Integrand size = 16, antiderivative size = 80 \[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=-\frac {4}{\sqrt [4]{x} \sqrt [4]{a-b x}}+\frac {4 \sqrt [4]{a-b x} \sqrt [4]{1-\frac {a}{a-b x}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a-b x}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{x}} \] Output:
-4/x^(1/4)/(-b*x+a)^(1/4)+4*(-b*x+a)^(1/4)*(1-a/(-b*x+a))^(1/4)*EllipticE( sin(1/2*arcsin(a^(1/2)/(-b*x+a)^(1/2))),2^(1/2))/a^(1/2)/x^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=-\frac {4 \sqrt [4]{1-\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},\frac {b x}{a}\right )}{\sqrt [4]{x} \sqrt [4]{a-b x}} \] Input:
Integrate[1/(x^(5/4)*(a - b*x)^(1/4)),x]
Output:
(-4*(1 - (b*x)/a)^(1/4)*Hypergeometric2F1[-1/4, 1/4, 3/4, (b*x)/a])/(x^(1/ 4)*(a - b*x)^(1/4))
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.46, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {61, 73, 840, 842, 858, 807, 226}
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.
\(\displaystyle \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {2 b \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a-b x}}dx}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {8 b \int \frac {\sqrt {x}}{\sqrt [4]{a-b x}}d\sqrt [4]{x}}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
\(\Big \downarrow \) 840 |
\(\displaystyle -\frac {8 b \left (-\frac {a \int \frac {1}{\sqrt {x} \sqrt [4]{a-b x}}d\sqrt [4]{x}}{2 b}-\frac {(a-b x)^{3/4}}{2 b \sqrt [4]{x}}\right )}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
\(\Big \downarrow \) 842 |
\(\displaystyle -\frac {8 b \left (-\frac {a \sqrt [4]{x} \sqrt [4]{1-\frac {a}{b x}} \int \frac {1}{\sqrt [4]{1-\frac {a}{b x}} x^{3/4}}d\sqrt [4]{x}}{2 b \sqrt [4]{a-b x}}-\frac {(a-b x)^{3/4}}{2 b \sqrt [4]{x}}\right )}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {8 b \left (\frac {a \sqrt [4]{x} \sqrt [4]{1-\frac {a}{b x}} \int \frac {1}{\sqrt [4]{x} \sqrt [4]{1-\frac {a x}{b}}}d\frac {1}{\sqrt [4]{x}}}{2 b \sqrt [4]{a-b x}}-\frac {(a-b x)^{3/4}}{2 b \sqrt [4]{x}}\right )}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {8 b \left (\frac {a \sqrt [4]{x} \sqrt [4]{1-\frac {a}{b x}} \int \frac {1}{\sqrt [4]{1-\frac {a \sqrt {x}}{b}}}d\sqrt {x}}{4 b \sqrt [4]{a-b x}}-\frac {(a-b x)^{3/4}}{2 b \sqrt [4]{x}}\right )}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle -\frac {8 b \left (\frac {\sqrt {a} \sqrt [4]{x} \sqrt [4]{1-\frac {a}{b x}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |2\right )}{2 \sqrt {b} \sqrt [4]{a-b x}}-\frac {(a-b x)^{3/4}}{2 b \sqrt [4]{x}}\right )}{a}-\frac {4 (a-b x)^{3/4}}{a \sqrt [4]{x}}\) |
Input:
Int[1/(x^(5/4)*(a - b*x)^(1/4)),x]
Output:
(-4*(a - b*x)^(3/4))/(a*x^(1/4)) - (8*b*(-1/2*(a - b*x)^(3/4)/(b*x^(1/4)) + (Sqrt[a]*(1 - a/(b*x))^(1/4)*x^(1/4)*EllipticE[ArcSin[(Sqrt[a]*Sqrt[x])/ Sqrt[b]]/2, 2])/(2*Sqrt[b]*(a - b*x)^(1/4))))/a
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[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)^(1/4), x_Symbol] :> Simp[(a + b*x^4)^(3/4) /(2*b*x), x] + Simp[a/(2*b) Int[1/(x^2*(a + b*x^4)^(1/4)), x], x] /; Free Q[{a, b}, x] && NegQ[b/a]
Int[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> Simp[x*((1 + a/(b* x^4))^(1/4)/(a + b*x^4)^(1/4)) Int[1/(x^3*(1 + a/(b*x^4))^(1/4)), x], x] /; FreeQ[{a, b}, x] && NegQ[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]
\[\int \frac {1}{x^{\frac {5}{4}} \left (-b x +a \right )^{\frac {1}{4}}}d x\]
Input:
int(1/x^(5/4)/(-b*x+a)^(1/4),x)
Output:
int(1/x^(5/4)/(-b*x+a)^(1/4),x)
\[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=\int { \frac {1}{{\left (-b x + a\right )}^{\frac {1}{4}} x^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/x^(5/4)/(-b*x+a)^(1/4),x, algorithm="fricas")
Output:
integral(-(-b*x + a)^(3/4)*x^(3/4)/(b*x^3 - a*x^2), x)
Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=\frac {2 i e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a}{b x}} \right )}}{\sqrt [4]{b} \sqrt {x}} \] Input:
integrate(1/x**(5/4)/(-b*x+a)**(1/4),x)
Output:
2*I*exp(I*pi/4)*hyper((1/4, 1/2), (3/2,), a/(b*x))/(b**(1/4)*sqrt(x))
\[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=\int { \frac {1}{{\left (-b x + a\right )}^{\frac {1}{4}} x^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/x^(5/4)/(-b*x+a)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((-b*x + a)^(1/4)*x^(5/4)), x)
\[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=\int { \frac {1}{{\left (-b x + a\right )}^{\frac {1}{4}} x^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/x^(5/4)/(-b*x+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((-b*x + a)^(1/4)*x^(5/4)), x)
Timed out. \[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=\int \frac {1}{x^{5/4}\,{\left (a-b\,x\right )}^{1/4}} \,d x \] Input:
int(1/(x^(5/4)*(a - b*x)^(1/4)),x)
Output:
int(1/(x^(5/4)*(a - b*x)^(1/4)), x)
\[ \int \frac {1}{x^{5/4} \sqrt [4]{a-b x}} \, dx=\int \frac {1}{x^{\frac {5}{4}} \left (-b x +a \right )^{\frac {1}{4}}}d x \] Input:
int(1/x^(5/4)/(-b*x+a)^(1/4),x)
Output:
int(1/(x**(1/4)*(a - b*x)**(1/4)*x),x)