Integrand size = 39, antiderivative size = 246 \[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=-\frac {e \arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} (a+b x)\right )}{\sqrt {c+d (a+b x)^3}}\right )}{3\ 2^{2/3} \sqrt {3} b c^{5/6} d^{2/3}}+\frac {e \arctan \left (\frac {\sqrt {c+d (a+b x)^3}}{\sqrt {3} \sqrt {c}}\right )}{3\ 2^{2/3} \sqrt {3} b c^{5/6} d^{2/3}}-\frac {e \text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} (a+b x)\right )}{\sqrt {c+d (a+b x)^3}}\right )}{3\ 2^{2/3} b c^{5/6} d^{2/3}}+\frac {e \text {arctanh}\left (\frac {\sqrt {c+d (a+b x)^3}}{\sqrt {c}}\right )}{9\ 2^{2/3} b c^{5/6} d^{2/3}} \] Output:
-1/18*e*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+2^(1/3)*d^(1/3)*(b*x+a))/(c+d*(b*x +a)^3)^(1/2))*2^(1/3)*3^(1/2)/b/c^(5/6)/d^(2/3)+1/18*e*arctan(1/3*(c+d*(b* x+a)^3)^(1/2)*3^(1/2)/c^(1/2))*2^(1/3)*3^(1/2)/b/c^(5/6)/d^(2/3)-1/6*e*arc tanh(c^(1/6)*(c^(1/3)-2^(1/3)*d^(1/3)*(b*x+a))/(c+d*(b*x+a)^3)^(1/2))*2^(1 /3)/b/c^(5/6)/d^(2/3)+1/18*e*arctanh((c+d*(b*x+a)^3)^(1/2)/c^(1/2))*2^(1/3 )/b/c^(5/6)/d^(2/3)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 15.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.36 \[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=\frac {e (a+b x)^2 \left (\frac {c+d (a+b x)^3}{c}\right )^{3/2} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d (a+b x)^3}{c},-\frac {d (a+b x)^3}{4 c}\right )}{8 b \left (c+d (a+b x)^3\right )^{3/2}} \] Input:
Integrate[(a*e + b*e*x)/(Sqrt[c + d*(a + b*x)^3]*(4*c + d*(a + b*x)^3)),x]
Output:
(e*(a + b*x)^2*((c + d*(a + b*x)^3)/c)^(3/2)*AppellF1[2/3, 1/2, 1, 5/3, -( (d*(a + b*x)^3)/c), -1/4*(d*(a + b*x)^3)/c])/(8*b*(c + d*(a + b*x)^3)^(3/2 ))
Time = 0.46 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1014, 986}
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.
| \(\displaystyle \int \frac {a e+b e x}{\sqrt {d (a+b x)^3+c} \left (d (a+b x)^3+4 c\right )} \, dx\) |
| \(\Big \downarrow \) 1014 |
| \(\displaystyle \frac {e \int \frac {a+b x}{\sqrt {d (a+b x)^3+c} \left (d (a+b x)^3+4 c\right )}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 986 |
| \(\displaystyle \frac {e \left (-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{d} (a+b x)+\sqrt [3]{c}\right )}{\sqrt {d (a+b x)^3+c}}\right )}{3\ 2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}+\frac {\arctan \left (\frac {\sqrt {d (a+b x)^3+c}}{\sqrt {3} \sqrt {c}}\right )}{3\ 2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} (a+b x)\right )}{\sqrt {d (a+b x)^3+c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt {d (a+b x)^3+c}}{\sqrt {c}}\right )}{9\ 2^{2/3} c^{5/6} d^{2/3}}\right )}{b}\) |
Input:
Int[(a*e + b*e*x)/(Sqrt[c + d*(a + b*x)^3]*(4*c + d*(a + b*x)^3)),x]
Output:
(e*(-1/3*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*(a + b*x)))/Sq rt[c + d*(a + b*x)^3]]/(2^(2/3)*Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTan[Sqrt[c + d*(a + b*x)^3]/(Sqrt[3]*Sqrt[c])]/(3*2^(2/3)*Sqrt[3]*c^(5/6)*d^(2/3)) - A rcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*d^(1/3)*(a + b*x)))/Sqrt[c + d*(a + b*x )^3]]/(3*2^(2/3)*c^(5/6)*d^(2/3)) + ArcTanh[Sqrt[c + d*(a + b*x)^3]/Sqrt[c ]]/(9*2^(2/3)*c^(5/6)*d^(2/3))))/b
