Integrand size = 13, antiderivative size = 121 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=-6 b^3 \sqrt {a+\frac {b}{\sqrt [3]{x}}}+\frac {87}{8} a b^2 \sqrt {a+\frac {b}{\sqrt [3]{x}}} \sqrt [3]{x}+\frac {19}{4} a^2 b \sqrt {a+\frac {b}{\sqrt [3]{x}}} x^{2/3}+a^3 \sqrt {a+\frac {b}{\sqrt [3]{x}}} x+\frac {105}{8} \sqrt {a} b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right ) \] Output:
-6*b^3*(a+b/x^(1/3))^(1/2)+87/8*a*b^2*(a+b/x^(1/3))^(1/2)*x^(1/3)+19/4*a^2 *b*(a+b/x^(1/3))^(1/2)*x^(2/3)+a^3*(a+b/x^(1/3))^(1/2)*x+105/8*a^(1/2)*b^3 *arctanh((a+b/x^(1/3))^(1/2)/a^(1/2))
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.69 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\frac {1}{8} \left (\sqrt {a+\frac {b}{\sqrt [3]{x}}} \left (-48 b^3+87 a b^2 \sqrt [3]{x}+38 a^2 b x^{2/3}+8 a^3 x\right )+105 \sqrt {a} b^3 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )\right ) \] Input:
Integrate[(a + b/x^(1/3))^(7/2),x]
Output:
(Sqrt[a + b/x^(1/3)]*(-48*b^3 + 87*a*b^2*x^(1/3) + 38*a^2*b*x^(2/3) + 8*a^ 3*x) + 105*Sqrt[a]*b^3*ArcTanh[Sqrt[a + b/x^(1/3)]/Sqrt[a]])/8
Time = 0.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {774, 798, 51, 51, 51, 60, 73, 221}
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.
| \(\displaystyle \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 3 \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} x^{2/3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle -3 \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{x^{4/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -3 \left (\frac {7}{6} b \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{5/2}}{x}d\frac {1}{\sqrt [3]{x}}-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{3 x}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -3 \left (\frac {7}{6} b \left (\frac {5}{4} b \int \frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{x^{2/3}}d\frac {1}{\sqrt [3]{x}}-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{5/2}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{3 x}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -3 \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \int \frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt [3]{x}}d\frac {1}{\sqrt [3]{x}}-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{\sqrt [3]{x}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{5/2}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{3 x}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -3 \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \left (a \int \frac {1}{\sqrt {a+\frac {b}{\sqrt [3]{x}}} \sqrt [3]{x}}d\frac {1}{\sqrt [3]{x}}+2 \sqrt {a+\frac {b}{\sqrt [3]{x}}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{\sqrt [3]{x}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{5/2}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{3 x}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -3 \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \left (\frac {2 a \int \frac {1}{\frac {x^{2/3}}{b}-\frac {a}{b}}d\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{b}+2 \sqrt {a+\frac {b}{\sqrt [3]{x}}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{\sqrt [3]{x}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{5/2}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{3 x}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -3 \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \left (2 \sqrt {a+\frac {b}{\sqrt [3]{x}}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{\sqrt [3]{x}}}}{\sqrt {a}}\right )\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{3/2}}{\sqrt [3]{x}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{5/2}}{2 x^{2/3}}\right )-\frac {\left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2}}{3 x}\right )\) |
