\(\int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx\) [786]

Optimal result

Integrand size = 24, antiderivative size = 130 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 a}{3 c e (e x)^{3/2} (c+d x)^{3/2}}+\frac {4 a d}{c^2 e^2 \sqrt {e x} (c+d x)^{3/2}}+\frac {2 \left (b c^2+8 a d^2\right ) \sqrt {e x}}{3 c^3 e^3 (c+d x)^{3/2}}+\frac {4 \left (b c^2+8 a d^2\right ) \sqrt {e x}}{3 c^4 e^3 \sqrt {c+d x}} \] Output:

-2/3*a/c/e/(e*x)^(3/2)/(d*x+c)^(3/2)+4*a*d/c^2/e^2/(e*x)^(1/2)/(d*x+c)^(3/ 
2)+2/3*(8*a*d^2+b*c^2)*(e*x)^(1/2)/c^3/e^3/(d*x+c)^(3/2)+4/3*(8*a*d^2+b*c^ 
2)*(e*x)^(1/2)/c^4/e^3/(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 x \left (b c^2 x^2 (3 c+2 d x)+a \left (-c^3+6 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )\right )}{3 c^4 (e x)^{5/2} (c+d x)^{3/2}} \] Input:

Integrate[(a + b*x^2)/((e*x)^(5/2)*(c + d*x)^(5/2)),x]
 

Output:

(2*x*(b*c^2*x^2*(3*c + 2*d*x) + a*(-c^3 + 6*c^2*d*x + 24*c*d^2*x^2 + 16*d^ 
3*x^3)))/(3*c^4*(e*x)^(5/2)*(c + d*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {519, 27, 87, 55, 48}

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.

\(\displaystyle \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 519

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {-\frac {\left (8 a d^2+b c^2\right ) \int \frac {1}{(e x)^{3/2} (c+d x)^{3/2}}dx}{3 c e}-\frac {2 \left (2 a d+\frac {b c^2}{d}\right )}{3 c e (e x)^{3/2} \sqrt {c+d x}}}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{3 c e (e x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {-\frac {\left (8 a d^2+b c^2\right ) \left (\frac {2 \int \frac {1}{(e x)^{3/2} \sqrt {c+d x}}dx}{c}+\frac {2}{c e \sqrt {e x} \sqrt {c+d x}}\right )}{3 c e}-\frac {2 \left (2 a d+\frac {b c^2}{d}\right )}{3 c e (e x)^{3/2} \sqrt {c+d x}}}{c d}+\frac {2 \left (a+\frac {b c^2}{d^2}\right )}{3 c e (e x)^{3/2} (c+d x)^{3/2}}\)

\(\Big \downarrow \) 48

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

Input:

Int[(a + b*x^2)/((e*x)^(5/2)*(c + d*x)^(5/2)),x]
 

Output:

(2*(a + (b*c^2)/d^2))/(3*c*e*(e*x)^(3/2)*(c + d*x)^(3/2)) + ((-2*((b*c^2)/ 
d + 2*a*d))/(3*c*e*(e*x)^(3/2)*Sqrt[c + d*x]) - ((b*c^2 + 8*a*d^2)*(2/(c*e 
*Sqrt[e*x]*Sqrt[c + d*x]) - (4*Sqrt[c + d*x])/(c^2*e*Sqrt[e*x])))/(3*c*e)) 
/(c*d)
 

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 48

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

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 519

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, c + d*x, x], R = 
 PolynomialRemainder[(a + b*x^2)^p, c + d*x, x]}, Simp[(-R)*(e*x)^(m + 1)*( 
(c + d*x)^(n + 1)/(c*e*(n + 1))), x] + Simp[1/(c*(n + 1))   Int[(e*x)^m*(c 
+ d*x)^(n + 1)*ExpandToSum[c*(n + 1)*Qx + R*(m + n + 2), x], x], x]] /; Fre 
eQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0] && LtQ[n, -1] &&  !IntegerQ[m]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.55

