\(\int \frac {1}{(b x)^{5/4} \sqrt {-c-d x}} \, dx\) [580]

Optimal result

Integrand size = 20, antiderivative size = 53 \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c-d x}} \, dx=\frac {4 \sqrt [4]{-\frac {d x}{c}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {-c-d x}}{\sqrt {c}}\right )\right |2\right )}{b \sqrt {c} \sqrt [4]{b x}} \] 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 = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c-d x}} \, dx=-\frac {4 x \sqrt {1+\frac {d x}{c}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {d x}{c}\right )}{(b x)^{5/4} \sqrt {-c-d x}} \] 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. \(343\) vs. \(2(53)=106\).

Time = 0.35 (sec) , antiderivative size = 343, normalized size of antiderivative = 6.47, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {61, 73, 834, 27, 761, 1510}

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.

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

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*(b*x)^(1/4)*Sqrt[-c - d 
*x])/(Sqrt[b]*Sqrt[c] + Sqrt[d]*Sqrt[b*x]) + (b^(1/4)*c^(1/4)*Sqrt[(b*c + 
b*d*x)/(Sqrt[b]*Sqrt[c] + Sqrt[d]*Sqrt[b*x])^2]*(Sqrt[b]*Sqrt[c] + Sqrt[d] 
*Sqrt[b*x])*EllipticE[2*ArcTan[(d^(1/4)*(b*x)^(1/4))/(b^(1/4)*c^(1/4))], 1 
/2])/(d^(1/4)*Sqrt[-c - d*x]))/Sqrt[d]) + (b^(1/4)*c^(1/4)*Sqrt[(b*c + b*d 
*x)/(Sqrt[b]*Sqrt[c] + Sqrt[d]*Sqrt[b*x])^2]*(Sqrt[b]*Sqrt[c] + Sqrt[d]*Sq 
rt[b*x])*EllipticF[2*ArcTan[(d^(1/4)*(b*x)^(1/4))/(b^(1/4)*c^(1/4))], 1/2] 
)/(2*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 761

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

rule 834

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

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

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 = 1.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c-d x}} \, dx=- \frac {i \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{b^{\frac {5}{4}} \sqrt {c} \sqrt [4]{x} \Gamma \left (\frac {3}{4}\right )} \] 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*exp_polar(I*pi)/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. \[ \int \frac {1}{(b x)^{5/4} \sqrt {-c-d x}} \, dx=\int \frac {1}{{\left (b\,x\right )}^{5/4}\,\sqrt {-c-d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

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