Integrand size = 15, antiderivative size = 127 \[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=-\frac {a^2 \sqrt [4]{x} \sqrt [4]{a+b x}}{6 b^2}+\frac {a x^{5/4} \sqrt [4]{a+b x}}{15 b}+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}-\frac {a^{5/2} \left (\frac {b x}{a+b x}\right )^{3/4} (a+b x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right ),2\right )}{6 b^3 x^{3/4}} \] Output:
-1/6*a^2*x^(1/4)*(b*x+a)^(1/4)/b^2+1/15*a*x^(5/4)*(b*x+a)^(1/4)/b+2/5*x^(9 /4)*(b*x+a)^(1/4)-1/6*a^(5/2)*(b*x/(b*x+a))^(3/4)*(b*x+a)^(3/4)*InverseJac obiAM(1/2*arcsin(a^(1/2)/(b*x+a)^(1/2)),2^(1/2))/b^3/x^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.37 \[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\frac {4 x^{9/4} \sqrt [4]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {9}{4},\frac {13}{4},-\frac {b x}{a}\right )}{9 \sqrt [4]{1+\frac {b x}{a}}} \] Input:
Integrate[x^(5/4)*(a + b*x)^(1/4),x]
Output:
(4*x^(9/4)*(a + b*x)^(1/4)*Hypergeometric2F1[-1/4, 9/4, 13/4, -((b*x)/a)]) /(9*(1 + (b*x)/a)^(1/4))
Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08, 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, 768, 858, 807, 229}
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.
\(\displaystyle \int x^{5/4} \sqrt [4]{a+b x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{10} a \int \frac {x^{5/4}}{(a+b x)^{3/4}}dx+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \int \frac {\sqrt [4]{x}}{(a+b x)^{3/4}}dx}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {a \int \frac {1}{x^{3/4} (a+b x)^{3/4}}dx}{2 b}\right )}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {2 a \int \frac {1}{(a+b x)^{3/4}}d\sqrt [4]{x}}{b}\right )}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {2 a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{3/4} x^{3/4}}d\sqrt [4]{x}}{b (a+b x)^{3/4}}\right )}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{3/4}}d\frac {1}{\sqrt [4]{x}}}{b (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{3/4}}d\sqrt {x}}{b (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \frac {1}{10} a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt {a} x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),2\right )}{\sqrt {b} (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )+\frac {2}{5} x^{9/4} \sqrt [4]{a+b x}\) |
Input:
Int[x^(5/4)*(a + b*x)^(1/4),x]
Output:
(2*x^(9/4)*(a + b*x)^(1/4))/5 + (a*((2*x^(5/4)*(a + b*x)^(1/4))/(3*b) - (5 *a*((2*x^(1/4)*(a + b*x)^(1/4))/b + (2*Sqrt[a]*(1 + a/(b*x))^(3/4)*x^(3/4) *EllipticF[ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/2, 2])/(Sqrt[b]*(a + b*x)^(3/ 4))))/(6*b)))/10
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)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int x^{\frac {5}{4}} \left (b x +a \right )^{\frac {1}{4}}d x\]
Input:
int(x^(5/4)*(b*x+a)^(1/4),x)
Output:
int(x^(5/4)*(b*x+a)^(1/4),x)
\[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {5}{4}} \,d x } \] Input:
integrate(x^(5/4)*(b*x+a)^(1/4),x, algorithm="fricas")
Output:
integral((b*x + a)^(1/4)*x^(5/4), x)
Result contains complex when optimal does not.
Time = 2.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.29 \[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\frac {\sqrt [4]{a} x^{\frac {9}{4}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {13}{4}\right )} \] Input:
integrate(x**(5/4)*(b*x+a)**(1/4),x)
Output:
a**(1/4)*x**(9/4)*gamma(9/4)*hyper((-1/4, 9/4), (13/4,), b*x*exp_polar(I*p i)/a)/gamma(13/4)
\[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {5}{4}} \,d x } \] Input:
integrate(x^(5/4)*(b*x+a)^(1/4),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/4)*x^(5/4), x)
\[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {5}{4}} \,d x } \] Input:
integrate(x^(5/4)*(b*x+a)^(1/4),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/4)*x^(5/4), x)
Timed out. \[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\int x^{5/4}\,{\left (a+b\,x\right )}^{1/4} \,d x \] Input:
int(x^(5/4)*(a + b*x)^(1/4),x)
Output:
int(x^(5/4)*(a + b*x)^(1/4), x)
\[ \int x^{5/4} \sqrt [4]{a+b x} \, dx=\frac {-20 x^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{2}+8 x^{\frac {5}{4}} \left (b x +a \right )^{\frac {1}{4}} a b +48 x^{\frac {9}{4}} \left (b x +a \right )^{\frac {1}{4}} b^{2}+5 \left (\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{\frac {3}{4}} a +x^{\frac {7}{4}} b}d x \right ) a^{3}}{120 b^{2}} \] Input:
int(x^(5/4)*(b*x+a)^(1/4),x)
Output:
( - 20*x**(1/4)*(a + b*x)**(1/4)*a**2 + 8*x**(1/4)*(a + b*x)**(1/4)*a*b*x + 48*x**(1/4)*(a + b*x)**(1/4)*b**2*x**2 + 5*int((a + b*x)**(1/4)/(x**(3/4 )*a + x**(3/4)*b*x),x)*a**3)/(120*b**2)