\(\int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx\) [149]

Optimal result

Integrand size = 26, antiderivative size = 944 \[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx =\text {Too large to display} \] Output:

6/7*e*(e*x)^(2/3)*(b*x+a)^(1/2)/b^2-192/7*a*e^2*(b*x+a)^(1/2)/b^(8/3)/((1+ 
3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))+8/3*2^(1/3)*a^(7/6)*e^(5/3)* 
arctan(1/3*(b*x+a)^(1/2)*3^(1/2)/a^(1/2))*3^(1/2)/b^(8/3)-8/3*2^(1/3)*a^(7 
/6)*e^(5/3)*arctan(3^(1/2)*a^(1/6)*(a^(1/3)*e^(1/3)+2^(1/3)*b^(1/3)*(e*x)^ 
(1/3))/e^(1/3)/(b*x+a)^(1/2))*3^(1/2)/b^(8/3)+8/3*2^(1/3)*a^(7/6)*e^(5/3)* 
arctanh((b*x+a)^(1/2)/a^(1/2))/b^(8/3)-8*2^(1/3)*a^(7/6)*e^(5/3)*arctanh(a 
^(1/6)*(a^(1/3)*e^(1/3)-2^(1/3)*b^(1/3)*(e*x)^(1/3))/e^(1/3)/(b*x+a)^(1/2) 
)/b^(8/3)+96/7*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(4/3)*e^(4/3)*(a^(1/3)* 
e^(1/3)+b^(1/3)*(e*x)^(1/3))*((a^(2/3)*e^(2/3)-a^(1/3)*b^(1/3)*e^(1/3)*(e* 
x)^(1/3)+b^(2/3)*(e*x)^(2/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^( 
1/3))^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3)) 
/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3)),I*3^(1/2)+2*I)/b^(8/3)/ 
(b*x+a)^(1/2)/(a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))/((1+3 
^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))^2)^(1/2)-64/7*2^(1/2)*3^(3/4) 
*a^(4/3)*e^(4/3)*(a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))*((a^(2/3)*e^(2/3)-a 
^(1/3)*b^(1/3)*e^(1/3)*(e*x)^(1/3)+b^(2/3)*(e*x)^(2/3))/((1+3^(1/2))*a^(1/ 
3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)*e^ 
(1/3)+b^(1/3)*(e*x)^(1/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3 
)),I*3^(1/2)+2*I)/b^(8/3)/(b*x+a)^(1/2)/(a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3)+ 
b^(1/3)*(e*x)^(1/3))/((1+3^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))^...
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.12 \[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=-\frac {6 e (e x)^{2/3} \left (-5 (a+b x)+5 a \sqrt {1+\frac {b x}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x}{a},-\frac {b x}{4 a}\right )+4 b x \sqrt {1+\frac {b x}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b x}{a},-\frac {b x}{4 a}\right )\right )}{35 b^2 \sqrt {a+b x}} \] Input:

Integrate[(e*x)^(5/3)/(Sqrt[a + b*x]*(4*a + b*x)),x]
 

Output:

(-6*e*(e*x)^(2/3)*(-5*(a + b*x) + 5*a*Sqrt[1 + (b*x)/a]*AppellF1[2/3, 1/2, 
 1, 5/3, -((b*x)/a), -1/4*(b*x)/a] + 4*b*x*Sqrt[1 + (b*x)/a]*AppellF1[5/3, 
 1/2, 1, 8/3, -((b*x)/a), -1/4*(b*x)/a]))/(35*b^2*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 953, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {148, 27, 979, 27, 1054, 2009}

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.

