3.492 \(\int (a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}})^{7/2} \, dx\)

Optimal. Leaf size=391 \[ \frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{105 a^3 b^4 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{126 a^2 b^5 \sqrt [6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}-\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}} \]

[Out]

(-6*b^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)])/((a + b/x^(1/6))*x^(1/6)) + (126*a^2*b^5*Sqrt[a^2 + b^2/x^(
1/3) + (2*a*b)/x^(1/6)]*x^(1/6))/(a + b/x^(1/6)) + (105*a^3*b^4*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1
/3))/(a + b/x^(1/6)) + (70*a^4*b^3*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Sqrt[x])/(a + b/x^(1/6)) + (63*a^
5*b^2*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(2/3))/(2*(a + b/x^(1/6))) + (42*a^6*b*Sqrt[a^2 + b^2/x^(1/3
) + (2*a*b)/x^(1/6)]*x^(5/6))/(5*(a + b/x^(1/6))) + (a^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x)/(a + b/x
^(1/6)) + (42*a*b^6*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Log[x^(1/6)])/(a + b/x^(1/6))

________________________________________________________________________________________

Rubi [A]  time = 0.179674, antiderivative size = 391, 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} \[ \frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{105 a^3 b^4 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{126 a^2 b^5 \sqrt [6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}-\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]

[Out]

(-6*b^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)])/((a + b/x^(1/6))*x^(1/6)) + (126*a^2*b^5*Sqrt[a^2 + b^2/x^(
1/3) + (2*a*b)/x^(1/6)]*x^(1/6))/(a + b/x^(1/6)) + (105*a^3*b^4*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(1
/3))/(a + b/x^(1/6)) + (70*a^4*b^3*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Sqrt[x])/(a + b/x^(1/6)) + (63*a^
5*b^2*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x^(2/3))/(2*(a + b/x^(1/6))) + (42*a^6*b*Sqrt[a^2 + b^2/x^(1/3
) + (2*a*b)/x^(1/6)]*x^(5/6))/(5*(a + b/x^(1/6))) + (a^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*x)/(a + b/x
^(1/6)) + (42*a*b^6*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*Log[x^(1/6)])/(a + b/x^(1/6))

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}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx &=6 \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{7/2} x^5 \, dx,x,\sqrt [6]{x}\right )\\ &=\frac{\left (6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}\right ) \operatorname{Subst}\left (\int \left (a b+\frac{b^2}{x}\right )^7 x^5 \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}\\ &=\frac{\left (6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}\right ) \operatorname{Subst}\left (\int \frac{\left (b^2+a b x\right )^7}{x^2} \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}\\ &=\frac{\left (6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}\right ) \operatorname{Subst}\left (\int \left (21 a^2 b^{12}+\frac{b^{14}}{x^2}+\frac{7 a b^{13}}{x}+35 a^3 b^{11} x+35 a^4 b^{10} x^2+21 a^5 b^9 x^3+7 a^6 b^8 x^4+a^7 b^7 x^5\right ) \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}\\ &=-\frac{6 b^8 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}}{\left (a b+\frac{b^2}{\sqrt [6]{x}}\right ) \sqrt [6]{x}}+\frac{126 a^2 b^6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \sqrt [6]{x}}{a b+\frac{b^2}{\sqrt [6]{x}}}+\frac{105 a^3 b^5 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \sqrt [3]{x}}{a b+\frac{b^2}{\sqrt [6]{x}}}+\frac{70 a^4 b^4 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \sqrt{x}}{a b+\frac{b^2}{\sqrt [6]{x}}}+\frac{63 a^5 b^3 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} x^{2/3}}{2 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}+\frac{42 a^6 b^2 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} x^{5/6}}{5 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}+\frac{a^7 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} x}{a+\frac{b}{\sqrt [6]{x}}}+\frac{7 a b^7 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \log (x)}{a b+\frac{b^2}{\sqrt [6]{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0701364, size = 124, normalized size = 0.32 \[ \frac{\sqrt{\frac{\left (a \sqrt [6]{x}+b\right )^2}{\sqrt [3]{x}}} \left (315 a^5 b^2 x^{5/6}+700 a^4 b^3 x^{2/3}+1050 a^3 b^4 \sqrt{x}+1260 a^2 b^5 \sqrt [3]{x}+84 a^6 b x+10 a^7 x^{7/6}+70 a b^6 \sqrt [6]{x} \log (x)-60 b^7\right )}{10 \left (a \sqrt [6]{x}+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6))^(7/2),x]

