\(\int \frac {(b d+2 c d x)^4}{(a+b x+c x^2)^{3/2}} \, dx\) [202]

Optimal result

Integrand size = 26, antiderivative size = 102 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}+12 c d^4 (b+2 c x) \sqrt {a+b x+c x^2}+6 \sqrt {c} \left (b^2-4 a c\right ) d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \] Output:

-2*d^4*(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)+12*c*d^4*(2*c*x+b)*(c*x^2+b*x+a)^(1 
/2)+6*c^(1/2)*(-4*a*c+b^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 = 0.82 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.91 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=d^4 \left (-\frac {2 (b+2 c x) \left (b^2-2 b c x-2 c \left (3 a+c x^2\right )\right )}{\sqrt {a+x (b+c x)}}+12 \sqrt {c} \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right ) \] Input:

Integrate[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2),x]
 

Output:

d^4*((-2*(b + 2*c*x)*(b^2 - 2*b*c*x - 2*c*(3*a + c*x^2)))/Sqrt[a + x*(b + 
c*x)] + 12*Sqrt[c]*(b^2 - 4*a*c)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + 
x*(b + c*x)])])
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1110, 27, 1116, 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.

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

\(\Big \downarrow \) 1110

\(\displaystyle 12 c d^2 \int \frac {d^2 (b+2 c x)^2}{\sqrt {c x^2+b x+a}}dx-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle 12 c d^4 \int \frac {(b+2 c x)^2}{\sqrt {c x^2+b x+a}}dx-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1116

\(\displaystyle 12 c d^4 \left (\frac {1}{2} \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+(b+2 c x) \sqrt {a+b x+c x^2}\right )-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle 12 c d^4 \left (\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle 12 c d^4 \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )-\frac {2 d^4 (b+2 c x)^3}{\sqrt {a+b x+c x^2}}\)

Input:

Int[(b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2),x]
 

Output:

(-2*d^4*(b + 2*c*x)^3)/Sqrt[a + b*x + c*x^2] + 12*c*d^4*((b + 2*c*x)*Sqrt[ 
a + b*x + c*x^2] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + 
b*x + c*x^2])])/(2*Sqrt[c]))
 

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]
 

rule 1116

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p 
 + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p 
+ 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(90)=180\).

Time = 1.22 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.25

method	result	size
risch	\(4 c \,d^{4} \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}+\left (-6 c^{2} \left (4 a c -b^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )-6 b c \left (4 a c -b^{2}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {2 b^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {16 a^{2} c^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 c a \,b^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right ) d^{4}\)	\(332\)
default	\(d^{4} \left (\frac {2 b^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+16 c^{4} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+32 b \,c^{3} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+24 b^{2} c^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+8 b^{3} c \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\right )\)	\(757\)

Input:

int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

4*c*d^4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)+(-6*c^2*(4*a*c-b^2)*(-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)))-6* 
b*c*(4*a*c-b^2)*(-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))+2*b^4*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-16*a^2*c^2* 
(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-4*c*a*b^2*(2*c*x+b)/(4*a*c-b^2)/ 
(c*x^2+b*x+a)^(1/2))*d^4
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.50 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{4} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 2 \, {\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x - {\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{c x^{2} + b x + a}, -\frac {2 \, {\left (3 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{4} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - {\left (4 \, c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 12 \, a c^{2} d^{4} x - {\left (b^{3} - 6 \, a b c\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}\right )}}{c x^{2} + b x + a}\right ] \] Input:

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

[-(3*((b^2*c - 4*a*c^2)*d^4*x^2 + (b^3 - 4*a*b*c)*d^4*x + (a*b^2 - 4*a^2*c 
)*d^4)*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) - 2*(4*c^3*d^4*x^3 + 6*b*c^2*d^4*x^2 + 12*a*c^2 
*d^4*x - (b^3 - 6*a*b*c)*d^4)*sqrt(c*x^2 + b*x + a))/(c*x^2 + b*x + a), -2 
*(3*((b^2*c - 4*a*c^2)*d^4*x^2 + (b^3 - 4*a*b*c)*d^4*x + (a*b^2 - 4*a^2*c) 
*d^4)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2* 
x^2 + b*c*x + a*c)) - (4*c^3*d^4*x^3 + 6*b*c^2*d^4*x^2 + 12*a*c^2*d^4*x - 
(b^3 - 6*a*b*c)*d^4)*sqrt(c*x^2 + b*x + a))/(c*x^2 + b*x + a)]
 

Sympy [F]

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

integrate((2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(3/2),x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),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. 247 vs. \(2 (90) = 180\).

