\(\int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx\) [579]

Optimal result

Integrand size = 19, antiderivative size = 51 \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=\frac {4 \sqrt [4]{\frac {d x}{c}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {-c+d x}}{\sqrt {c}}\right )\right |2\right )}{b \sqrt {c} \sqrt [4]{b x}} \] Output:

4*(d*x/c)^(1/4)*EllipticE(sin(1/2*arctan((d*x-c)^(1/2)/c^(1/2))),2^(1/2))/ 
b/c^(1/2)/(b*x)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=-\frac {4 x \sqrt {1-\frac {d x}{c}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {d x}{c}\right )}{(b x)^{5/4} \sqrt {-c+d x}} \] Input:

Integrate[1/((b*x)^(5/4)*Sqrt[-c + d*x]),x]
 

Output:

(-4*x*Sqrt[1 - (d*x)/c]*Hypergeometric2F1[-1/4, 1/2, 3/4, (d*x)/c])/((b*x) 
^(5/4)*Sqrt[-c + d*x])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(170\) vs. \(2(51)=102\).

Time = 0.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.33, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {61, 73, 836, 27, 765, 762, 1390, 1389, 327}

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.

\(\displaystyle \int \frac {1}{(b x)^{5/4} \sqrt {d x-c}} \, dx\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {d \int \frac {1}{\sqrt [4]{b x} \sqrt {d x-c}}dx}{b c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \int \frac {\sqrt {b x}}{\sqrt {d x-c}}d\sqrt [4]{b x}}{b^2 c}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {\sqrt {b} \sqrt {c} \int \frac {\sqrt {b} \sqrt {c}+\sqrt {d} \sqrt {b x}}{\sqrt {b} \sqrt {c} \sqrt {d x-c}}d\sqrt [4]{b x}}{\sqrt {d}}-\frac {\sqrt {b} \sqrt {c} \int \frac {1}{\sqrt {d x-c}}d\sqrt [4]{b x}}{\sqrt {d}}\right )}{b^2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {\int \frac {\sqrt {b} \sqrt {c}+\sqrt {d} \sqrt {b x}}{\sqrt {d x-c}}d\sqrt [4]{b x}}{\sqrt {d}}-\frac {\sqrt {b} \sqrt {c} \int \frac {1}{\sqrt {d x-c}}d\sqrt [4]{b x}}{\sqrt {d}}\right )}{b^2 c}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {\int \frac {\sqrt {b} \sqrt {c}+\sqrt {d} \sqrt {b x}}{\sqrt {d x-c}}d\sqrt [4]{b x}}{\sqrt {d}}-\frac {\sqrt {b} \sqrt {c} \sqrt {1-\frac {d x}{c}} \int \frac {1}{\sqrt {1-\frac {d x}{c}}}d\sqrt [4]{b x}}{\sqrt {d} \sqrt {d x-c}}\right )}{b^2 c}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {\int \frac {\sqrt {b} \sqrt {c}+\sqrt {d} \sqrt {b x}}{\sqrt {d x-c}}d\sqrt [4]{b x}}{\sqrt {d}}-\frac {b^{3/4} c^{3/4} \sqrt {1-\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{b x}}{\sqrt [4]{b} \sqrt [4]{c}}\right ),-1\right )}{d^{3/4} \sqrt {d x-c}}\right )}{b^2 c}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {\sqrt {1-\frac {d x}{c}} \int \frac {\sqrt {b} \sqrt {c}+\sqrt {d} \sqrt {b x}}{\sqrt {1-\frac {d x}{c}}}d\sqrt [4]{b x}}{\sqrt {d} \sqrt {d x-c}}-\frac {b^{3/4} c^{3/4} \sqrt {1-\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{b x}}{\sqrt [4]{b} \sqrt [4]{c}}\right ),-1\right )}{d^{3/4} \sqrt {d x-c}}\right )}{b^2 c}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {\sqrt {b} \sqrt {c} \sqrt {1-\frac {d x}{c}} \int \frac {\sqrt {\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b} \sqrt {c}}+1}}{\sqrt {1-\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b} \sqrt {c}}}}d\sqrt [4]{b x}}{\sqrt {d} \sqrt {d x-c}}-\frac {b^{3/4} c^{3/4} \sqrt {1-\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{b x}}{\sqrt [4]{b} \sqrt [4]{c}}\right ),-1\right )}{d^{3/4} \sqrt {d x-c}}\right )}{b^2 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {4 \sqrt {d x-c}}{b c \sqrt [4]{b x}}-\frac {4 d \left (\frac {b^{3/4} c^{3/4} \sqrt {1-\frac {d x}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{b x}}{\sqrt [4]{b} \sqrt [4]{c}}\right )\right |-1\right )}{d^{3/4} \sqrt {d x-c}}-\frac {b^{3/4} c^{3/4} \sqrt {1-\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{b x}}{\sqrt [4]{b} \sqrt [4]{c}}\right ),-1\right )}{d^{3/4} \sqrt {d x-c}}\right )}{b^2 c}\)

