Integrand size = 15, antiderivative size = 131 \[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=-\frac {7 a^2 x^{3/4} \sqrt [4]{a+b x}}{96 b^2}+\frac {a x^{7/4} \sqrt [4]{a+b x}}{24 b}+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}-\frac {7 a^3 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{64 b^{11/4}}+\frac {7 a^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{64 b^{11/4}} \] Output:
-7/96*a^2*x^(3/4)*(b*x+a)^(1/4)/b^2+1/24*a*x^(7/4)*(b*x+a)^(1/4)/b+1/3*x^( 11/4)*(b*x+a)^(1/4)-7/64*a^3*arctan(b^(1/4)*x^(1/4)/(b*x+a)^(1/4))/b^(11/4 )+7/64*a^3*arctanh(b^(1/4)*x^(1/4)/(b*x+a)^(1/4))/b^(11/4)
Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.78 \[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=\frac {2 b^{3/4} x^{3/4} \sqrt [4]{a+b x} \left (-7 a^2+4 a b x+32 b^2 x^2\right )-21 a^3 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )+21 a^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{192 b^{11/4}} \] Input:
Integrate[x^(7/4)*(a + b*x)^(1/4),x]
Output:
(2*b^(3/4)*x^(3/4)*(a + b*x)^(1/4)*(-7*a^2 + 4*a*b*x + 32*b^2*x^2) - 21*a^ 3*ArcTan[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)] + 21*a^3*ArcTanh[(b^(1/4)*x^(1 /4))/(a + b*x)^(1/4)])/(192*b^(11/4))
Time = 0.22 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {60, 60, 60, 73, 854, 827, 216, 219}
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.
| \(\displaystyle \int x^{7/4} \sqrt [4]{a+b x} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{12} a \int \frac {x^{7/4}}{(a+b x)^{3/4}}dx+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \int \frac {x^{3/4}}{(a+b x)^{3/4}}dx}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}}dx}{4 b}\right )}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {\sqrt {x}}{(a+b x)^{3/4}}d\sqrt [4]{x}}{b}\right )}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {\sqrt {x}}{1-b x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{b}\right )}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\int \frac {1}{1-\sqrt {b} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}-\frac {\int \frac {1}{\sqrt {b} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}\right )}{b}\right )}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\int \frac {1}{1-\sqrt {b} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )}{b}\right )}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} a \left (\frac {x^{7/4} \sqrt [4]{a+b x}}{2 b}-\frac {7 a \left (\frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )}{b}\right )}{8 b}\right )+\frac {1}{3} x^{11/4} \sqrt [4]{a+b x}\) |
Input:
Int[x^(7/4)*(a + b*x)^(1/4),x]
Output:
(x^(11/4)*(a + b*x)^(1/4))/3 + (a*((x^(7/4)*(a + b*x)^(1/4))/(2*b) - (7*a* ((x^(3/4)*(a + b*x)^(1/4))/b - (3*a*(-1/2*ArcTan[(b^(1/4)*x^(1/4))/(a + b* x)^(1/4)]/b^(3/4) + ArcTanh[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)]/(2*b^(3/4)) ))/b))/(8*b)))/12
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
\[\int x^{\frac {7}{4}} \left (b x +a \right )^{\frac {1}{4}}d x\]
Input:
int(x^(7/4)*(b*x+a)^(1/4),x)
Output:
int(x^(7/4)*(b*x+a)^(1/4),x)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.82 \[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=\frac {21 \, \left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{2} \log \left (\frac {7 \, {\left (\left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{3} x + {\left (b x + a\right )}^{\frac {1}{4}} a^{3} x^{\frac {3}{4}}\right )}}{x}\right ) - 21 \, \left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{2} \log \left (-\frac {7 \, {\left (\left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{3} x - {\left (b x + a\right )}^{\frac {1}{4}} a^{3} x^{\frac {3}{4}}\right )}}{x}\right ) - 21 i \, \left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{2} \log \left (-\frac {7 \, {\left (i \, \left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{3} x - {\left (b x + a\right )}^{\frac {1}{4}} a^{3} x^{\frac {3}{4}}\right )}}{x}\right ) + 21 i \, \left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{2} \log \left (-\frac {7 \, {\left (-i \, \left (\frac {a^{12}}{b^{11}}\right )^{\frac {1}{4}} b^{3} x - {\left (b x + a\right )}^{\frac {1}{4}} a^{3} x^{\frac {3}{4}}\right )}}{x}\right ) + 4 \, {\left (32 \, b^{2} x^{2} + 4 \, a b x - 7 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{384 \, b^{2}} \] Input:
integrate(x^(7/4)*(b*x+a)^(1/4),x, algorithm="fricas")
Output:
1/384*(21*(a^12/b^11)^(1/4)*b^2*log(7*((a^12/b^11)^(1/4)*b^3*x + (b*x + a) ^(1/4)*a^3*x^(3/4))/x) - 21*(a^12/b^11)^(1/4)*b^2*log(-7*((a^12/b^11)^(1/4 )*b^3*x - (b*x + a)^(1/4)*a^3*x^(3/4))/x) - 21*I*(a^12/b^11)^(1/4)*b^2*log (-7*(I*(a^12/b^11)^(1/4)*b^3*x - (b*x + a)^(1/4)*a^3*x^(3/4))/x) + 21*I*(a ^12/b^11)^(1/4)*b^2*log(-7*(-I*(a^12/b^11)^(1/4)*b^3*x - (b*x + a)^(1/4)*a ^3*x^(3/4))/x) + 4*(32*b^2*x^2 + 4*a*b*x - 7*a^2)*(b*x + a)^(1/4)*x^(3/4)) /b^2
Result contains complex when optimal does not.
