\(\int \frac {\sqrt {c+d x}}{(e x)^{7/2} (a+b x^2)} \, dx\) [859]

Optimal result

Integrand size = 26, antiderivative size = 274 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=-\frac {2 \sqrt {c+d x}}{5 a e (e x)^{5/2}}-\frac {2 d \sqrt {c+d x}}{15 a c e^2 (e x)^{3/2}}+\frac {2 \left (15 b c^2+2 a d^2\right ) \sqrt {c+d x}}{15 a^2 c^2 e^3 \sqrt {e x}}+\frac {b \sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {\sqrt {b} c-\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} e^{7/2}}-\frac {b \sqrt {\sqrt {b} c+\sqrt {-a} d} \text {arctanh}\left (\frac {\sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {e x}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} e^{7/2}} \] Output:

-2/5*(d*x+c)^(1/2)/a/e/(e*x)^(5/2)-2/15*d*(d*x+c)^(1/2)/a/c/e^2/(e*x)^(3/2 
)+2/15*(2*a*d^2+15*b*c^2)*(d*x+c)^(1/2)/a^2/c^2/e^3/(e*x)^(1/2)+b*(b^(1/2) 
*c-(-a)^(1/2)*d)^(1/2)*arctan((b^(1/2)*c-(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/( 
-a)^(1/4)/e^(1/2)/(d*x+c)^(1/2))/(-a)^(9/4)/e^(7/2)-b*(b^(1/2)*c+(-a)^(1/2 
)*d)^(1/2)*arctanh((b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(e*x)^(1/2)/(-a)^(1/4)/e 
^(1/2)/(d*x+c)^(1/2))/(-a)^(9/4)/e^(7/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.33 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\frac {x \left (2 \sqrt {c+d x} \left (15 b c^2 x^2-a \left (3 c^2+c d x-2 d^2 x^2\right )\right )+15 b c^2 \sqrt {d} x^{5/2} \text {RootSum}\left [b c^4-4 b c^3 \text {$\#$1}+6 b c^2 \text {$\#$1}^2+16 a d^2 \text {$\#$1}^2-4 b c \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {b c^3 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right )-2 b c^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}-4 a d^2 \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}+b c \log \left (c+2 d x-2 \sqrt {d} \sqrt {x} \sqrt {c+d x}-\text {$\#$1}\right ) \text {$\#$1}^2}{b c^3-3 b c^2 \text {$\#$1}-8 a d^2 \text {$\#$1}+3 b c \text {$\#$1}^2-b \text {$\#$1}^3}\&\right ]\right )}{15 a^2 c^2 (e x)^{7/2}} \] Input:

Integrate[Sqrt[c + d*x]/((e*x)^(7/2)*(a + b*x^2)),x]
 

Output:

(x*(2*Sqrt[c + d*x]*(15*b*c^2*x^2 - a*(3*c^2 + c*d*x - 2*d^2*x^2)) + 15*b* 
c^2*Sqrt[d]*x^(5/2)*RootSum[b*c^4 - 4*b*c^3*#1 + 6*b*c^2*#1^2 + 16*a*d^2*# 
1^2 - 4*b*c*#1^3 + b*#1^4 & , (b*c^3*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqr 
t[c + d*x] - #1] - 2*b*c^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] 
 - #1]*#1 - 4*a*d^2*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]* 
#1 + b*c*Log[c + 2*d*x - 2*Sqrt[d]*Sqrt[x]*Sqrt[c + d*x] - #1]*#1^2)/(b*c^ 
3 - 3*b*c^2*#1 - 8*a*d^2*#1 + 3*b*c*#1^2 - b*#1^3) & ]))/(15*a^2*c^2*(e*x) 
^(7/2))
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {610, 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.

\(\displaystyle \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 610

\(\displaystyle \frac {\int \left (\frac {c e^3}{a (e x)^{7/2} \sqrt {c+d x}}+\frac {d e^2}{a (e x)^{5/2} \sqrt {c+d x}}-\frac {b c e}{a^2 (e x)^{3/2} \sqrt {c+d x}}-\frac {b (a d-b c x)}{a^2 \sqrt {e x} \sqrt {c+d x} \left (b x^2+a\right )}\right )dx}{e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {2 b \sqrt {c+d x}}{a^2 \sqrt {e x}}+\frac {b \sqrt {\sqrt {b} c-\sqrt {-a} d} \arctan \left (\frac {\sqrt {e x} \sqrt {\sqrt {b} c-\sqrt {-a} d}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} \sqrt {e}}-\frac {b \sqrt {\sqrt {-a} d+\sqrt {b} c} \text {arctanh}\left (\frac {\sqrt {e x} \sqrt {\sqrt {-a} d+\sqrt {b} c}}{\sqrt [4]{-a} \sqrt {e} \sqrt {c+d x}}\right )}{(-a)^{9/4} \sqrt {e}}+\frac {4 d^2 \sqrt {c+d x}}{15 a c^2 \sqrt {e x}}-\frac {2 e^2 \sqrt {c+d x}}{5 a (e x)^{5/2}}-\frac {2 d e \sqrt {c+d x}}{15 a c (e x)^{3/2}}}{e^3}\)