Input:
Int[(a + b/x^(1/3))^(7/2),x]
Output:
-3*(-1/3*(a + b/x^(1/3))^(7/2)/x + (7*b*(-1/2*(a + b/x^(1/3))^(5/2)/x^(2/3 ) + (5*b*(-((a + b/x^(1/3))^(3/2)/x^(1/3)) + (3*b*(2*Sqrt[a + b/x^(1/3)] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b/x^(1/3)]/Sqrt[a]]))/2))/4))/6)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(\frac {\sqrt {\frac {b +a \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}}\, \left (16 a^{\frac {7}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} x^{\frac {2}{3}}+60 a^{\frac {7}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b x +270 a^{\frac {5}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{2} x^{\frac {2}{3}}+105 a^{2} \ln \left (\frac {2 \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{3} x^{\frac {2}{3}}-96 a^{\frac {3}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{2}\right )}{16 x^{\frac {1}{3}} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, a^{\frac {3}{2}}}\) | \(165\) |
| derivativedivides | \(\frac {\left (\frac {b +a \,x^{\frac {1}{3}}}{x^{\frac {1}{3}}}\right )^{\frac {7}{2}} x^{\frac {2}{3}} \left (16 a^{\frac {7}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} x^{\frac {2}{3}}+60 a^{\frac {7}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b x +270 a^{\frac {5}{2}} \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, b^{2} x^{\frac {2}{3}}+105 a^{2} \ln \left (\frac {2 \sqrt {a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}}\, \sqrt {a}+2 a \,x^{\frac {1}{3}}+b}{2 \sqrt {a}}\right ) b^{3} x^{\frac {2}{3}}-96 a^{\frac {3}{2}} \left (a \,x^{\frac {2}{3}}+b \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{2}\right )}{16 \left (b +a \,x^{\frac {1}{3}}\right )^{3} \sqrt {\left (b +a \,x^{\frac {1}{3}}\right ) x^{\frac {1}{3}}}\, a^{\frac {3}{2}}}\) | \(174\) |
Input:
int((a+b/x^(1/3))^(7/2),x,method=_RETURNVERBOSE)
Output:
1/16*((b+a*x^(1/3))/x^(1/3))^(1/2)/x^(1/3)*(16*a^(7/2)*(a*x^(2/3)+b*x^(1/3 ))^(3/2)*x^(2/3)+60*a^(7/2)*(a*x^(2/3)+b*x^(1/3))^(1/2)*b*x+270*a^(5/2)*(a *x^(2/3)+b*x^(1/3))^(1/2)*b^2*x^(2/3)+105*a^2*ln(1/2*(2*(a*x^(2/3)+b*x^(1/ 3))^(1/2)*a^(1/2)+2*a*x^(1/3)+b)/a^(1/2))*b^3*x^(2/3)-96*a^(3/2)*(a*x^(2/3 )+b*x^(1/3))^(3/2)*b^2)/((b+a*x^(1/3))*x^(1/3))^(1/2)/a^(3/2)
Timed out. \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\text {Timed out} \] Input:
integrate((a+b/x^(1/3))^(7/2),x, algorithm="fricas")
Output:
Timed out
Time = 16.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.40 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\frac {105 \sqrt {a} b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt [6]{x}}{\sqrt {b}} \right )}}{8} + \frac {a^{4} x^{\frac {7}{6}}}{\sqrt {b} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {23 a^{3} \sqrt {b} x^{\frac {5}{6}}}{4 \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {125 a^{2} b^{\frac {3}{2}} \sqrt {x}}{8 \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} + \frac {39 a b^{\frac {5}{2}} \sqrt [6]{x}}{8 \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} - \frac {6 b^{\frac {7}{2}}}{\sqrt [6]{x} \sqrt {\frac {a \sqrt [3]{x}}{b} + 1}} \] Input:
integrate((a+b/x**(1/3))**(7/2),x)
Output:
105*sqrt(a)*b**3*asinh(sqrt(a)*x**(1/6)/sqrt(b))/8 + a**4*x**(7/6)/(sqrt(b )*sqrt(a*x**(1/3)/b + 1)) + 23*a**3*sqrt(b)*x**(5/6)/(4*sqrt(a*x**(1/3)/b + 1)) + 125*a**2*b**(3/2)*sqrt(x)/(8*sqrt(a*x**(1/3)/b + 1)) + 39*a*b**(5/ 2)*x**(1/6)/(8*sqrt(a*x**(1/3)/b + 1)) - 6*b**(7/2)/(x**(1/6)*sqrt(a*x**(1 /3)/b + 1))