method	result	size
gosper	\(-\frac {2 x \left (-16 a \,x^{3} d^{3}-2 b \,c^{2} d \,x^{3}-24 a \,d^{2} x^{2} c -3 b \,c^{3} x^{2}-6 a d x \,c^{2}+c^{3} a \right )}{3 \left (d x +c \right )^{\frac {3}{2}} c^{4} \left (e x \right )^{\frac {5}{2}}}\)	\(71\)
orering	\(-\frac {2 x \left (-16 a \,x^{3} d^{3}-2 b \,c^{2} d \,x^{3}-24 a \,d^{2} x^{2} c -3 b \,c^{3} x^{2}-6 a d x \,c^{2}+c^{3} a \right )}{3 \left (d x +c \right )^{\frac {3}{2}} c^{4} \left (e x \right )^{\frac {5}{2}}}\)	\(71\)
default	\(-\frac {2 \left (-16 a \,x^{3} d^{3}-2 b \,c^{2} d \,x^{3}-24 a \,d^{2} x^{2} c -3 b \,c^{3} x^{2}-6 a d x \,c^{2}+c^{3} a \right )}{3 \sqrt {e x}\, e^{2} \left (d x +c \right )^{\frac {3}{2}} c^{4} x}\)	\(76\)
risch	\(-\frac {2 \sqrt {d x +c}\, a \left (-8 d x +c \right )}{3 c^{4} x \,e^{2} \sqrt {e x}}+\frac {2 \left (8 a x \,d^{3}+2 b \,c^{2} d x +9 a \,d^{2} c +3 b \,c^{3}\right ) x}{3 \left (d x +c \right )^{\frac {3}{2}} c^{4} e^{2} \sqrt {e x}}\)	\(82\)

Input:

int((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3*x*(-16*a*d^3*x^3-2*b*c^2*d*x^3-24*a*c*d^2*x^2-3*b*c^3*x^2-6*a*c^2*d*x 
+a*c^3)/(d*x+c)^(3/2)/c^4/(e*x)^(5/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (6 \, a c^{2} d x - a c^{3} + 2 \, {\left (b c^{2} d + 8 \, a d^{3}\right )} x^{3} + 3 \, {\left (b c^{3} + 8 \, a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{3 \, {\left (c^{4} d^{2} e^{3} x^{4} + 2 \, c^{5} d e^{3} x^{3} + c^{6} e^{3} x^{2}\right )}} \] Input:

integrate((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

2/3*(6*a*c^2*d*x - a*c^3 + 2*(b*c^2*d + 8*a*d^3)*x^3 + 3*(b*c^3 + 8*a*c*d^ 
2)*x^2)*sqrt(d*x + c)*sqrt(e*x)/(c^4*d^2*e^3*x^4 + 2*c^5*d*e^3*x^3 + c^6*e 
^3*x^2)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (128) = 256\).

Time = 28.01 (sec) , antiderivative size = 556, normalized size of antiderivative = 4.28 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=a \left (- \frac {2 c^{4} d^{\frac {19}{2}} \sqrt {\frac {c}{d x} + 1}}{3 c^{7} d^{9} e^{\frac {5}{2}} x + 9 c^{6} d^{10} e^{\frac {5}{2}} x^{2} + 9 c^{5} d^{11} e^{\frac {5}{2}} x^{3} + 3 c^{4} d^{12} e^{\frac {5}{2}} x^{4}} + \frac {10 c^{3} d^{\frac {21}{2}} x \sqrt {\frac {c}{d x} + 1}}{3 c^{7} d^{9} e^{\frac {5}{2}} x + 9 c^{6} d^{10} e^{\frac {5}{2}} x^{2} + 9 c^{5} d^{11} e^{\frac {5}{2}} x^{3} + 3 c^{4} d^{12} e^{\frac {5}{2}} x^{4}} + \frac {60 c^{2} d^{\frac {23}{2}} x^{2} \sqrt {\frac {c}{d x} + 1}}{3 c^{7} d^{9} e^{\frac {5}{2}} x + 9 c^{6} d^{10} e^{\frac {5}{2}} x^{2} + 9 c^{5} d^{11} e^{\frac {5}{2}} x^{3} + 3 c^{4} d^{12} e^{\frac {5}{2}} x^{4}} + \frac {80 c d^{\frac {25}{2}} x^{3} \sqrt {\frac {c}{d x} + 1}}{3 c^{7} d^{9} e^{\frac {5}{2}} x + 9 c^{6} d^{10} e^{\frac {5}{2}} x^{2} + 9 c^{5} d^{11} e^{\frac {5}{2}} x^{3} + 3 c^{4} d^{12} e^{\frac {5}{2}} x^{4}} + \frac {32 d^{\frac {27}{2}} x^{4} \sqrt {\frac {c}{d x} + 1}}{3 c^{7} d^{9} e^{\frac {5}{2}} x + 9 c^{6} d^{10} e^{\frac {5}{2}} x^{2} + 9 c^{5} d^{11} e^{\frac {5}{2}} x^{3} + 3 c^{4} d^{12} e^{\frac {5}{2}} x^{4}}\right ) + b \left (\frac {6 c}{3 c^{3} \sqrt {d} e^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1} + 3 c^{2} d^{\frac {3}{2}} e^{\frac {5}{2}} x \sqrt {\frac {c}{d x} + 1}} + \frac {4 d x}{3 c^{3} \sqrt {d} e^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1} + 3 c^{2} d^{\frac {3}{2}} e^{\frac {5}{2}} x \sqrt {\frac {c}{d x} + 1}}\right ) \] Input:

integrate((b*x**2+a)/(e*x)**(5/2)/(d*x+c)**(5/2),x)
 

