\(\int \frac {(b d+2 c d x)^4}{(a+b x+c x^2)^{5/2}} \, dx\) [212]

Optimal result

Integrand size = 26, antiderivative size = 96 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 c d^4 (b+2 c x)}{\sqrt {a+b x+c x^2}}+16 c^{3/2} d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:

-2/3*d^4*(2*c*x+b)^3/(c*x^2+b*x+a)^(3/2)-8*c*d^4*(2*c*x+b)/(c*x^2+b*x+a)^( 
1/2)+16*c^(3/2)*d^4*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=d^4 \left (-\frac {2 (b+2 c x) \left (b^2+16 b c x+4 c \left (3 a+4 c x^2\right )\right )}{3 (a+x (b+c x))^{3/2}}+32 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right ) \] Input:

Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2),x]
 

Output:

d^4*((-2*(b + 2*c*x)*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)))/(3*(a + x*(b 
+ c*x))^(3/2)) + 32*c^(3/2)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b 
+ c*x)])])
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1110, 27, 1110, 1092, 219}

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.

\(\displaystyle \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1110

\(\displaystyle 4 c d^4 \left (4 c \int \frac {1}{\sqrt {c x^2+b x+a}}dx-\frac {2 (b+2 c x)}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle 4 c d^4 \left (8 c \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}-\frac {2 (b+2 c x)}{\sqrt {a+b x+c x^2}}\right )-\frac {2 d^4 (b+2 c x)^3}{3 \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

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

Input:

Int[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2),x]
 

Output:

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

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1110

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), 
x] - Simp[d*e*((m - 1)/(b*(p + 1)))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 
2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N 
eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1136\) vs. \(2(82)=164\).

Time = 1.19 (sec) , antiderivative size = 1137, normalized size of antiderivative = 11.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1137\)

Input:

int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

d^4*(b^4*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2) 
^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))+16*c^4*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1 
/2*b/c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1 
/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x 
^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a 
/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2* 
c*x+b)/(c*x^2+b*x+a)^(1/2)))+2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/ 
3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b) 
/(c*x^2+b*x+a)^(1/2))))+1/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2 
+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln( 
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+32*b*c^3*(-x^2/c/(c*x^2+b*x+a)^ 
(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^ 
(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c 
-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2) 
/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+ 
2*a/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^ 
2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+24*b^ 
2*c^2*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/ 
2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2* 
(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (82) = 164\).

Time = 0.42 (sec) , antiderivative size = 440, normalized size of antiderivative = 4.58 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\left [\frac {2 \, {\left (12 \, {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - {\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, -\frac {2 \, {\left (24 \, {\left (c^{3} d^{4} x^{4} + 2 \, b c^{2} d^{4} x^{3} + 2 \, a b c d^{4} x + a^{2} c d^{4} + {\left (b^{2} c + 2 \, a c^{2}\right )} d^{4} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (32 \, c^{3} d^{4} x^{3} + 48 \, b c^{2} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{4} x + {\left (b^{3} + 12 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \] Input:

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[2/3*(12*(c^3*d^4*x^4 + 2*b*c^2*d^4*x^3 + 2*a*b*c*d^4*x + a^2*c*d^4 + (b^2 
*c + 2*a*c^2)*d^4*x^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x 
^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - (32*c^3*d^4*x^3 + 48*b*c^2*d^ 
4*x^2 + 6*(3*b^2*c + 4*a*c^2)*d^4*x + (b^3 + 12*a*b*c)*d^4)*sqrt(c*x^2 + b 
*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), -2/3*( 
24*(c^3*d^4*x^4 + 2*b*c^2*d^4*x^3 + 2*a*b*c*d^4*x + a^2*c*d^4 + (b^2*c + 2 
*a*c^2)*d^4*x^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqr 
t(-c)/(c^2*x^2 + b*c*x + a*c)) + (32*c^3*d^4*x^3 + 48*b*c^2*d^4*x^2 + 6*(3 
*b^2*c + 4*a*c^2)*d^4*x + (b^3 + 12*a*b*c)*d^4)*sqrt(c*x^2 + b*x + a))/(c^ 
2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)]
 

