\(\int \frac {1}{\sqrt {d x} (-a+b x)^{5/4}} \, dx\) [680]

Optimal result

Integrand size = 19, antiderivative size = 240 \[ \int \frac {1}{\sqrt {d x} (-a+b x)^{5/4}} \, dx=-\frac {4 \sqrt {d x}}{a d \sqrt [4]{-a+b x}}+\frac {4 \sqrt {d x} \sqrt [4]{-a+b x}}{a d \left (\sqrt {a}+\sqrt {-a+b x}\right )}-\frac {4 \sqrt {\frac {b x}{\left (\sqrt {a}+\sqrt {-a+b x}\right )^2}} \left (\sqrt {a}+\sqrt {-a+b x}\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{-a+b x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} b \sqrt {d x}}+\frac {2 \sqrt {\frac {b x}{\left (\sqrt {a}+\sqrt {-a+b x}\right )^2}} \left (\sqrt {a}+\sqrt {-a+b x}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-a+b x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{3/4} b \sqrt {d x}} \] Output:

-4*(d*x)^(1/2)/a/d/(b*x-a)^(1/4)+4*(d*x)^(1/2)*(b*x-a)^(1/4)/a/d/(a^(1/2)+ 
(b*x-a)^(1/2))-4*(b*x/(a^(1/2)+(b*x-a)^(1/2))^2)^(1/2)*(a^(1/2)+(b*x-a)^(1 
/2))*EllipticE(sin(2*arctan((b*x-a)^(1/4)/a^(1/4))),1/2*2^(1/2))/a^(3/4)/b 
/(d*x)^(1/2)+2*(b*x/(a^(1/2)+(b*x-a)^(1/2))^2)^(1/2)*(a^(1/2)+(b*x-a)^(1/2 
))*InverseJacobiAM(2*arctan((b*x-a)^(1/4)/a^(1/4)),1/2*2^(1/2))/a^(3/4)/b/ 
(d*x)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.22 \[ \int \frac {1}{\sqrt {d x} (-a+b x)^{5/4}} \, dx=-\frac {2 x \sqrt [4]{1-\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\frac {b x}{a}\right )}{a \sqrt {d x} \sqrt [4]{-a+b x}} \] Input:

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

Output:

(-2*x*(1 - (b*x)/a)^(1/4)*Hypergeometric2F1[1/2, 5/4, 3/2, (b*x)/a])/(a*Sq 
rt[d*x]*(-a + b*x)^(1/4))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, 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.

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

(-4*Sqrt[d*x])/(a*d*(-a + b*x)^(1/4)) + (4*((b*(-a + b*x)^(1/4)*Sqrt[(a*d) 
/b + (d*(-a + b*x))/b])/(d*(Sqrt[a] + Sqrt[-a + b*x])) - (a^(1/4)*Sqrt[(b* 
x)/(Sqrt[a] + Sqrt[-a + b*x])^2]*(Sqrt[a] + Sqrt[-a + b*x])*EllipticE[2*Ar 
cTan[(-a + b*x)^(1/4)/a^(1/4)], 1/2])/Sqrt[(a*d)/b + (d*(-a + b*x))/b] + ( 
a^(1/4)*Sqrt[(b*x)/(Sqrt[a] + Sqrt[-a + b*x])^2]*(Sqrt[a] + Sqrt[-a + b*x] 
)*EllipticF[2*ArcTan[(-a + b*x)^(1/4)/a^(1/4)], 1/2])/(2*Sqrt[(a*d)/b + (d 
*(-a + b*x))/b])))/(a*b)
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\sqrt {d x} (-a+b x)^{5/4}} \, dx=\frac {2 \sqrt {x} e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x}{a}} \right )}}{a^{\frac {5}{4}} \sqrt {d}} \] Input:

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

Output:

2*sqrt(x)*exp(3*I*pi/4)*hyper((1/2, 5/4), (3/2,), b*x/a)/(a**(5/4)*sqrt(d) 
)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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