Output:

a*(-2*c**4*d**(19/2)*sqrt(c/(d*x) + 1)/(3*c**7*d**9*e**(5/2)*x + 9*c**6*d* 
*10*e**(5/2)*x**2 + 9*c**5*d**11*e**(5/2)*x**3 + 3*c**4*d**12*e**(5/2)*x** 
4) + 10*c**3*d**(21/2)*x*sqrt(c/(d*x) + 1)/(3*c**7*d**9*e**(5/2)*x + 9*c** 
6*d**10*e**(5/2)*x**2 + 9*c**5*d**11*e**(5/2)*x**3 + 3*c**4*d**12*e**(5/2) 
*x**4) + 60*c**2*d**(23/2)*x**2*sqrt(c/(d*x) + 1)/(3*c**7*d**9*e**(5/2)*x 
+ 9*c**6*d**10*e**(5/2)*x**2 + 9*c**5*d**11*e**(5/2)*x**3 + 3*c**4*d**12*e 
**(5/2)*x**4) + 80*c*d**(25/2)*x**3*sqrt(c/(d*x) + 1)/(3*c**7*d**9*e**(5/2 
)*x + 9*c**6*d**10*e**(5/2)*x**2 + 9*c**5*d**11*e**(5/2)*x**3 + 3*c**4*d** 
12*e**(5/2)*x**4) + 32*d**(27/2)*x**4*sqrt(c/(d*x) + 1)/(3*c**7*d**9*e**(5 
/2)*x + 9*c**6*d**10*e**(5/2)*x**2 + 9*c**5*d**11*e**(5/2)*x**3 + 3*c**4*d 
**12*e**(5/2)*x**4)) + b*(6*c/(3*c**3*sqrt(d)*e**(5/2)*sqrt(c/(d*x) + 1) + 
 3*c**2*d**(3/2)*e**(5/2)*x*sqrt(c/(d*x) + 1)) + 4*d*x/(3*c**3*sqrt(d)*e** 
(5/2)*sqrt(c/(d*x) + 1) + 3*c**2*d**(3/2)*e**(5/2)*x*sqrt(c/(d*x) + 1)))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.28 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, b x}{3 \, {\left (d e x^{2} + c e x\right )}^{\frac {3}{2}} d e} - \frac {4 \, a d x}{3 \, {\left (d e x^{2} + c e x\right )}^{\frac {3}{2}} c^{2} e} - \frac {2 \, a}{3 \, {\left (d e x^{2} + c e x\right )}^{\frac {3}{2}} c e} + \frac {4 \, b x}{3 \, \sqrt {d e x^{2} + c e x} c^{2} e^{2}} + \frac {32 \, a d^{2} x}{3 \, \sqrt {d e x^{2} + c e x} c^{4} e^{2}} + \frac {2 \, b}{3 \, \sqrt {d e x^{2} + c e x} c d e^{2}} + \frac {16 \, a d}{3 \, \sqrt {d e x^{2} + c e x} c^{3} e^{2}} \] Input:

integrate((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

-2/3*b*x/((d*e*x^2 + c*e*x)^(3/2)*d*e) - 4/3*a*d*x/((d*e*x^2 + c*e*x)^(3/2 
)*c^2*e) - 2/3*a/((d*e*x^2 + c*e*x)^(3/2)*c*e) + 4/3*b*x/(sqrt(d*e*x^2 + c 
*e*x)*c^2*e^2) + 32/3*a*d^2*x/(sqrt(d*e*x^2 + c*e*x)*c^4*e^2) + 2/3*b/(sqr 
t(d*e*x^2 + c*e*x)*c*d*e^2) + 16/3*a*d/(sqrt(d*e*x^2 + c*e*x)*c^3*e^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (108) = 216\).