Input:

Int[Sqrt[c + d*x]/((e*x)^(7/2)*(a + b*x^2)),x]
 

Output:

((-2*e^2*Sqrt[c + d*x])/(5*a*(e*x)^(5/2)) - (2*d*e*Sqrt[c + d*x])/(15*a*c* 
(e*x)^(3/2)) + (2*b*Sqrt[c + d*x])/(a^2*Sqrt[e*x]) + (4*d^2*Sqrt[c + d*x]) 
/(15*a*c^2*Sqrt[e*x]) + (b*Sqrt[Sqrt[b]*c - Sqrt[-a]*d]*ArcTan[(Sqrt[Sqrt[ 
b]*c - Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^( 
9/4)*Sqrt[e]) - (b*Sqrt[Sqrt[b]*c + Sqrt[-a]*d]*ArcTanh[(Sqrt[Sqrt[b]*c + 
Sqrt[-a]*d]*Sqrt[e*x])/((-a)^(1/4)*Sqrt[e]*Sqrt[c + d*x])])/((-a)^(9/4)*Sq 
rt[e]))/e^3
 

Defintions of rubi rules used

rule 610

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S 
ymbol] :> Simp[e^(m + 1/2)   Int[ExpandIntegrand[1/(Sqrt[e*x]*Sqrt[c + d*x] 
), x^(m + 1/2)*((c + d*x)^(n + 1/2)/(a + b*x^2)), x], x], x] /; FreeQ[{a, b 
, c, d, e}, x] && IGtQ[n + 1/2, 0] && ILtQ[m - 1/2, 0]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(208)=416\).

Time = 0.36 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.78

method	result	size
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-2 a \,d^{2} x^{2}-15 b \,c^{2} x^{2}+a d x c +3 a \,c^{2}\right )}{15 a^{2} c^{2} x^{2} e^{3} \sqrt {e x}}+\frac {b \left (-\frac {\left (-a d +c \sqrt {-a b}\right ) \ln \left (\frac {-\frac {2 e \left (a d -c \sqrt {-a b}\right )}{b}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (2 \sqrt {-a b}\, d +b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {e \left (a d -c \sqrt {-a b}\right )}{b}}}-\frac {\left (a d +c \sqrt {-a b}\right ) \ln \left (\frac {-\frac {2 e \left (a d +c \sqrt {-a b}\right )}{b}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {d e \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {e \left (-2 \sqrt {-a b}\, d +b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\right ) \sqrt {\left (d x +c \right ) e x}}{a^{2} e^{3} \sqrt {e x}\, \sqrt {d x +c}}\)	\(488\)
default	\(-\frac {\sqrt {d x +c}\, \left (15 \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) a b \,c^{2} d e \,x^{3}+15 \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \ln \left (\frac {-2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b -c e \sqrt {-a b}}{b x +\sqrt {-a b}}\right ) b \,c^{3} e \,x^{3}-15 \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, a b \,c^{2} d e \,x^{3}+15 \sqrt {-a b}\, \ln \left (\frac {2 \sqrt {-a b}\, d e x +b c e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, b +c e \sqrt {-a b}}{b x -\sqrt {-a b}}\right ) \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\, b \,c^{3} e \,x^{3}-8 a \,d^{2} x^{2} \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}-60 b \,c^{2} x^{2} \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}+4 a c d x \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}+12 a \,c^{2} \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {\left (d x +c \right ) e x}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}\right )}{30 e^{3} x^{2} \sqrt {e x}\, \sqrt {\left (d x +c \right ) e x}\, a^{2} c^{2} \sqrt {-a b}\, \sqrt {\frac {e \left (-a d +c \sqrt {-a b}\right )}{b}}\, \sqrt {-\frac {e \left (a d +c \sqrt {-a b}\right )}{b}}}\)	\(772\)

Input:

int((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2/15*(d*x+c)^(1/2)*(-2*a*d^2*x^2-15*b*c^2*x^2+a*c*d*x+3*a*c^2)/a^2/c^2/x^ 
2/e^3/(e*x)^(1/2)+1/a^2*b*(-1/2*(-a*d+c*(-a*b)^(1/2))/(-a*b)^(1/2)/(-e*(a* 
d-c*(-a*b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d-c*(-a*b)^(1/2))/b+e*(2*(-a*b)^(1/ 
2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)+2*(-e*(a*d-c*(-a*b)^(1/2))/b)^(1/2)*(d*e*(x 
-(-a*b)^(1/2)/b)^2+e*(2*(-a*b)^(1/2)*d+b*c)/b*(x-(-a*b)^(1/2)/b)-e*(a*d-c* 
(-a*b)^(1/2))/b)^(1/2))/(x-(-a*b)^(1/2)/b))-1/2*(a*d+c*(-a*b)^(1/2))/(-a*b 
)^(1/2)/(-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2)*ln((-2*e*(a*d+c*(-a*b)^(1/2))/b+ 
e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^(1/2)/b)+2*(-e*(a*d+c*(-a*b)^(1/2))/ 
b)^(1/2)*(d*e*(x+(-a*b)^(1/2)/b)^2+e*(-2*(-a*b)^(1/2)*d+b*c)/b*(x+(-a*b)^( 
1/2)/b)-e*(a*d+c*(-a*b)^(1/2))/b)^(1/2))/(x+(-a*b)^(1/2)/b)))/e^3*((d*x+c) 
*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (208) = 416\).