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi th[{q = Rt[d/c, 3]}, Simp[q*(ArcTanh[Sqrt[c + d*x^3]/Rt[c, 2]]/(9*2^(2/3)*b *Rt[c, 2])), x] + (-Simp[q*(ArcTanh[Rt[c, 2]*((1 - 2^(1/3)*q*x)/Sqrt[c + d* x^3])]/(3*2^(2/3)*b*Rt[c, 2])), x] + Simp[q*(ArcTan[Sqrt[c + d*x^3]/(Sqrt[3 ]*Rt[c, 2])]/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2])), x] - Simp[q*(ArcTan[Sqrt[3]*R t[c, 2]*((1 + 2^(1/3)*q*x)/Sqrt[c + d*x^3])]/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2]) ), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && PosQ[c]
Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^(n_))^(p_.)*((c_.) + (d_.)*(v_)^(n_))^(q _.), x_Symbol] :> Simp[u^m/(Coefficient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x, v], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && LinearPairQ[u, v, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.65 (sec) , antiderivative size = 795, normalized size of antiderivative = 3.23
| method | result | size |
| default | \(\frac {e \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b^{3} d \,\textit {\_Z}^{3}+3 d a \,b^{2} \textit {\_Z}^{2}+3 a^{2} b d \textit {\_Z} +a^{3} d +4 c \right )}{\sum }\left (-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {6}\, \sqrt {\frac {i b \left (\frac {2 a}{b}+2 x -\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}-\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d b}\right ) d \sqrt {3}}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d b}+\frac {a}{b}\right ) d}{i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (\frac {2 a}{b}+2 x -\frac {-\left (-c \,d^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d b}\right ) d \sqrt {3}}{6 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 d^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2}\right )+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b d +i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b d -\left (-c \,d^{2}\right )^{\frac {1}{3}} a d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}}{\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}-\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}}}, \frac {2 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d +4 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a b d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b +2 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} a^{2} d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}} a +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} a -3 c d}{6 d c}, \sqrt {\frac {\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}-\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}}{\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}-\frac {\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}-a}{b}}}\right )}{3 \left (\underline {\hspace {1.25 ex}}\alpha b +a \right ) d \sqrt {b^{3} d \,x^{3}+3 a \,b^{2} d \,x^{2}+3 a^{2} b d x +a^{3} d +c}}\right )\right )}{18 c \,d^{2} b}\) | \(795\) |
