\(\int \frac {1}{(e x)^{4/3} \sqrt {a+b x} (4 a+b x)} \, dx\) [152]

Optimal result

Integrand size = 26, antiderivative size = 954 \[ \int \frac {1}{(e x)^{4/3} \sqrt {a+b x} (4 a+b x)} \, dx =\text {Too large to display} \] Output:

-3/4*(b*x+a)^(1/2)/a^2/e/(e*x)^(1/3)+3/4*b^(1/3)*(b*x+a)^(1/2)/a^2/e/((1+3 
^(1/2))*a^(1/3)*e^(1/3)+b^(1/3)*(e*x)^(1/3))-1/24*b^(1/3)*arctan(1/3*(b*x+ 
a)^(1/2)*3^(1/2)/a^(1/2))*2^(1/3)*3^(1/2)/a^(11/6)/e^(4/3)+1/24*b^(1/3)*ar 
ctan(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))*2^(1/3)*3^(1/2)/a^(11/6)/e^(4/3)-1/24*b^(1/3)*arctanh((b*x 
+a)^(1/2)/a^(1/2))*2^(1/3)/a^(11/6)/e^(4/3)+1/8*b^(1/3)*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))*2^(1/3) 
/a^(11/6)/e^(4/3)-3/8*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*b^(1/3)*(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)/a^(5/3)/e 
^(5/3)/(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)+1/4*3^(3/4)*b 
^(1/3)*(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)*2^(1/2)/a^(5/3)/e^(5/3)/(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)^(...
 

Mathematica [C] (verified)

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

Time = 11.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.13 \[ \int \frac {1}{(e x)^{4/3} \sqrt {a+b x} (4 a+b x)} \, dx=\frac {3 x \left (-40 a (a+b x)+5 a b x \sqrt {1+\frac {b x}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x}{a},-\frac {b x}{4 a}\right )+b^2 x^2 \sqrt {1+\frac {b x}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b x}{a},-\frac {b x}{4 a}\right )\right )}{160 a^3 (e x)^{4/3} \sqrt {a+b x}} \] Input:

Integrate[1/((e*x)^(4/3)*Sqrt[a + b*x]*(4*a + b*x)),x]
 

Output:

(3*x*(-40*a*(a + b*x) + 5*a*b*x*Sqrt[1 + (b*x)/a]*AppellF1[2/3, 1/2, 1, 5/ 
3, -((b*x)/a), -1/4*(b*x)/a] + b^2*x^2*Sqrt[1 + (b*x)/a]*AppellF1[5/3, 1/2 
, 1, 8/3, -((b*x)/a), -1/4*(b*x)/a]))/(160*a^3*(e*x)^(4/3)*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 957, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {148, 27, 980, 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.

\(\displaystyle \int \frac {1}{(e x)^{4/3} \sqrt {a+b x} (4 a+b x)} \, dx\)

\(\Big \downarrow \) 148

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

\(\Big \downarrow \) 27

\(\displaystyle 3 \int \frac {1}{(e x)^{2/3} \sqrt {a+b x} (4 a e+b x e)}d\sqrt [3]{e x}\)

\(\Big \downarrow \) 980

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (\frac {b \left (\frac {2 \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 {\sqrt [3]{2} \sqrt [6]{a} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {3} \sqrt {a}}\right ) e^{2/3}}{3 \sqrt {3} b^{2/3}}+\frac {\sqrt [3]{2} \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 \sqrt {3} b^{2/3}}-\frac {\sqrt [3]{2} \sqrt [6]{a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) e^{2/3}}{9 b^{2/3}}+\frac {\sqrt [3]{2} \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 b^{2/3}}-\frac {\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 {2 \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 )}{8 a^2 e^2}-\frac {\sqrt {a+b x}}{4 a^2 e \sqrt [3]{e x}}\right )\)

Input:

Int[1/((e*x)^(4/3)*Sqrt[a + b*x]*(4*a + b*x)),x]
 

Output:

3*(-1/4*Sqrt[a + b*x]/(a^2*e*(e*x)^(1/3)) + (b*((2*e*Sqrt[a + b*x])/(b^(2/ 
3)*((1 + Sqrt[3])*a^(1/3)*e^(1/3) + b^(1/3)*(e*x)^(1/3))) - (2^(1/3)*a^(1/ 
6)*e^(2/3)*ArcTan[Sqrt[a + b*x]/(Sqrt[3]*Sqrt[a])])/(3*Sqrt[3]*b^(2/3)) + 
(2^(1/3)*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*Sqrt[3]*b^(2/3)) - (2 
^(1/3)*a^(1/6)*e^(2/3)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(9*b^(2/3)) + (2^(1 
/3)*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*b^(2/3)) - (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]*EllipticE[ArcSin[((1 
- Sqrt[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]])/(b^(2/3)*Sqrt[a + b*x]*Sq 
rt[(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]) + (2*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 + Sqrt[3])*a^(1/3)*e^ 
(1/3) + b^(1/3)*(e*x)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[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*x]*Sqrt[(a^(1/3...
 

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 980

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

\[\int \frac {1}{\left (e x \right )^{\frac {4}{3}} \sqrt {b x +a}\, \left (b x +4 a \right )}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{4/3} \sqrt {a+b x} (4 a+b x)} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(1/((e*x)**(4/3)*sqrt(a + b*x)*(4*a + b*x)), x)
 

Maxima [F]

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

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

Output:

integrate(1/((b*x + 4*a)*sqrt(b*x + a)*(e*x)^(4/3)), x)
 

Giac [F]

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

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

Output:

integrate(1/((b*x + 4*a)*sqrt(b*x + a)*(e*x)^(4/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{4/3} \sqrt {a+b x} (4 a+b x)} \, dx=\int \frac {1}{{\left (e\,x\right )}^{4/3}\,\left (4\,a+b\,x\right )\,\sqrt {a+b\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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)/(e**(1/3)*e)