Sympy [F]

\[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=d^{4} \left (\int \frac {b^{4}}{a^{2} \sqrt {a + b x + c x^{2}} + 2 a b x \sqrt {a + b x + c x^{2}} + 2 a c x^{2} \sqrt {a + b x + c x^{2}} + b^{2} x^{2} \sqrt {a + b x + c x^{2}} + 2 b c x^{3} \sqrt {a + b x + c x^{2}} + c^{2} x^{4} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {16 c^{4} x^{4}}{a^{2} \sqrt {a + b x + c x^{2}} + 2 a b x \sqrt {a + b x + c x^{2}} + 2 a c x^{2} \sqrt {a + b x + c x^{2}} + b^{2} x^{2} \sqrt {a + b x + c x^{2}} + 2 b c x^{3} \sqrt {a + b x + c x^{2}} + c^{2} x^{4} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {32 b c^{3} x^{3}}{a^{2} \sqrt {a + b x + c x^{2}} + 2 a b x \sqrt {a + b x + c x^{2}} + 2 a c x^{2} \sqrt {a + b x + c x^{2}} + b^{2} x^{2} \sqrt {a + b x + c x^{2}} + 2 b c x^{3} \sqrt {a + b x + c x^{2}} + c^{2} x^{4} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {24 b^{2} c^{2} x^{2}}{a^{2} \sqrt {a + b x + c x^{2}} + 2 a b x \sqrt {a + b x + c x^{2}} + 2 a c x^{2} \sqrt {a + b x + c x^{2}} + b^{2} x^{2} \sqrt {a + b x + c x^{2}} + 2 b c x^{3} \sqrt {a + b x + c x^{2}} + c^{2} x^{4} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {8 b^{3} c x}{a^{2} \sqrt {a + b x + c x^{2}} + 2 a b x \sqrt {a + b x + c x^{2}} + 2 a c x^{2} \sqrt {a + b x + c x^{2}} + b^{2} x^{2} \sqrt {a + b x + c x^{2}} + 2 b c x^{3} \sqrt {a + b x + c x^{2}} + c^{2} x^{4} \sqrt {a + b x + c x^{2}}}\, dx\right ) \] Input:

integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(5/2),x)
 

Output:

d**4*(Integral(b**4/(a**2*sqrt(a + b*x + c*x**2) + 2*a*b*x*sqrt(a + b*x + 
c*x**2) + 2*a*c*x**2*sqrt(a + b*x + c*x**2) + b**2*x**2*sqrt(a + b*x + c*x 
**2) + 2*b*c*x**3*sqrt(a + b*x + c*x**2) + c**2*x**4*sqrt(a + b*x + c*x**2 
)), x) + Integral(16*c**4*x**4/(a**2*sqrt(a + b*x + c*x**2) + 2*a*b*x*sqrt 
(a + b*x + c*x**2) + 2*a*c*x**2*sqrt(a + b*x + c*x**2) + b**2*x**2*sqrt(a 
+ b*x + c*x**2) + 2*b*c*x**3*sqrt(a + b*x + c*x**2) + c**2*x**4*sqrt(a + b 
*x + c*x**2)), x) + Integral(32*b*c**3*x**3/(a**2*sqrt(a + b*x + c*x**2) + 
 2*a*b*x*sqrt(a + b*x + c*x**2) + 2*a*c*x**2*sqrt(a + b*x + c*x**2) + b**2 
*x**2*sqrt(a + b*x + c*x**2) + 2*b*c*x**3*sqrt(a + b*x + c*x**2) + c**2*x* 
*4*sqrt(a + b*x + c*x**2)), x) + Integral(24*b**2*c**2*x**2/(a**2*sqrt(a + 
 b*x + c*x**2) + 2*a*b*x*sqrt(a + b*x + c*x**2) + 2*a*c*x**2*sqrt(a + b*x 
+ c*x**2) + b**2*x**2*sqrt(a + b*x + c*x**2) + 2*b*c*x**3*sqrt(a + b*x + c 
*x**2) + c**2*x**4*sqrt(a + b*x + c*x**2)), x) + Integral(8*b**3*c*x/(a**2 
*sqrt(a + b*x + c*x**2) + 2*a*b*x*sqrt(a + b*x + c*x**2) + 2*a*c*x**2*sqrt 
(a + b*x + c*x**2) + b**2*x**2*sqrt(a + b*x + c*x**2) + 2*b*c*x**3*sqrt(a 
+ b*x + c*x**2) + c**2*x**4*sqrt(a + b*x + c*x**2)), x))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (82) = 164\).