Time = 10.77 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.28 \[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=\frac {\sqrt [4]{a} x^{\frac {11}{4}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {15}{4}\right )} \] Input:
integrate(x**(7/4)*(b*x+a)**(1/4),x)
Output:
a**(1/4)*x**(11/4)*gamma(11/4)*hyper((-1/4, 11/4), (15/4,), b*x*exp_polar( I*pi)/a)/gamma(15/4)
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.34 \[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=-\frac {\frac {21 \, {\left (b x + a\right )}^{\frac {1}{4}} a^{3} b^{2}}{x^{\frac {1}{4}}} + \frac {18 \, {\left (b x + a\right )}^{\frac {5}{4}} a^{3} b}{x^{\frac {5}{4}}} - \frac {7 \, {\left (b x + a\right )}^{\frac {9}{4}} a^{3}}{x^{\frac {9}{4}}}}{96 \, {\left (b^{5} - \frac {3 \, {\left (b x + a\right )} b^{4}}{x} + \frac {3 \, {\left (b x + a\right )}^{2} b^{3}}{x^{2}} - \frac {{\left (b x + a\right )}^{3} b^{2}}{x^{3}}\right )}} + \frac {7 \, {\left (\frac {2 \, a^{3} \arctan \left (\frac {{\left (b x + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a^{3} \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}{b^{\frac {1}{4}} + \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}\right )}{b^{\frac {3}{4}}}\right )}}{128 \, b^{2}} \] Input:
integrate(x^(7/4)*(b*x+a)^(1/4),x, algorithm="maxima")
Output:
-1/96*(21*(b*x + a)^(1/4)*a^3*b^2/x^(1/4) + 18*(b*x + a)^(5/4)*a^3*b/x^(5/ 4) - 7*(b*x + a)^(9/4)*a^3/x^(9/4))/(b^5 - 3*(b*x + a)*b^4/x + 3*(b*x + a) ^2*b^3/x^2 - (b*x + a)^3*b^2/x^3) + 7/128*(2*a^3*arctan((b*x + a)^(1/4)/(b ^(1/4)*x^(1/4)))/b^(3/4) - a^3*log(-(b^(1/4) - (b*x + a)^(1/4)/x^(1/4))/(b ^(1/4) + (b*x + a)^(1/4)/x^(1/4)))/b^(3/4))/b^2
\[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {7}{4}} \,d x } \] Input:
integrate(x^(7/4)*(b*x+a)^(1/4),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/4)*x^(7/4), x)
Timed out. \[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=\int x^{7/4}\,{\left (a+b\,x\right )}^{1/4} \,d x \] Input:
int(x^(7/4)*(a + b*x)^(1/4),x)
Output:
int(x^(7/4)*(a + b*x)^(1/4), x)
\[ \int x^{7/4} \sqrt [4]{a+b x} \, dx=\frac {-28 x^{\frac {3}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{2}+16 x^{\frac {7}{4}} \left (b x +a \right )^{\frac {1}{4}} a b +128 x^{\frac {11}{4}} \left (b x +a \right )^{\frac {1}{4}} b^{2}+21 \left (\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{\frac {1}{4}} a +x^{\frac {5}{4}} b}d x \right ) a^{3}}{384 b^{2}} \] Input:
int(x^(7/4)*(b*x+a)^(1/4),x)
Output:
( - 28*x**(3/4)*(a + b*x)**(1/4)*a**2 + 16*x**(3/4)*(a + b*x)**(1/4)*a*b*x + 128*x**(3/4)*(a + b*x)**(1/4)*b**2*x**2 + 21*int((a + b*x)**(1/4)/(x**( 1/4)*a + x**(1/4)*b*x),x)*a**3)/(384*b**2)