Time = 0.21 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.22 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {d x + c} {\left (\frac {8 \, {\left (d x + c\right )} a d^{2} {\left | d \right |}}{c^{4} e} - \frac {9 \, a d^{2} {\left | d \right |}}{c^{3} e}\right )}}{3 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {3}{2}}} + \frac {8 \, {\left (\sqrt {d e} b c^{4} d^{3} e^{2} + 4 \, \sqrt {d e} a c^{2} d^{5} e^{2} + 3 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} b c^{3} d^{2} e + 9 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2} a c d^{4} e + 3 \, \sqrt {d e} {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{4} a d^{3}\right )}}{3 \, {\left (c d e + {\left (\sqrt {d e} \sqrt {d x + c} - \sqrt {{\left (d x + c\right )} d e - c d e}\right )}^{2}\right )}^{3} c^{3} e^{2} {\left | d \right |}} \] Input:

integrate((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

2/3*sqrt(d*x + c)*(8*(d*x + c)*a*d^2*abs(d)/(c^4*e) - 9*a*d^2*abs(d)/(c^3* 
e))/((d*x + c)*d*e - c*d*e)^(3/2) + 8/3*(sqrt(d*e)*b*c^4*d^3*e^2 + 4*sqrt( 
d*e)*a*c^2*d^5*e^2 + 3*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt((d*x + c) 
*d*e - c*d*e))^2*b*c^3*d^2*e + 9*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c) - sqrt 
((d*x + c)*d*e - c*d*e))^2*a*c*d^4*e + 3*sqrt(d*e)*(sqrt(d*e)*sqrt(d*x + c 
) - sqrt((d*x + c)*d*e - c*d*e))^4*a*d^3)/((c*d*e + (sqrt(d*e)*sqrt(d*x + 
c) - sqrt((d*x + c)*d*e - c*d*e))^2)^3*c^3*e^2*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.02 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (6\,b\,c^3+48\,a\,c\,d^2\right )}{3\,c^4\,d^2\,e^2}-\frac {2\,a}{3\,c\,d^2\,e^2}+\frac {x^3\,\left (4\,b\,c^2\,d+32\,a\,d^3\right )}{3\,c^4\,d^2\,e^2}+\frac {4\,a\,x}{c^2\,d\,e^2}\right )}{x^3\,\sqrt {e\,x}+\frac {2\,c\,x^2\,\sqrt {e\,x}}{d}+\frac {c^2\,x\,\sqrt {e\,x}}{d^2}} \] Input:

int((a + b*x^2)/((e*x)^(5/2)*(c + d*x)^(5/2)),x)
 

Output:

((c + d*x)^(1/2)*((x^2*(6*b*c^3 + 48*a*c*d^2))/(3*c^4*d^2*e^2) - (2*a)/(3* 
c*d^2*e^2) + (x^3*(32*a*d^3 + 4*b*c^2*d))/(3*c^4*d^2*e^2) + (4*a*x)/(c^2*d 
*e^2)))/(x^3*(e*x)^(1/2) + (2*c*x^2*(e*x)^(1/2))/d + (c^2*x*(e*x)^(1/2))/d 
^2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.32 \[ \int \frac {a+b x^2}{(e x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {e}\, \left (-16 \sqrt {d}\, \sqrt {d x +c}\, a c \,d^{2} x^{2}-16 \sqrt {d}\, \sqrt {d x +c}\, a \,d^{3} x^{3}-6 \sqrt {d}\, \sqrt {d x +c}\, b \,c^{3} x^{2}-6 \sqrt {d}\, \sqrt {d x +c}\, b \,c^{2} d \,x^{3}-\sqrt {x}\, a \,c^{3} d +6 \sqrt {x}\, a \,c^{2} d^{2} x +24 \sqrt {x}\, a c \,d^{3} x^{2}+16 \sqrt {x}\, a \,d^{4} x^{3}+3 \sqrt {x}\, b \,c^{3} d \,x^{2}+2 \sqrt {x}\, b \,c^{2} d^{2} x^{3}\right )}{3 \sqrt {d x +c}\, c^{4} d \,e^{3} x^{2} \left (d x +c \right )} \] Input:

int((b*x^2+a)/(e*x)^(5/2)/(d*x+c)^(5/2),x)
 

Output:

(2*sqrt(e)*( - 16*sqrt(d)*sqrt(c + d*x)*a*c*d**2*x**2 - 16*sqrt(d)*sqrt(c 
+ d*x)*a*d**3*x**3 - 6*sqrt(d)*sqrt(c + d*x)*b*c**3*x**2 - 6*sqrt(d)*sqrt( 
c + d*x)*b*c**2*d*x**3 - sqrt(x)*a*c**3*d + 6*sqrt(x)*a*c**2*d**2*x + 24*s 
qrt(x)*a*c*d**3*x**2 + 16*sqrt(x)*a*d**4*x**3 + 3*sqrt(x)*b*c**3*d*x**2 + 
2*sqrt(x)*b*c**2*d**2*x**3))/(3*sqrt(c + d*x)*c**4*d*e**3*x**2*(c + d*x))