[Out]

(Sqrt[(b + a*x^(1/6))^2/x^(1/3)]*(-60*b^7 + 1260*a^2*b^5*x^(1/3) + 1050*a^3*b^4*Sqrt[x] + 700*a^4*b^3*x^(2/3)
+ 315*a^5*b^2*x^(5/6) + 84*a^6*b*x + 10*a^7*x^(7/6) + 70*a*b^6*x^(1/6)*Log[x]))/(10*(b + a*x^(1/6)))

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 116, normalized size = 0.3 \begin{align*}{\frac{1}{10}\sqrt{{ \left ( \sqrt{x}{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}\sqrt [6]{x} \right ){\frac{1}{\sqrt{x}}}}} \left ( 84\,{a}^{6}bx+315\,{a}^{5}{b}^{2}{x}^{5/6}+70\,a{b}^{6}\ln \left ( x \right ) \sqrt [6]{x}+1050\,{a}^{3}{b}^{4}\sqrt{x}+1260\,{a}^{2}{b}^{5}\sqrt [3]{x}+700\,{a}^{4}{b}^{3}{x}^{2/3}+10\,{a}^{7}{x}^{7/6}-60\,{b}^{7} \right ) \left ( a\sqrt [6]{x}+b \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x)

[Out]

1/10*((x^(1/2)*a^2+2*a*b*x^(1/3)+b^2*x^(1/6))/x^(1/2))^(1/2)*(84*a^6*b*x+315*a^5*b^2*x^(5/6)+70*a*b^6*ln(x)*x^
(1/6)+1050*a^3*b^4*x^(1/2)+1260*a^2*b^5*x^(1/3)+700*a^4*b^3*x^(2/3)+10*a^7*x^(7/6)-60*b^7)/(a*x^(1/6)+b)

________________________________________________________________________________________

Maxima [A]  time = 0.993768, size = 107, normalized size = 0.27 \begin{align*} 7 \, a b^{6} \log \left (x\right ) + \frac{10 \, a^{7} x^{\frac{7}{6}} + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac{5}{6}} + 700 \, a^{4} b^{3} x^{\frac{2}{3}} + 1050 \, a^{3} b^{4} \sqrt{x} + 1260 \, a^{2} b^{5} x^{\frac{1}{3}} - 60 \, b^{7}}{10 \, x^{\frac{1}{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x, algorithm="maxima")

[Out]

7*a*b^6*log(x) + 1/10*(10*a^7*x^(7/6) + 84*a^6*b*x + 315*a^5*b^2*x^(5/6) + 700*a^4*b^3*x^(2/3) + 1050*a^3*b^4*
sqrt(x) + 1260*a^2*b^5*x^(1/3) - 60*b^7)/x^(1/6)

________________________________________________________________________________________

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^(1/3)+2*a*b/x^(1/6))^(7/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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**(1/3)+2*a*b/x**(1/6))**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.36387, size = 232, normalized size = 0.59 \begin{align*} a^{7} x \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + \frac{42}{5} \, a^{6} b x^{\frac{5}{6}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + \frac{63}{2} \, a^{5} b^{2} x^{\frac{2}{3}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 70 \, a^{4} b^{3} \sqrt{x} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 105 \, a^{3} b^{4} x^{\frac{1}{3}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac{1}{6}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) - \frac{6 \, b^{7} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right )}{x^{\frac{1}{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(7/2),x, algorithm="giac")

[Out]

a^7*x*sgn(a*x + b*x^(5/6))*sgn(x) + 7*a*b^6*log(abs(x))*sgn(a*x + b*x^(5/6))*sgn(x) + 42/5*a^6*b*x^(5/6)*sgn(a
*x + b*x^(5/6))*sgn(x) + 63/2*a^5*b^2*x^(2/3)*sgn(a*x + b*x^(5/6))*sgn(x) + 70*a^4*b^3*sqrt(x)*sgn(a*x + b*x^(
5/6))*sgn(x) + 105*a^3*b^4*x^(1/3)*sgn(a*x + b*x^(5/6))*sgn(x) + 126*a^2*b^5*x^(1/6)*sgn(a*x + b*x^(5/6))*sgn(
x) - 6*b^7*sgn(a*x + b*x^(5/6))*sgn(x)/x^(1/6)