Time = 0.11 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=-\frac {15 \, a^{2} c^{2} e^{4} x^{3} \sqrt {\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} + b^{2} d}{a^{4} e^{7}}} \log \left (\frac {a^{2} e^{4} x \sqrt {\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} + b^{2} d}{a^{4} e^{7}}} + \sqrt {d x + c} \sqrt {e x} b}{x}\right ) - 15 \, a^{2} c^{2} e^{4} x^{3} \sqrt {\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} + b^{2} d}{a^{4} e^{7}}} \log \left (-\frac {a^{2} e^{4} x \sqrt {\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} + b^{2} d}{a^{4} e^{7}}} - \sqrt {d x + c} \sqrt {e x} b}{x}\right ) + 15 \, a^{2} c^{2} e^{4} x^{3} \sqrt {-\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} - b^{2} d}{a^{4} e^{7}}} \log \left (\frac {a^{2} e^{4} x \sqrt {-\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} - b^{2} d}{a^{4} e^{7}}} + \sqrt {d x + c} \sqrt {e x} b}{x}\right ) - 15 \, a^{2} c^{2} e^{4} x^{3} \sqrt {-\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} - b^{2} d}{a^{4} e^{7}}} \log \left (-\frac {a^{2} e^{4} x \sqrt {-\frac {a^{4} e^{7} \sqrt {-\frac {b^{5} c^{2}}{a^{9} e^{14}}} - b^{2} d}{a^{4} e^{7}}} - \sqrt {d x + c} \sqrt {e x} b}{x}\right ) + 4 \, {\left (a c d x + 3 \, a c^{2} - {\left (15 \, b c^{2} + 2 \, a d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{30 \, a^{2} c^{2} e^{4} x^{3}} \] Input:

integrate((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

-1/30*(15*a^2*c^2*e^4*x^3*sqrt((a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) + b^2*d) 
/(a^4*e^7))*log((a^2*e^4*x*sqrt((a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) + b^2*d 
)/(a^4*e^7)) + sqrt(d*x + c)*sqrt(e*x)*b)/x) - 15*a^2*c^2*e^4*x^3*sqrt((a^ 
4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) + b^2*d)/(a^4*e^7))*log(-(a^2*e^4*x*sqrt(( 
a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) + b^2*d)/(a^4*e^7)) - sqrt(d*x + c)*sqrt 
(e*x)*b)/x) + 15*a^2*c^2*e^4*x^3*sqrt(-(a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) 
- b^2*d)/(a^4*e^7))*log((a^2*e^4*x*sqrt(-(a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14) 
) - b^2*d)/(a^4*e^7)) + sqrt(d*x + c)*sqrt(e*x)*b)/x) - 15*a^2*c^2*e^4*x^3 
*sqrt(-(a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) - b^2*d)/(a^4*e^7))*log(-(a^2*e^ 
4*x*sqrt(-(a^4*e^7*sqrt(-b^5*c^2/(a^9*e^14)) - b^2*d)/(a^4*e^7)) - sqrt(d* 
x + c)*sqrt(e*x)*b)/x) + 4*(a*c*d*x + 3*a*c^2 - (15*b*c^2 + 2*a*d^2)*x^2)* 
sqrt(d*x + c)*sqrt(e*x))/(a^2*c^2*e^4*x^3)
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {c + d x}}{\left (e x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )}\, dx \] Input:

integrate((d*x+c)**(1/2)/(e*x)**(7/2)/(b*x**2+a),x)
                                                                                    
                                                                                    
 

Output:

Integral(sqrt(c + d*x)/((e*x)**(7/2)*(a + b*x**2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x^{2} + a\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x + c)/((b*x^2 + a)*(e*x)^(7/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x}}{(e x)^{7/2} \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {c+d\,x}}{{\left (e\,x\right )}^{7/2}\,\left (b\,x^2+a\right )} \,d x \] Input:

int((c + d*x)^(1/2)/((e*x)^(7/2)*(a + b*x^2)),x)
 

Output:

int((c + d*x)^(1/2)/((e*x)^(7/2)*(a + b*x^2)), x)
 

Reduce [F]

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

int((d*x+c)^(1/2)/(e*x)^(7/2)/(b*x^2+a),x)
 

Output:

int(sqrt(c + d*x)/(sqrt(x)*a*x**3 + sqrt(x)*b*x**5),x)/(sqrt(e)*e**3)