\(\displaystyle \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx\)

\(\Big \downarrow \) 148

\(\displaystyle \frac {3 \int \frac {e (e x)^{7/3}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \int \frac {(e x)^{7/3}}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}\)

\(\Big \downarrow \) 979

\(\displaystyle 3 \left (\frac {2 e (e x)^{2/3} \sqrt {a+b x}}{7 b^2}-\frac {2 e \int \frac {8 a \sqrt [3]{e x} (a e+2 b x e)}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{7 b^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \left (\frac {2 e (e x)^{2/3} \sqrt {a+b x}}{7 b^2}-\frac {16 a e \int \frac {\sqrt [3]{e x} (a e+2 b x e)}{\sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}}{7 b^2}\right )\)

\(\Big \downarrow \) 1054

\(\displaystyle 3 \left (\frac {2 e (e x)^{2/3} \sqrt {a+b x}}{7 b^2}-\frac {16 a e \int \left (\frac {2 \sqrt [3]{e x}}{\sqrt {a+b x}}-\frac {7 a e \sqrt [3]{e x}}{\sqrt {a+b x} (4 a e+b x e)}\right )d\sqrt [3]{e x}}{7 b^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {2 e (e x)^{2/3} \sqrt {a+b x}}{7 b^2}-\frac {16 a e \left (\frac {4 \sqrt {a+b x} e}{b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}-\frac {7 \sqrt [6]{a} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {3} \sqrt {a}}\right ) e^{2/3}}{3\ 2^{2/3} \sqrt {3} b^{2/3}}+\frac {7 \sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{e} \sqrt {a+b x}}\right ) e^{2/3}}{3\ 2^{2/3} \sqrt {3} b^{2/3}}-\frac {7 \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) e^{2/3}}{9\ 2^{2/3} b^{2/3}}+\frac {7 \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt [6]{a} \left (\sqrt [3]{a} \sqrt [3]{e}-\sqrt [3]{2} \sqrt [3]{b} \sqrt [3]{e x}\right )}{\sqrt [3]{e} \sqrt {a+b x}}\right ) e^{2/3}}{3\ 2^{2/3} b^{2/3}}-\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right ) \sqrt {\frac {a^{2/3} e^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{e x} \sqrt [3]{e}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}\right )|-7-4 \sqrt {3}\right ) \sqrt [3]{e}}{b^{2/3} \sqrt {a+b x} \sqrt {\frac {\sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}}}+\frac {4 \sqrt {2} \sqrt [3]{a} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right ) \sqrt {\frac {a^{2/3} e^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{e x} \sqrt [3]{e}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}}\right ),-7-4 \sqrt {3}\right ) \sqrt [3]{e}}{\sqrt [4]{3} b^{2/3} \sqrt {a+b x} \sqrt {\frac {\sqrt [3]{a} \sqrt [3]{e} \left (\sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{e}+\sqrt [3]{b} \sqrt [3]{e x}\right )^2}}}\right )}{7 b^2}\right )\)

Input:

Int[(e*x)^(5/3)/(Sqrt[a + b*x]*(4*a + b*x)),x]
 

Output:

3*((2*e*(e*x)^(2/3)*Sqrt[a + b*x])/(7*b^2) - (16*a*e*((4*e*Sqrt[a + b*x])/ 
(b^(2/3)*((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))) - (7*a^(1/ 
6)*e^(2/3)*ArcTan[Sqrt[a + b*x]/(Sqrt[3]*Sqrt[a])])/(3*2^(2/3)*Sqrt[3]*b^( 
2/3)) + (7*a^(1/6)*e^(2/3)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3)*e^(1/3) + 2^(1 
/3)*b^(1/3)*(e*x)^(1/3)))/(e^(1/3)*Sqrt[a + b*x])])/(3*2^(2/3)*Sqrt[3]*b^( 
2/3)) - (7*a^(1/6)*e^(2/3)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(9*2^(2/3)*b^(2 
/3)) + (7*a^(1/6)*e^(2/3)*ArcTanh[(a^(1/6)*(a^(1/3)*e^(1/3) - 2^(1/3)*b^(1 
/3)*(e*x)^(1/3)))/(e^(1/3)*Sqrt[a + b*x])])/(3*2^(2/3)*b^(2/3)) - (2*3^(1/ 
4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3 
))*Sqrt[(a^(2/3)*e^(2/3) - a^(1/3)*b^(1/3)*e^(1/3)*(e*x)^(1/3) + b^(2/3)*( 
e*x)^(2/3))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))^2]*Ellip 
ticE[ArcSin[((1 - Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))/((1 + Sq 
rt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))], -7 - 4*Sqrt[3]])/(b^(2/3)* 
Sqrt[a + b*x]*Sqrt[(a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3) 
))/((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))^2]) + (4*Sqrt[2]* 
a^(1/3)*e^(1/3)*(a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))*Sqrt[(a^(2/3)*e^(2 
/3) - a^(1/3)*b^(1/3)*e^(1/3)*(e*x)^(1/3) + b^(2/3)*(e*x)^(2/3))/((1 + Sqr 
t[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqr 
t[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))/((1 + Sqrt[3])*a^(1/3)*e^(1/3 
) + b^(1/3)*(e*x)^(1/3))], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*Sqrt[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 148

