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))^...
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])
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...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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 ]
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]
\[\int \frac {\left (e x \right )^{\frac {5}{3}}}{\sqrt {b x +a}\, \left (b x +4 a \right )}d x\]
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)
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
\[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int \frac {\left (e x\right )^{\frac {5}{3}}}{\sqrt {a + b x} \left (4 a + b x\right )}\, dx \] 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)
\[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{3}}}{{\left (b x + 4 \, a\right )} \sqrt {b x + a}} \,d x } \] 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)
\[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{3}}}{{\left (b x + 4 \, a\right )} \sqrt {b x + a}} \,d x } \] 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)
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)
\[ \int \frac {(e x)^{5/3}}{\sqrt {a+b x} (4 a+b x)} \, dx=\frac {2 e^{\frac {5}{3}} \left (-96 \sqrt {b x +a}\, a +3 \sqrt {b x +a}\, b x -128 x^{\frac {1}{3}} \left (\int \frac {\sqrt {b x +a}}{4 x^{\frac {4}{3}} a^{2}+5 x^{\frac {7}{3}} a b +x^{\frac {10}{3}} b^{2}}d x \right ) a^{3}+24 x^{\frac {1}{3}} \left (\int \frac {\sqrt {b x +a}}{4 x^{\frac {1}{3}} a^{2}+5 x^{\frac {4}{3}} a b +x^{\frac {7}{3}} b^{2}}d x \right ) a^{2} b \right )}{7 x^{\frac {1}{3}} b^{3}} \] 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)