Input:

Int[1/((b*x)^(5/4)*Sqrt[-c + d*x]),x]
 

Output:

(4*Sqrt[-c + d*x])/(b*c*(b*x)^(1/4)) - (4*d*((b^(3/4)*c^(3/4)*Sqrt[1 - (d* 
x)/c]*EllipticE[ArcSin[(d^(1/4)*(b*x)^(1/4))/(b^(1/4)*c^(1/4))], -1])/(d^( 
3/4)*Sqrt[-c + d*x]) - (b^(3/4)*c^(3/4)*Sqrt[1 - (d*x)/c]*EllipticF[ArcSin 
[(d^(1/4)*(b*x)^(1/4))/(b^(1/4)*c^(1/4))], -1])/(d^(3/4)*Sqrt[-c + d*x]))) 
/(b^2*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 61

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]
 

rule 73

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]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [F]

\[\int \frac {1}{\left (b x \right )^{\frac {5}{4}} \sqrt {x d -c}}d x\]

Input:

int(1/(b*x)^(5/4)/(d*x-c)^(1/2),x)
 

Output:

int(1/(b*x)^(5/4)/(d*x-c)^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=\int { \frac {1}{\left (b x\right )^{\frac {5}{4}} \sqrt {d x - c}} \,d x } \] Input:

integrate(1/(b*x)^(5/4)/(d*x-c)^(1/2),x, algorithm="fricas")
 

Output:

integral((b*x)^(3/4)*sqrt(d*x - c)/(b^2*d*x^3 - b^2*c*x^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=- \frac {i \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x}{c}} \right )}}{b^{\frac {5}{4}} \sqrt {c} \sqrt [4]{x} \Gamma \left (\frac {3}{4}\right )} \] Input:

integrate(1/(b*x)**(5/4)/(d*x-c)**(1/2),x)
 

Output:

-I*gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), d*x/c)/(b**(5/4)*sqrt(c)*x**(1/4 
)*gamma(3/4))
 

Maxima [F]

\[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=\int { \frac {1}{\left (b x\right )^{\frac {5}{4}} \sqrt {d x - c}} \,d x } \] Input:

integrate(1/(b*x)^(5/4)/(d*x-c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x)^(5/4)*sqrt(d*x - c)), x)
 

Giac [F]

\[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=\int { \frac {1}{\left (b x\right )^{\frac {5}{4}} \sqrt {d x - c}} \,d x } \] Input:

integrate(1/(b*x)^(5/4)/(d*x-c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x)^(5/4)*sqrt(d*x - c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=\int \frac {1}{{\left (b\,x\right )}^{5/4}\,\sqrt {d\,x-c}} \,d x \] Input:

int(1/((b*x)^(5/4)*(d*x - c)^(1/2)),x)
 

Output:

int(1/((b*x)^(5/4)*(d*x - c)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(b x)^{5/4} \sqrt {-c+d x}} \, dx=-\frac {\int \frac {x^{\frac {3}{4}} \sqrt {d x -c}}{-d \,x^{3}+c \,x^{2}}d x}{b^{\frac {3}{4}} \sqrt {b}} \] Input:

int(1/(b*x)^(5/4)/(d*x-c)^(1/2),x)
 

Output:

( - b**(1/4)*int((x**(3/4)*sqrt( - c + d*x))/(c*x**2 - d*x**3),x))/(sqrt(b 
)*b)