Time = 0.39 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.42 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {6 \, {\left (b^{2} c d^{4} - 4 \, a c^{2} d^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{\sqrt {c}} + \frac {2 \, {\left (2 \, {\left ({\left (\frac {2 \, {\left (b^{2} c^{5} d^{4} - 4 \, a c^{6} d^{4}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, {\left (b^{3} c^{4} d^{4} - 4 \, a b c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac {6 \, {\left (a b^{2} c^{4} d^{4} - 4 \, a^{2} c^{5} d^{4}\right )}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x - \frac {b^{5} c^{2} d^{4} - 10 \, a b^{3} c^{3} d^{4} + 24 \, a^{2} b c^{4} d^{4}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt {c x^{2} + b x + a}} \] Input:

integrate((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

-6*(b^2*c*d^4 - 4*a*c^2*d^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
*sqrt(c) + b))/sqrt(c) + 2*(2*((2*(b^2*c^5*d^4 - 4*a*c^6*d^4)*x/(b^2*c^2 - 
 4*a*c^3) + 3*(b^3*c^4*d^4 - 4*a*b*c^5*d^4)/(b^2*c^2 - 4*a*c^3))*x + 6*(a* 
b^2*c^4*d^4 - 4*a^2*c^5*d^4)/(b^2*c^2 - 4*a*c^3))*x - (b^5*c^2*d^4 - 10*a* 
b^3*c^3*d^4 + 24*a^2*b*c^4*d^4)/(b^2*c^2 - 4*a*c^3))/sqrt(c*x^2 + b*x + a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:

int((b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2),x)
 

Output:

int((b*d + 2*c*d*x)^4/(a + b*x + c*x^2)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 434, normalized size of antiderivative = 4.25 \[ \int \frac {(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {d^{4} \left (24 \sqrt {c \,x^{2}+b x +a}\, a b c +48 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} x -4 \sqrt {c \,x^{2}+b x +a}\, b^{3}+24 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} x^{2}+16 \sqrt {c \,x^{2}+b x +a}\, c^{3} x^{3}-48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c +12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2}-48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b c x -48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{2} x^{2}+12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} x +12 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c \,x^{2}+36 \sqrt {c}\, a^{2} c -9 \sqrt {c}\, a \,b^{2}+36 \sqrt {c}\, a b c x +36 \sqrt {c}\, a \,c^{2} x^{2}-9 \sqrt {c}\, b^{3} x -9 \sqrt {c}\, b^{2} c \,x^{2}\right )}{2 c \,x^{2}+2 b x +2 a} \] Input:

int((2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x)
 

Output:

(d**4*(24*sqrt(a + b*x + c*x**2)*a*b*c + 48*sqrt(a + b*x + c*x**2)*a*c**2* 
x - 4*sqrt(a + b*x + c*x**2)*b**3 + 24*sqrt(a + b*x + c*x**2)*b*c**2*x**2 
+ 16*sqrt(a + b*x + c*x**2)*c**3*x**3 - 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**2*c + 12*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**2 - 
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 + 12*sqrt(c)*log((2*sqrt(c)*sqrt(a + 
 b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*x + 12*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 + 36*sqrt(c)*a**2*c - 9*sqrt(c)*a*b**2 + 36*sqrt(c)*a*b*c*x + 36*sqrt(c 
)*a*c**2*x**2 - 9*sqrt(c)*b**3*x - 9*sqrt(c)*b**2*c*x**2))/(2*(a + b*x + c 
*x**2))