{2}} \left (12 a^{5} x +12 b^{5} \ln \relax (x )+75 a^{4} b \,x^{\frac {4}{5}}+200 a^{3} b^{2} x^{\frac {3}{5}}+300 a^{2} b^{3} x^{\frac {2}{5}}+300 a \,b^{4} x^{\frac {1}{5}}\right ) x}{12 \left (a \,x^{\frac {1}{5}}+b \right )^{5}} \]

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)+12*b^5*ln(x)+300*a*b^4*x^(1/5)+12*a^5*x)/(x^(1/5)*a+b)^5

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maxima [A]  time = 0.94, size = 52, normalized size = 0.18 \[ a^{5} x + b^{5} \log \relax (x) + \frac {25}{4} \, a^{4} b x^{\frac {4}{5}} + \frac {50}{3} \, a^{3} b^{2} x^{\frac {3}{5}} + 25 \, a^{2} b^{3} x^{\frac {2}{5}} + 25 \, a b^{4} x^{\frac {1}{5}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a^2+\frac {b^2}{x^{2/5}}+\frac {2\,a\,b}{x^{1/5}}\right )}^{5/2} \,d x \]

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]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{2} + \frac {2 a b}{\sqrt [5]{x}} + \frac {b^{2}}{x^{\frac {2}{5}}}\right )^{\frac {5}{2}}\, dx \]

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]

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

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