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), 
x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1) - 1)*(c 
 + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, 
 d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
 

rule 979

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 
 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d 
*(m + n*(p + q) + 1))   Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim 
p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x 
^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I 
GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x 
]
 

rule 1054

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, p}, x] && IGtQ[n, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (e x \right )^{\frac {5}{3}}}{\sqrt {b x +a}\, \left (b x +4 a \right )}d x\]

Input:

int((e*x)^(5/3)/(b*x+a)^(1/2)/(b*x+4*a),x)
 

Output:

int((e*x)^(5/3)/(b*x+a)^(1/2)/(b*x+4*a),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(5/3)/(b*x+a)^(1/2)/(b*x+4*a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(5/3)/(b*x+a)**(1/2)/(b*x+4*a),x)
 

Output:

Integral((e*x)**(5/3)/(sqrt(a + b*x)*(4*a + b*x)), x)
 

Maxima [F]

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

integrate((e*x)^(5/3)/(b*x+a)^(1/2)/(b*x+4*a),x, algorithm="maxima")
 

Output:

integrate((e*x)^(5/3)/((b*x + 4*a)*sqrt(b*x + a)), x)
 

Giac [F]

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

integrate((e*x)^(5/3)/(b*x+a)^(1/2)/(b*x+4*a),x, algorithm="giac")
 

Output:

integrate((e*x)^(5/3)/((b*x + 4*a)*sqrt(b*x + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int \frac {{\left (e\,x\right )}^{5/3}}{\left (4\,a+b\,x\right )\,\sqrt {a+b\,x}} \,d x \] Input:

int((e*x)^(5/3)/((4*a + b*x)*(a + b*x)^(1/2)),x)
 

Output:

int((e*x)^(5/3)/((4*a + b*x)*(a + b*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\frac {2 e^{\frac {5}{3}} \left (-96 \sqrt {b x +a}\, a +3 \sqrt {b x +a}\, b x -128 x^{\frac {1}{3}} \left (\int \frac {\sqrt {b x +a}}{4 x^{\frac {4}{3}} a^{2}+5 x^{\frac {7}{3}} a b +x^{\frac {10}{3}} b^{2}}d x \right ) a^{3}+24 x^{\frac {1}{3}} \left (\int \frac {\sqrt {b x +a}}{4 x^{\frac {1}{3}} a^{2}+5 x^{\frac {4}{3}} a b +x^{\frac {7}{3}} b^{2}}d x \right ) a^{2} b \right )}{7 x^{\frac {1}{3}} b^{3}} \] Input:

int((e*x)^(5/3)/(b*x+a)^(1/2)/(b*x+4*a),x)
 

Output:

(2*e**(2/3)*e*( - 96*sqrt(a + b*x)*a + 3*sqrt(a + b*x)*b*x - 128*x**(1/3)* 
int(sqrt(a + b*x)/(4*x**(1/3)*a**2*x + 5*x**(1/3)*a*b*x**2 + x**(1/3)*b**2 
*x**3),x)*a**3 + 24*x**(1/3)*int(sqrt(a + b*x)/(4*x**(1/3)*a**2 + 5*x**(1/ 
3)*a*b*x + x**(1/3)*b**2*x**2),x)*a**2*b))/(7*x**(1/3)*b**3)