Time = 0.36 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.26 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-16 \, c^{\frac {3}{2}} d^{4} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right ) - \frac {2 \, {\left (2 \, {\left (8 \, {\left (\frac {2 \, {\left (b^{4} c^{3} d^{4} - 8 \, a b^{2} c^{4} d^{4} + 16 \, a^{2} c^{5} d^{4}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (b^{5} c^{2} d^{4} - 8 \, a b^{3} c^{3} d^{4} + 16 \, a^{2} b c^{4} d^{4}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (3 \, b^{6} c d^{4} - 20 \, a b^{4} c^{2} d^{4} + 16 \, a^{2} b^{2} c^{3} d^{4} + 64 \, a^{3} c^{4} d^{4}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{7} d^{4} + 4 \, a b^{5} c d^{4} - 80 \, a^{2} b^{3} c^{2} d^{4} + 192 \, a^{3} b c^{3} d^{4}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

-16*c^(3/2)*d^4*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b) 
) - 2/3*(2*(8*(2*(b^4*c^3*d^4 - 8*a*b^2*c^4*d^4 + 16*a^2*c^5*d^4)*x/(b^4 - 
 8*a*b^2*c + 16*a^2*c^2) + 3*(b^5*c^2*d^4 - 8*a*b^3*c^3*d^4 + 16*a^2*b*c^4 
*d^4)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 3*(3*b^6*c*d^4 - 20*a*b^4*c^2*d^ 
4 + 16*a^2*b^2*c^3*d^4 + 64*a^3*c^4*d^4)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x 
 + (b^7*d^4 + 4*a*b^5*c*d^4 - 80*a^2*b^3*c^2*d^4 + 192*a^3*b*c^3*d^4)/(b^4 
 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int((b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 431, normalized size of antiderivative = 4.49 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 d^{4} \left (-12 \sqrt {c \,x^{2}+b x +a}\, a b c -24 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} x -\sqrt {c \,x^{2}+b x +a}\, b^{3}-18 \sqrt {c \,x^{2}+b x +a}\, b^{2} c x -48 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} x^{2}-32 \sqrt {c \,x^{2}+b x +a}\, c^{3} x^{3}+24 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c +48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b c x +48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} x^{2}+24 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c \,x^{2}+48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{2} x^{3}+24 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{3} x^{4}\right )}{3 c^{2} x^{4}+6 b c \,x^{3}+6 a c \,x^{2}+3 b^{2} x^{2}+6 a b x +3 a^{2}} \] Input:

int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x)
 

Output:

(2*d**4*( - 12*sqrt(a + b*x + c*x**2)*a*b*c - 24*sqrt(a + b*x + c*x**2)*a* 
c**2*x - sqrt(a + b*x + c*x**2)*b**3 - 18*sqrt(a + b*x + c*x**2)*b**2*c*x 
- 48*sqrt(a + b*x + c*x**2)*b*c**2*x**2 - 32*sqrt(a + b*x + c*x**2)*c**3*x 
**3 + 24*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4 
*a*c - b**2))*a**2*c + 48*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + 
b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c*x + 48*sqrt(c)*log((2*sqrt(c)*sqrt(a 
+ b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**2*x**2 + 24*sqrt(c)* 
log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b** 
2*c*x**2 + 48*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s 
qrt(4*a*c - b**2))*b*c**2*x**3 + 24*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + 
c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*c**3*x**4))/(3*(a**2 + 2*a*b*x + 
2*a*c*x**2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4))