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.22 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=-\frac {105}{16} \, \sqrt {a} b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x^{\frac {1}{3}}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{\frac {1}{3}}}} + \sqrt {a}}\right ) - 6 \, \sqrt {a + \frac {b}{x^{\frac {1}{3}}}} b^{3} + \frac {87 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {5}{2}} a b^{3} - 136 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{\frac {3}{2}} a^{2} b^{3} + 57 \, \sqrt {a + \frac {b}{x^{\frac {1}{3}}}} a^{3} b^{3}}{8 \, {\left ({\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} - 3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a + 3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{2} - a^{3}\right )}} \] Input:
integrate((a+b/x^(1/3))^(7/2),x, algorithm="maxima")
Output:
-105/16*sqrt(a)*b^3*log((sqrt(a + b/x^(1/3)) - sqrt(a))/(sqrt(a + b/x^(1/3 )) + sqrt(a))) - 6*sqrt(a + b/x^(1/3))*b^3 + 1/8*(87*(a + b/x^(1/3))^(5/2) *a*b^3 - 136*(a + b/x^(1/3))^(3/2)*a^2*b^3 + 57*sqrt(a + b/x^(1/3))*a^3*b^ 3)/((a + b/x^(1/3))^3 - 3*(a + b/x^(1/3))^2*a + 3*(a + b/x^(1/3))*a^2 - a^ 3)
Time = 0.38 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=-\frac {105 \, \sqrt {a} b^{3} \log \left ({\left (\sqrt {a} x^{\frac {1}{6}} - \sqrt {a x^{\frac {1}{3}} + b}\right )}^{2}\right )}{16 \, \mathrm {sgn}\left (x\right )^{\frac {1}{3}}} + \frac {12 \, \sqrt {a} b^{4}}{{\left ({\left (\sqrt {a} x^{\frac {1}{6}} - \sqrt {a x^{\frac {1}{3}} + b}\right )}^{2} - b\right )} \mathrm {sgn}\left (x\right )^{\frac {1}{3}}} + \frac {1}{8} \, {\left (\frac {87 \, a b^{2}}{\mathrm {sgn}\left (x\right )^{\frac {1}{3}}} + 2 \, {\left (\frac {4 \, a^{3} x^{\frac {1}{3}}}{\mathrm {sgn}\left (x\right )^{\frac {1}{3}}} + \frac {19 \, a^{2} b}{\mathrm {sgn}\left (x\right )^{\frac {1}{3}}}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x^{\frac {1}{3}} + b} x^{\frac {1}{6}} \] Input:
integrate((a+b/x^(1/3))^(7/2),x, algorithm="giac")
Output:
-105/16*sqrt(a)*b^3*log((sqrt(a)*x^(1/6) - sqrt(a*x^(1/3) + b))^2)/sgn(x)^ (1/3) + 12*sqrt(a)*b^4/(((sqrt(a)*x^(1/6) - sqrt(a*x^(1/3) + b))^2 - b)*sg n(x)^(1/3)) + 1/8*(87*a*b^2/sgn(x)^(1/3) + 2*(4*a^3*x^(1/3)/sgn(x)^(1/3) + 19*a^2*b/sgn(x)^(1/3))*x^(1/3))*sqrt(a*x^(1/3) + b)*x^(1/6)
Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.31 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=-\frac {6\,x\,{\left (a+\frac {b}{x^{1/3}}\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {a\,x^{1/3}}{b}\right )}{{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{7/2}} \] Input:
int((a + b/x^(1/3))^(7/2),x)
Output:
-(6*x*(a + b/x^(1/3))^(7/2)*hypergeom([-7/2, -1/2], 1/2, -(a*x^(1/3))/b))/ ((a*x^(1/3))/b + 1)^(7/2)
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\frac {304 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b +696 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{2}+64 \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} x -384 \sqrt {x^{\frac {1}{3}} a +b}\, b^{3}+840 x^{\frac {1}{6}} \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}+x^{\frac {1}{6}} \sqrt {a}}{\sqrt {b}}\right ) b^{3}-525 x^{\frac {1}{6}} \sqrt {a}\, b^{3}}{64 x^{\frac {1}{6}}} \] Input:
int((a+b/x^(1/3))^(7/2),x)
Output:
(304*x**(2/3)*sqrt(x**(1/3)*a + b)*a**2*b + 696*x**(1/3)*sqrt(x**(1/3)*a + b)*a*b**2 + 64*sqrt(x**(1/3)*a + b)*a**3*x - 384*sqrt(x**(1/3)*a + b)*b** 3 + 840*x**(1/6)*sqrt(a)*log((sqrt(x**(1/3)*a + b) + x**(1/6)*sqrt(a))/sqr t(b))*b**3 - 525*x**(1/6)*sqrt(a)*b**3)/(64*x**(1/6))