\(\int x^{9/4} \sqrt [4]{a+b x} \, dx\) [691]

Optimal result

Integrand size = 15, antiderivative size = 151 \[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\frac {3 a^3 \sqrt [4]{x} \sqrt [4]{a+b x}}{28 b^3}-\frac {3 a^2 x^{5/4} \sqrt [4]{a+b x}}{70 b^2}+\frac {a x^{9/4} \sqrt [4]{a+b x}}{35 b}+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}+\frac {3 a^{7/2} \left (\frac {b x}{a+b x}\right )^{3/4} (a+b x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right ),2\right )}{28 b^4 x^{3/4}} \] Output:

3/28*a^3*x^(1/4)*(b*x+a)^(1/4)/b^3-3/70*a^2*x^(5/4)*(b*x+a)^(1/4)/b^2+1/35 
*a*x^(9/4)*(b*x+a)^(1/4)/b+2/7*x^(13/4)*(b*x+a)^(1/4)+3/28*a^(7/2)*(b*x/(b 
*x+a))^(3/4)*(b*x+a)^(3/4)*InverseJacobiAM(1/2*arcsin(a^(1/2)/(b*x+a)^(1/2 
)),2^(1/2))/b^4/x^(3/4)
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.31 \[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\frac {4 x^{13/4} \sqrt [4]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {13}{4},\frac {17}{4},-\frac {b x}{a}\right )}{13 \sqrt [4]{1+\frac {b x}{a}}} \] Input:

Integrate[x^(9/4)*(a + b*x)^(1/4),x]
 

Output:

(4*x^(13/4)*(a + b*x)^(1/4)*Hypergeometric2F1[-1/4, 13/4, 17/4, -((b*x)/a) 
])/(13*(1 + (b*x)/a)^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {60, 60, 60, 60, 73, 768, 858, 807, 229}

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.

\(\displaystyle \int x^{9/4} \sqrt [4]{a+b x} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{14} a \int \frac {x^{9/4}}{(a+b x)^{3/4}}dx+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \int \frac {x^{5/4}}{(a+b x)^{3/4}}dx}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \int \frac {\sqrt [4]{x}}{(a+b x)^{3/4}}dx}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {a \int \frac {1}{x^{3/4} (a+b x)^{3/4}}dx}{2 b}\right )}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {2 a \int \frac {1}{(a+b x)^{3/4}}d\sqrt [4]{x}}{b}\right )}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 768

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}-\frac {2 a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{3/4} x^{3/4}}d\sqrt [4]{x}}{b (a+b x)^{3/4}}\right )}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 858

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{3/4}}d\frac {1}{\sqrt [4]{x}}}{b (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {a x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{3/4}}d\sqrt {x}}{b (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

\(\Big \downarrow \) 229

\(\displaystyle \frac {1}{14} a \left (\frac {2 x^{9/4} \sqrt [4]{a+b x}}{5 b}-\frac {9 a \left (\frac {2 x^{5/4} \sqrt [4]{a+b x}}{3 b}-\frac {5 a \left (\frac {2 \sqrt {a} x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),2\right )}{\sqrt {b} (a+b x)^{3/4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a+b x}}{b}\right )}{6 b}\right )}{10 b}\right )+\frac {2}{7} x^{13/4} \sqrt [4]{a+b x}\)

Input:

Int[x^(9/4)*(a + b*x)^(1/4),x]
 

Output:

(2*x^(13/4)*(a + b*x)^(1/4))/7 + (a*((2*x^(9/4)*(a + b*x)^(1/4))/(5*b) - ( 
9*a*((2*x^(5/4)*(a + b*x)^(1/4))/(3*b) - (5*a*((2*x^(1/4)*(a + b*x)^(1/4)) 
/b + (2*Sqrt[a]*(1 + a/(b*x))^(3/4)*x^(3/4)*EllipticF[ArcTan[(Sqrt[a]*Sqrt 
[x])/Sqrt[b]]/2, 2])/(Sqrt[b]*(a + b*x)^(3/4))))/(6*b)))/(10*b)))/14
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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 229

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

rule 768

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

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 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]

\[\int x^{\frac {9}{4}} \left (b x +a \right )^{\frac {1}{4}}d x\]

Input:

int(x^(9/4)*(b*x+a)^(1/4),x)
 

Output:

int(x^(9/4)*(b*x+a)^(1/4),x)
 

Fricas [F]

\[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {9}{4}} \,d x } \] Input:

integrate(x^(9/4)*(b*x+a)^(1/4),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(1/4)*x^(9/4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.92 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.25 \[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\frac {\sqrt [4]{a} x^{\frac {13}{4}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {17}{4}\right )} \] Input:

integrate(x**(9/4)*(b*x+a)**(1/4),x)
 

Output:

a**(1/4)*x**(13/4)*gamma(13/4)*hyper((-1/4, 13/4), (17/4,), b*x*exp_polar( 
I*pi)/a)/gamma(17/4)
 

Maxima [F]

\[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {9}{4}} \,d x } \] Input:

integrate(x^(9/4)*(b*x+a)^(1/4),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/4)*x^(9/4), x)
 

Giac [F]

\[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {9}{4}} \,d x } \] Input:

integrate(x^(9/4)*(b*x+a)^(1/4),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/4)*x^(9/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\int x^{9/4}\,{\left (a+b\,x\right )}^{1/4} \,d x \] Input:

int(x^(9/4)*(a + b*x)^(1/4),x)
 

Output:

int(x^(9/4)*(a + b*x)^(1/4), x)
 

Reduce [F]

\[ \int x^{9/4} \sqrt [4]{a+b x} \, dx=\frac {60 x^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{3}-24 x^{\frac {5}{4}} \left (b x +a \right )^{\frac {1}{4}} a^{2} b +16 x^{\frac {9}{4}} \left (b x +a \right )^{\frac {1}{4}} a \,b^{2}+160 x^{\frac {13}{4}} \left (b x +a \right )^{\frac {1}{4}} b^{3}-15 \left (\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{\frac {3}{4}} a +x^{\frac {7}{4}} b}d x \right ) a^{4}}{560 b^{3}} \] Input:

int(x^(9/4)*(b*x+a)^(1/4),x)
 

Output:

(60*x**(1/4)*(a + b*x)**(1/4)*a**3 - 24*x**(1/4)*(a + b*x)**(1/4)*a**2*b*x 
 + 16*x**(1/4)*(a + b*x)**(1/4)*a*b**2*x**2 + 160*x**(1/4)*(a + b*x)**(1/4 
)*b**3*x**3 - 15*int((a + b*x)**(1/4)/(x**(3/4)*a + x**(3/4)*b*x),x)*a**4) 
/(560*b**3)