Integrand size = 15, antiderivative size = 63 \[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\frac {\sqrt {2} a \sqrt [4]{-\frac {b x}{a}-\frac {b^2 x^2}{a^2}} E\left (\left .\frac {1}{2} \arcsin \left (1+\frac {2 b x}{a}\right )\right |2\right )}{b \sqrt [4]{x} \sqrt [4]{a+b x}} \] Output:
2^(1/2)*a*(-b*x/a-b^2*x^2/a^2)^(1/4)*EllipticE(sin(1/2*arcsin(1+2*b*x/a)), 2^(1/2))/b/x^(1/4)/(b*x+a)^(1/4)
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 = 0.75 \[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\frac {4 x^{3/4} \sqrt [4]{1+\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},-\frac {b x}{a}\right )}{3 \sqrt [4]{a+b x}} \] Input:
Integrate[1/(x^(1/4)*(a + b*x)^(1/4)),x]
Output:
(4*x^(3/4)*(1 + (b*x)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x)/a) ])/(3*(a + b*x)^(1/4))
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {73, 839, 813, 858, 807, 212}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 4 \int \frac {\sqrt {x}}{\sqrt [4]{a+b x}}d\sqrt [4]{x}\) |
\(\Big \downarrow \) 839 |
\(\displaystyle 4 \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {1}{2} a \int \frac {\sqrt {x}}{(a+b x)^{5/4}}d\sqrt [4]{x}\right )\) |
\(\Big \downarrow \) 813 |
\(\displaystyle 4 \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{5/4} x^{3/4}}d\sqrt [4]{x}}{2 b \sqrt [4]{a+b x}}\right )\) |
\(\Big \downarrow \) 858 |
\(\displaystyle 4 \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{x}}}{2 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )\) |
\(\Big \downarrow \) 807 |
\(\displaystyle 4 \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{5/4}}d\sqrt {x}}{4 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )\) |
\(\Big \downarrow \) 212 |
\(\displaystyle 4 \left (\frac {\sqrt {a} \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |2\right )}{2 \sqrt {b} \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )\) |
Input:
Int[1/(x^(1/4)*(a + b*x)^(1/4)),x]
Output:
4*(x^(3/4)/(2*(a + b*x)^(1/4)) + (Sqrt[a]*(1 + a/(b*x))^(1/4)*x^(1/4)*Elli pticE[ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/2, 2])/(2*Sqrt[b]*(a + b*x)^(1/4)) )
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)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
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_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) )^(1/4)/(b*(a + b*x^4)^(1/4))) Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 )^(1/4)), x] - Simp[a/2 Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} , x] && PosQ[b/a]
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]
\[\int \frac {1}{x^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}}}d x\]
Input:
int(1/x^(1/4)/(b*x+a)^(1/4),x)
Output:
int(1/x^(1/4)/(b*x+a)^(1/4),x)
\[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/x^(1/4)/(b*x+a)^(1/4),x, algorithm="fricas")
Output:
integral((b*x + a)^(3/4)*x^(3/4)/(b*x^2 + a*x), x)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\frac {x^{\frac {3}{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt [4]{a} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate(1/x**(1/4)/(b*x+a)**(1/4),x)
Output:
x**(3/4)*gamma(3/4)*hyper((1/4, 3/4), (7/4,), b*x*exp_polar(I*pi)/a)/(a**( 1/4)*gamma(7/4))
\[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/x^(1/4)/(b*x+a)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(1/4)*x^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/x^(1/4)/(b*x+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(1/4)*x^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\int \frac {1}{x^{1/4}\,{\left (a+b\,x\right )}^{1/4}} \,d x \] Input:
int(1/(x^(1/4)*(a + b*x)^(1/4)),x)
Output:
int(1/(x^(1/4)*(a + b*x)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}} \, dx=\int \frac {1}{x^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}}}d x \] Input:
int(1/x^(1/4)/(b*x+a)^(1/4),x)
Output:
int(1/(x**(1/4)*(a + b*x)**(1/4)),x)