| elliptic | \(\frac {e \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b^{3} d \,\textit {\_Z}^{3}+3 d a \,b^{2} \textit {\_Z}^{2}+3 a^{2} b d \textit {\_Z} +a^{3} d +4 c \right )}{\sum }\left (-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {6}\, \sqrt {\frac {i b \left (\frac {2 a}{b}+2 x -\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}-\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d b}\right ) d \sqrt {3}}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d b}+\frac {a}{b}\right ) d}{i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (\frac {2 a}{b}+2 x -\frac {-\left (-c \,d^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d b}\right ) d \sqrt {3}}{6 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 d^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}+2 \underline {\hspace {1.25 ex}}\alpha a b +a^{2}\right )+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b d +i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b d -\left (-c \,d^{2}\right )^{\frac {1}{3}} a d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}}{\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}-\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}}}, \frac {2 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d +4 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a b d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b +2 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} a^{2} d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}} a +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} a -3 c d}{6 d c}, \sqrt {\frac {\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}-\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}}{\frac {-\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-a}{b}-\frac {\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}-a}{b}}}\right )}{3 \left (\underline {\hspace {1.25 ex}}\alpha b +a \right ) d \sqrt {b^{3} d \,x^{3}+3 a \,b^{2} d \,x^{2}+3 a^{2} b d x +a^{3} d +c}}\right )\right )}{18 c \,d^{2} b}\) | \(795\) |
Input:
int((b*e*x+a*e)/(c+d*(b*x+a)^3)^(1/2)/(4*c+d*(b*x+a)^3),x,method=_RETURNVE RBOSE)
Output:
1/18*e/c/d^2/b*2^(1/2)*sum(-2*I/(_alpha*b+a)/d*3^(1/2)*(-c*d^2)^(1/3)*(1/6 *I*b*(2*a/b+2*x-1/d/b*(I*3^(1/2)*(-c*d^2)^(1/3)-(-c*d^2)^(1/3)))*d*3^(1/2) /(-c*d^2)^(1/3))^(1/2)*(b*(x-1/d/b*(-c*d^2)^(1/3)+a/b)*d/(I*3^(1/2)*(-c*d^ 2)^(1/3)-3*(-c*d^2)^(1/3)))^(1/2)*(-1/6*I*b*(2*a/b+2*x-1/d/b*(-(-c*d^2)^(1 /3)-I*3^(1/2)*(-c*d^2)^(1/3)))*d*3^(1/2)/(-c*d^2)^(1/3))^(1/2)/(b^3*d*x^3+ 3*a*b^2*d*x^2+3*a^2*b*d*x+a^3*d+c)^(1/2)*(2*d^2*(_alpha^2*b^2+2*_alpha*a*b +a^2)+I*(-c*d^2)^(1/3)*3^(1/2)*_alpha*b*d+I*(-c*d^2)^(1/3)*3^(1/2)*a*d-I*( -c*d^2)^(2/3)*3^(1/2)-(-c*d^2)^(1/3)*_alpha*b*d-(-c*d^2)^(1/3)*a*d-(-c*d^2 )^(2/3))*EllipticPi(((x-(-1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1 /3)-a)/b)/((-1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)-a)/b-(-1/ 2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)-a)/b))^(1/2),1/6/d*(2*I* 3^(1/2)*(-c*d^2)^(1/3)*_alpha^2*b^2*d+4*I*3^(1/2)*(-c*d^2)^(1/3)*_alpha*a* b*d-I*3^(1/2)*(-c*d^2)^(2/3)*_alpha*b+2*I*3^(1/2)*(-c*d^2)^(1/3)*a^2*d-I*3 ^(1/2)*(-c*d^2)^(2/3)*a+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha*b-3*(-c*d^2) ^(2/3)*a-3*c*d)/c,(((-1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)- a)/b-(-1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)-a)/b)/((-1/2/d* (-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)-a)/b-(1/d*(-c*d^2)^(1/3)-a)/ b))^(1/2)),_alpha=RootOf(_Z^3*b^3*d+3*_Z^2*a*b^2*d+3*_Z*a^2*b*d+a^3*d+4*c) )
