Integrand size = 26, antiderivative size = 391 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=-\frac {6 b^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}}{\left (a+\frac {b}{\sqrt [6]{x}}\right ) \sqrt [6]{x}}+\frac {126 a^2 b^5 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [6]{x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {105 a^3 b^4 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {70 a^4 b^3 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt {x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {63 a^5 b^2 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {42 a^6 b \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{5/6}}{5 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {a^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x}{a+\frac {b}{\sqrt [6]{x}}}+\frac {42 a b^6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \log \left (\sqrt [6]{x}\right )}{a+\frac {b}{\sqrt [6]{x}}} \]
-6*b^7*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))/x^(1/6)+126*a^2 *b^5*x^(1/6)*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+105*a^3*b ^4*x^(1/3)*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+63/2*a^5*b^ 2*x^(2/3)*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+42/5*a^6*b*x ^(5/6)*(a^2+b^2/x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+a^7*x*(a^2+b^2/ x^(1/3)+2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+7*a*b^6*ln(x)*(a^2+b^2/x^(1/3)+ 2*a*b/x^(1/6))^(1/2)/(a+b/x^(1/6))+70*a^4*b^3*(a^2+b^2/x^(1/3)+2*a*b/x^(1/ 6))^(1/2)*x^(1/2)/(a+b/x^(1/6))
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.32 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\frac {\sqrt {\frac {\left (b+a \sqrt [6]{x}\right )^2}{\sqrt [3]{x}}} \left (-60 b^7+1260 a^2 b^5 \sqrt [3]{x}+1050 a^3 b^4 \sqrt {x}+700 a^4 b^3 x^{2/3}+315 a^5 b^2 x^{5/6}+84 a^6 b x+10 a^7 x^{7/6}+70 a b^6 \sqrt [6]{x} \log (x)\right )}{10 \left (b+a \sqrt [6]{x}\right )} \]
(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 + 1 0*a^7*x^(7/6) + 70*a*b^6*x^(1/6)*Log[x]))/(10*(b + a*x^(1/6)))
Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.38, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1384, 774, 27, 795, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {\sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}} \int \left (\frac {b^2}{\sqrt [6]{x}}+a b\right )^7dx}{a b^7+\frac {b^8}{\sqrt [6]{x}}}\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle \frac {6 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}} \int b^7 \left (a+\frac {b}{\sqrt [6]{x}}\right )^7 x^{5/6}d\sqrt [6]{x}}{a b^7+\frac {b^8}{\sqrt [6]{x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}} \int \left (a+\frac {b}{\sqrt [6]{x}}\right )^7 x^{5/6}d\sqrt [6]{x}}{a b^7+\frac {b^8}{\sqrt [6]{x}}}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle \frac {6 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}} \int \frac {\left (\sqrt [6]{x} a+b\right )^7}{\sqrt [3]{x}}d\sqrt [6]{x}}{a b^7+\frac {b^8}{\sqrt [6]{x}}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {6 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}} \int \left (x^{5/6} a^7+7 b x^{2/3} a^6+21 b^2 \sqrt {x} a^5+35 b^3 \sqrt [3]{x} a^4+35 b^4 \sqrt [6]{x} a^3+21 b^5 a^2+\frac {7 b^6 a}{\sqrt [6]{x}}+\frac {b^7}{\sqrt [3]{x}}\right )d\sqrt [6]{x}}{a b^7+\frac {b^8}{\sqrt [6]{x}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}} \left (\frac {a^7 x}{6}+\frac {7}{5} a^6 b x^{5/6}+\frac {21}{4} a^5 b^2 x^{2/3}+\frac {35}{3} a^4 b^3 \sqrt {x}+\frac {35}{2} a^3 b^4 \sqrt [3]{x}+21 a^2 b^5 \sqrt [6]{x}+7 a b^6 \log \left (\sqrt [6]{x}\right )-\frac {b^7}{\sqrt [6]{x}}\right )}{a b^7+\frac {b^8}{\sqrt [6]{x}}}\) |
(6*b^7*Sqrt[a^2 + b^2/x^(1/3) + (2*a*b)/x^(1/6)]*(-(b^7/x^(1/6)) + 21*a^2* b^5*x^(1/6) + (35*a^3*b^4*x^(1/3))/2 + (35*a^4*b^3*Sqrt[x])/3 + (21*a^5*b^ 2*x^(2/3))/4 + (7*a^6*b*x^(5/6))/5 + (a^7*x)/6 + 7*a*b^6*Log[x^(1/6)]))/(a *b^7 + b^8/x^(1/6))
3.5.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
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]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.30
\[\frac {\left (\frac {a^{2} \sqrt {x}+2 a b \,x^{\frac {1}{3}}+x^{\frac {1}{6}} b^{2}}{\sqrt {x}}\right )^{\frac {7}{2}} x \left (10 a^{7} x^{\frac {7}{6}}+84 a^{6} b x +315 b^{2} a^{5} x^{\frac {5}{6}}+700 a^{4} b^{3} x^{\frac {2}{3}}+1050 b^{4} a^{3} \sqrt {x}+70 a \,b^{6} \ln \left (x \right ) x^{\frac {1}{6}}+1260 a^{2} b^{5} x^{\frac {1}{3}}-60 b^{7}\right )}{10 \left (a \,x^{\frac {1}{6}}+b \right )^{7}}\]
1/10*((a^2*x^(1/2)+2*a*b*x^(1/3)+x^(1/6)*b^2)/x^(1/2))^(7/2)*x*(10*a^7*x^( 7/6)+84*a^6*b*x+315*b^2*a^5*x^(5/6)+700*a^4*b^3*x^(2/3)+1050*b^4*a^3*x^(1/ 2)+70*a*b^6*ln(x)*x^(1/6)+1260*a^2*b^5*x^(1/3)-60*b^7)/(a*x^(1/6)+b)^7
Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\text {Timed out} \]
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.20 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=7 \, a b^{6} \log \left (x\right ) + \frac {10 \, a^{7} x^{\frac {7}{6}} + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac {5}{6}} + 700 \, a^{4} b^{3} x^{\frac {2}{3}} + 1050 \, a^{3} b^{4} \sqrt {x} + 1260 \, a^{2} b^{5} x^{\frac {1}{3}} - 60 \, b^{7}}{10 \, x^{\frac {1}{6}}} \]
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)
Time = 0.43 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.44 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=a^{7} x \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + \frac {42}{5} \, a^{6} b x^{\frac {5}{6}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + \frac {63}{2} \, a^{5} b^{2} x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 70 \, a^{4} b^{3} \sqrt {x} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 105 \, a^{3} b^{4} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac {1}{6}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) - \frac {6 \, b^{7} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right )}{x^{\frac {1}{6}}} \]
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)
Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\int {\left (a^2+\frac {b^2}{x^{1/3}}+\frac {2\,a\,b}{x^{1/6}}\right )}^{7/2} \,d x \]