\(\int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx\) [740]

Optimal result

Integrand size = 15, antiderivative size = 149 \[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}-\frac {32 b^2}{3 a^3 \sqrt [4]{x} \sqrt [4]{a+b x}}-\frac {40 (a+b x)^{3/4}}{9 a^2 x^{9/4}}+\frac {16 b (a+b x)^{3/4}}{3 a^3 x^{5/4}}+\frac {32 b^2 \sqrt [4]{\frac {b x}{a+b x}} \sqrt [4]{a+b x} E\left (\left .\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{x}} \] Output:

4/a/x^(9/4)/(b*x+a)^(1/4)-32/3*b^2/a^3/x^(1/4)/(b*x+a)^(1/4)-40/9*(b*x+a)^ 
(3/4)/a^2/x^(9/4)+16/3*b*(b*x+a)^(3/4)/a^3/x^(5/4)+32/3*b^2*(b*x/(b*x+a))^ 
(1/4)*(b*x+a)^(1/4)*EllipticE(sin(1/2*arcsin(a^(1/2)/(b*x+a)^(1/2))),2^(1/ 
2))/a^(7/2)/x^(1/4)
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.34 \[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=-\frac {4 \sqrt [4]{1+\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {5}{4},-\frac {5}{4},-\frac {b x}{a}\right )}{9 a x^{9/4} \sqrt [4]{a+b x}} \] Input:

Integrate[1/(x^(13/4)*(a + b*x)^(5/4)),x]
 

Output:

(-4*(1 + (b*x)/a)^(1/4)*Hypergeometric2F1[-9/4, 5/4, -5/4, -((b*x)/a)])/(9 
*a*x^(9/4)*(a + b*x)^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {61, 61, 61, 61, 73, 839, 813, 858, 807, 212}

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.

\(\displaystyle \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {10 \int \frac {1}{x^{13/4} \sqrt [4]{a+b x}}dx}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {10 \left (-\frac {2 b \int \frac {1}{x^{9/4} \sqrt [4]{a+b x}}dx}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \int \frac {1}{x^{5/4} \sqrt [4]{a+b x}}dx}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {2 b \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a+b x}}dx}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \int \frac {\sqrt {x}}{\sqrt [4]{a+b x}}d\sqrt [4]{x}}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 839

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {1}{2} a \int \frac {\sqrt {x}}{(a+b x)^{5/4}}d\sqrt [4]{x}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 813

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}-\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{5/4} x^{3/4}}d\sqrt [4]{x}}{2 b \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 858

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{5/4}}d\frac {1}{\sqrt [4]{x}}}{2 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {a \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{5/4}}d\sqrt {x}}{4 b \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {10 \left (-\frac {2 b \left (-\frac {2 b \left (\frac {8 b \left (\frac {\sqrt {a} \sqrt [4]{x} \sqrt [4]{\frac {a}{b x}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |2\right )}{2 \sqrt {b} \sqrt [4]{a+b x}}+\frac {x^{3/4}}{2 \sqrt [4]{a+b x}}\right )}{a}-\frac {4 (a+b x)^{3/4}}{a \sqrt [4]{x}}\right )}{5 a}-\frac {4 (a+b x)^{3/4}}{5 a x^{5/4}}\right )}{3 a}-\frac {4 (a+b x)^{3/4}}{9 a x^{9/4}}\right )}{a}+\frac {4}{a x^{9/4} \sqrt [4]{a+b x}}\)

Input:

Int[1/(x^(13/4)*(a + b*x)^(5/4)),x]
 

Output:

4/(a*x^(9/4)*(a + b*x)^(1/4)) + (10*((-4*(a + b*x)^(3/4))/(9*a*x^(9/4)) - 
(2*b*((-4*(a + b*x)^(3/4))/(5*a*x^(5/4)) - (2*b*((-4*(a + b*x)^(3/4))/(a*x 
^(1/4)) + (8*b*(x^(3/4)/(2*(a + b*x)^(1/4)) + (Sqrt[a]*(1 + a/(b*x))^(1/4) 
*x^(1/4)*EllipticE[ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/2, 2])/(2*Sqrt[b]*(a 
+ b*x)^(1/4))))/a))/(5*a)))/(3*a)))/a
 

Defintions of rubi rules used

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 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 807

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]
 

rule 813

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Simp[x*((1 + a/(b*x^4) 
)^(1/4)/(b*(a + b*x^4)^(1/4)))   Int[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] 
/; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 839

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4 
)^(1/4)), x] - Simp[a/2   Int[x^2/(a + b*x^4)^(5/4), x], x] /; FreeQ[{a, b} 
, x] && PosQ[b/a]
 

rule 858

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]
 

Maple [F]

\[\int \frac {1}{x^{\frac {13}{4}} \left (b x +a \right )^{\frac {5}{4}}}d x\]

Input:

int(1/x^(13/4)/(b*x+a)^(5/4),x)
 

Output:

int(1/x^(13/4)/(b*x+a)^(5/4),x)
 

Fricas [F]

\[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} x^{\frac {13}{4}}} \,d x } \] Input:

integrate(1/x^(13/4)/(b*x+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(3/4)*x^(3/4)/(b^2*x^6 + 2*a*b*x^5 + a^2*x^4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 110.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.28 \[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\frac {\Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, \frac {5}{4} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {5}{4}} x^{\frac {9}{4}} \Gamma \left (- \frac {5}{4}\right )} \] Input:

integrate(1/x**(13/4)/(b*x+a)**(5/4),x)
 

Output:

gamma(-9/4)*hyper((-9/4, 5/4), (-5/4,), b*x*exp_polar(I*pi)/a)/(a**(5/4)*x 
**(9/4)*gamma(-5/4))
 

Maxima [F]

\[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} x^{\frac {13}{4}}} \,d x } \] Input:

integrate(1/x^(13/4)/(b*x+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate(1/((b*x + a)^(5/4)*x^(13/4)), x)
 

Giac [F]

\[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{4}} x^{\frac {13}{4}}} \,d x } \] Input:

integrate(1/x^(13/4)/(b*x+a)^(5/4),x, algorithm="giac")
 

Output:

integrate(1/((b*x + a)^(5/4)*x^(13/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\int \frac {1}{x^{13/4}\,{\left (a+b\,x\right )}^{5/4}} \,d x \] Input:

int(1/(x^(13/4)*(a + b*x)^(5/4)),x)
 

Output:

int(1/(x^(13/4)*(a + b*x)^(5/4)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^{13/4} (a+b x)^{5/4}} \, dx=\frac {4 \left (b x +a \right )^{\frac {1}{4}} \left (32 b^{2} x^{2}+8 a b x -3 a^{2}\right )}{21 x^{\frac {7}{4}} \sqrt {x}\, \sqrt {b x +a}\, a^{3}} \] Input:

int(1/x^(13/4)/(b*x+a)^(5/4),x)
 

Output:

(4*x**(1/4)*(a + b*x)**(1/4)*( - 3*a**2 + 8*a*b*x + 32*b**2*x**2))/(21*sqr 
t(x)*sqrt(a + b*x)*a**3*x**2)