Timed out. \[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=\text {Timed out} \] Input:
integrate((b*e*x+a*e)/(c+d*(b*x+a)^3)^(1/2)/(4*c+d*(b*x+a)^3),x, algorithm ="fricas")
Output:
Timed out
Timed out. \[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=\text {Timed out} \] Input:
integrate((b*e*x+a*e)/(c+d*(b*x+a)**3)**(1/2)/(4*c+d*(b*x+a)**3),x)
Output:
Timed out
\[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=\int { \frac {b e x + a e}{{\left ({\left (b x + a\right )}^{3} d + 4 \, c\right )} \sqrt {{\left (b x + a\right )}^{3} d + c}} \,d x } \] Input:
integrate((b*e*x+a*e)/(c+d*(b*x+a)^3)^(1/2)/(4*c+d*(b*x+a)^3),x, algorithm ="maxima")
Output:
integrate((b*e*x + a*e)/(((b*x + a)^3*d + 4*c)*sqrt((b*x + a)^3*d + c)), x )
\[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=\int { \frac {b e x + a e}{{\left ({\left (b x + a\right )}^{3} d + 4 \, c\right )} \sqrt {{\left (b x + a\right )}^{3} d + c}} \,d x } \] Input:
integrate((b*e*x+a*e)/(c+d*(b*x+a)^3)^(1/2)/(4*c+d*(b*x+a)^3),x, algorithm ="giac")
Output:
integrate((b*e*x + a*e)/(((b*x + a)^3*d + 4*c)*sqrt((b*x + a)^3*d + c)), x )
Timed out. \[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=\text {Hanged} \] Input:
int((a*e + b*e*x)/((4*c + d*(a + b*x)^3)*(c + d*(a + b*x)^3)^(1/2)),x)
Output:
\text{Hanged}
\[ \int \frac {a e+b e x}{\sqrt {c+d (a+b x)^3} \left (4 c+d (a+b x)^3\right )} \, dx=e \left (\left (\int \frac {\sqrt {b^{3} d \,x^{3}+3 a \,b^{2} d \,x^{2}+3 a^{2} b d x +a^{3} d +c}}{b^{6} d^{2} x^{6}+6 a \,b^{5} d^{2} x^{5}+15 a^{2} b^{4} d^{2} x^{4}+20 a^{3} b^{3} d^{2} x^{3}+15 a^{4} b^{2} d^{2} x^{2}+6 a^{5} b \,d^{2} x +a^{6} d^{2}+5 b^{3} c d \,x^{3}+15 a \,b^{2} c d \,x^{2}+15 a^{2} b c d x +5 a^{3} c d +4 c^{2}}d x \right ) a +\left (\int \frac {\sqrt {b^{3} d \,x^{3}+3 a \,b^{2} d \,x^{2}+3 a^{2} b d x +a^{3} d +c}\, x}{b^{6} d^{2} x^{6}+6 a \,b^{5} d^{2} x^{5}+15 a^{2} b^{4} d^{2} x^{4}+20 a^{3} b^{3} d^{2} x^{3}+15 a^{4} b^{2} d^{2} x^{2}+6 a^{5} b \,d^{2} x +a^{6} d^{2}+5 b^{3} c d \,x^{3}+15 a \,b^{2} c d \,x^{2}+15 a^{2} b c d x +5 a^{3} c d +4 c^{2}}d x \right ) b \right ) \] Input:
int((b*e*x+a*e)/(c+d*(b*x+a)^3)^(1/2)/(4*c+d*(b*x+a)^3),x)
Output:
e*(int(sqrt(a**3*d + 3*a**2*b*d*x + 3*a*b**2*d*x**2 + b**3*d*x**3 + c)/(a* *6*d**2 + 6*a**5*b*d**2*x + 15*a**4*b**2*d**2*x**2 + 20*a**3*b**3*d**2*x** 3 + 5*a**3*c*d + 15*a**2*b**4*d**2*x**4 + 15*a**2*b*c*d*x + 6*a*b**5*d**2* x**5 + 15*a*b**2*c*d*x**2 + b**6*d**2*x**6 + 5*b**3*c*d*x**3 + 4*c**2),x)* a + int((sqrt(a**3*d + 3*a**2*b*d*x + 3*a*b**2*d*x**2 + b**3*d*x**3 + c)*x )/(a**6*d**2 + 6*a**5*b*d**2*x + 15*a**4*b**2*d**2*x**2 + 20*a**3*b**3*d** 2*x**3 + 5*a**3*c*d + 15*a**2*b**4*d**2*x**4 + 15*a**2*b*c*d*x + 6*a*b**5* d**2*x**5 + 15*a*b**2*c*d*x**2 + b**6*d**2*x**6 + 5*b**3*c*d*x**3 + 4*c**2 ),x)*b)