Integrand size = 21, antiderivative size = 80 \[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\frac {4 \sqrt [4]{\frac {2-b x}{a-b x}} \sqrt [4]{a-b x} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {2-a}}{\sqrt {a-b x}}\right )\right |2\right )}{\sqrt {2-a} b \sqrt [4]{2-b x}} \] Output:
4*((-b*x+2)/(-b*x+a))^(1/4)*(-b*x+a)^(1/4)*EllipticE(sin(1/2*arctan((2-a)^ (1/2)/(-b*x+a)^(1/2))),2^(1/2))/(2-a)^(1/2)/b/(-b*x+2)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\frac {4 \sqrt [4]{\frac {a-b x}{-2+a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},\frac {-2+b x}{-2+a}\right )}{b \sqrt [4]{2-b x} \sqrt [4]{a-b x}} \] Input:
Integrate[1/((2 - b*x)^(5/4)*(a - b*x)^(1/4)),x]
Output:
(4*((a - b*x)/(-2 + a))^(1/4)*Hypergeometric2F1[-1/4, 1/4, 3/4, (-2 + b*x) /(-2 + a)])/(b*(2 - b*x)^(1/4)*(a - b*x)^(1/4))
Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.
\(\displaystyle \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {2 \int \frac {1}{\sqrt [4]{2-b x} \sqrt [4]{a-b x}}dx}{2-a}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {8 \int \frac {\sqrt {2-b x}}{\sqrt [4]{a-b x}}d\sqrt [4]{2-b x}}{(2-a) b}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
\(\Big \downarrow \) 840 |
\(\displaystyle \frac {8 \left (\frac {(a-b x)^{3/4}}{2 \sqrt [4]{2-b x}}-\frac {1}{2} (2-a) \int \frac {1}{\sqrt {2-b x} \sqrt [4]{a-b x}}d\sqrt [4]{2-b x}\right )}{(2-a) b}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
\(\Big \downarrow \) 842 |
\(\displaystyle \frac {8 \left (\frac {(a-b x)^{3/4}}{2 \sqrt [4]{2-b x}}-\frac {(2-a) \sqrt [4]{2-b x} \sqrt [4]{1-\frac {2-a}{2-b x}} \int \frac {1}{(2-b x)^{3/4} \sqrt [4]{1-\frac {2-a}{2-b x}}}d\sqrt [4]{2-b x}}{2 \sqrt [4]{a-b x}}\right )}{(2-a) b}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {8 \left (\frac {(2-a) \sqrt [4]{2-b x} \sqrt [4]{1-\frac {2-a}{2-b x}} \int \frac {1}{\sqrt [4]{2-b x} \sqrt [4]{(a-2) (2-b x)+1}}d\frac {1}{\sqrt [4]{2-b x}}}{2 \sqrt [4]{a-b x}}+\frac {(a-b x)^{3/4}}{2 \sqrt [4]{2-b x}}\right )}{(2-a) b}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {8 \left (\frac {(2-a) \sqrt [4]{2-b x} \sqrt [4]{1-\frac {2-a}{2-b x}} \int \frac {1}{\sqrt [4]{1-(2-a) \sqrt {2-b x}}}d\sqrt {2-b x}}{4 \sqrt [4]{a-b x}}+\frac {(a-b x)^{3/4}}{2 \sqrt [4]{2-b x}}\right )}{(2-a) b}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle \frac {8 \left (\frac {\sqrt {2-a} \sqrt [4]{2-b x} \sqrt [4]{1-\frac {2-a}{2-b x}} E\left (\left .\frac {1}{2} \arcsin \left (\sqrt {2-a} \sqrt {2-b x}\right )\right |2\right )}{2 \sqrt [4]{a-b x}}+\frac {(a-b x)^{3/4}}{2 \sqrt [4]{2-b x}}\right )}{(2-a) b}-\frac {4 (a-b x)^{3/4}}{(2-a) b \sqrt [4]{2-b x}}\) |
Input:
Int[1/((2 - b*x)^(5/4)*(a - b*x)^(1/4)),x]
Output:
(-4*(a - b*x)^(3/4))/((2 - a)*b*(2 - b*x)^(1/4)) + (8*((a - b*x)^(3/4)/(2* (2 - b*x)^(1/4)) + (Sqrt[2 - a]*(2 - b*x)^(1/4)*(1 - (2 - a)/(2 - b*x))^(1 /4)*EllipticE[ArcSin[Sqrt[2 - a]*Sqrt[2 - b*x]]/2, 2])/(2*(a - b*x)^(1/4)) ))/((2 - a)*b)
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]
\[\int \frac {1}{\left (-b x +2\right )^{\frac {5}{4}} \left (-b x +a \right )^{\frac {1}{4}}}d x\]
Input:
int(1/(-b*x+2)^(5/4)/(-b*x+a)^(1/4),x)
Output:
int(1/(-b*x+2)^(5/4)/(-b*x+a)^(1/4),x)
\[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\int { \frac {1}{{\left (-b x + a\right )}^{\frac {1}{4}} {\left (-b x + 2\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/(-b*x+2)^(5/4)/(-b*x+a)^(1/4),x, algorithm="fricas")
Output:
integral(-(-b*x + a)^(3/4)*(-b*x + 2)^(3/4)/(b^3*x^3 - (a + 4)*b^2*x^2 + 4 *(a + 1)*b*x - 4*a), x)
\[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\int \frac {1}{\sqrt [4]{a - b x} \left (- b x + 2\right )^{\frac {5}{4}}}\, dx \] Input:
integrate(1/(-b*x+2)**(5/4)/(-b*x+a)**(1/4),x)
Output:
Integral(1/((a - b*x)**(1/4)*(-b*x + 2)**(5/4)), x)
\[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\int { \frac {1}{{\left (-b x + a\right )}^{\frac {1}{4}} {\left (-b x + 2\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/(-b*x+2)^(5/4)/(-b*x+a)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((-b*x + a)^(1/4)*(-b*x + 2)^(5/4)), x)
\[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\int { \frac {1}{{\left (-b x + a\right )}^{\frac {1}{4}} {\left (-b x + 2\right )}^{\frac {5}{4}}} \,d x } \] Input:
integrate(1/(-b*x+2)^(5/4)/(-b*x+a)^(1/4),x, algorithm="giac")
Output:
integrate(1/((-b*x + a)^(1/4)*(-b*x + 2)^(5/4)), x)
Timed out. \[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=\int \frac {1}{{\left (2-b\,x\right )}^{5/4}\,{\left (a-b\,x\right )}^{1/4}} \,d x \] Input:
int(1/((2 - b*x)^(5/4)*(a - b*x)^(1/4)),x)
Output:
int(1/((2 - b*x)^(5/4)*(a - b*x)^(1/4)), x)
\[ \int \frac {1}{(2-b x)^{5/4} \sqrt [4]{a-b x}} \, dx=-\left (\int \frac {1}{\left (-b x +a \right )^{\frac {1}{4}} \left (-b x +2\right )^{\frac {1}{4}} b x -2 \left (-b x +a \right )^{\frac {1}{4}} \left (-b x +2\right )^{\frac {1}{4}}}d x \right ) \] Input:
int(1/(-b*x+2)^(5/4)/(-b*x+a)^(1/4),x)
Output:
- int(1/((a - b*x)**(1/4)*( - b*x + 2)**(1/4)*b*x - 2*(a - b*x)**(1/4)*( - b*x + 2)**(1/4)),x)