\(\int \frac {\sqrt {b d+2 c d x}}{(a+b x+c x^2)^3} \, dx\) [139]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac {5 c (b d+2 c d x)^{3/2}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac {5 c^2 \sqrt {d} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4}}-\frac {5 c^2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{9/4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx=\left (\frac {1}{2}+\frac {i}{2}\right ) c^2 \sqrt {d (b+2 c x)} \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x) \left (b^2-5 b c x-c \left (9 a+5 c x^2\right )\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {5 \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4} \sqrt {b+2 c x}}+\frac {5 \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/4} \sqrt {b+2 c x}}-\frac {5 \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{9/4} \sqrt {b+2 c x}}\right ) \] Input:

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

Output:

(1/2 + I/2)*c^2*Sqrt[d*(b + 2*c*x)]*(((-1/2 + I/2)*(b + 2*c*x)*(b^2 - 5*b* 
c*x - c*(9*a + 5*c*x^2)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - (5*A 
rcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/((b^2 - 4*a*c)^( 
9/4)*Sqrt[b + 2*c*x]) + (5*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a 
*c)^(1/4)])/((b^2 - 4*a*c)^(9/4)*Sqrt[b + 2*c*x]) - (5*ArcTanh[((1 + I)*(b 
^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(Sqrt[b^2 - 4*a*c] + I*(b + 2*c*x))])/( 
(b^2 - 4*a*c)^(9/4)*Sqrt[b + 2*c*x]))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1111, 1111, 1118, 27, 25, 266, 827, 216, 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.

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

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {5 c \int \frac {\sqrt {b d+2 c x d}}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {5 c \left (-\frac {c \int \frac {\sqrt {b d+2 c x d}}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1118

\(\displaystyle -\frac {5 c \left (-\frac {\int \frac {4 c d^2 \sqrt {b d+2 c x d}}{\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2}d(b d+2 c x d)}{2 d \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 c \left (-\frac {2 c d \int -\frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)}{b^2-4 a c}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 c \left (\frac {2 c d \int \frac {\sqrt {b d+2 c x d}}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d(b d+2 c x d)}{b^2-4 a c}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {5 c \left (\frac {4 c d \int \frac {b d+2 c x d}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}}{b^2-4 a c}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 827

\(\displaystyle -\frac {5 c \left (\frac {4 c d \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {1}{2} \int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}\right )}{b^2-4 a c}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {5 c \left (\frac {4 c d \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{b^2-4 a c}-\frac {(b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {(b d+2 c d x)^{3/2}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 1111

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

rule 1118

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(339\) vs. \(2(166)=332\).

Time = 1.38 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.75

method	result	size
pseudoelliptic	\(\frac {\left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{8 \left (a +x \left (c x +b \right )\right )^{2} \left (a c -\frac {b^{2}}{4}\right ) d}+\frac {5 c \left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{32 \left (c \,x^{2}+b x +a \right ) \left (a c -\frac {b^{2}}{4}\right )^{2} d}+\frac {5 c^{2} d^{5} \sqrt {2}\, \ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )}{4 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {9}{4}}}+\frac {5 c^{2} d^{5} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {9}{4}}}+\frac {5 c^{2} d^{5} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {9}{4}}}\)	\(340\)
derivativedivides	\(64 c^{2} d^{5} \left (\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {\frac {5 \left (2 c d x +b d \right )^{\frac {3}{2}}}{32 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {5}{4}}}}{4 a \,d^{2} c -b^{2} d^{2}}\right )\)	\(377\)
default	\(64 c^{2} d^{5} \left (\frac {\left (2 c d x +b d \right )^{\frac {3}{2}}}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {\frac {5 \left (2 c d x +b d \right )^{\frac {3}{2}}}{32 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {5}{4}}}}{4 a \,d^{2} c -b^{2} d^{2}}\right )\)	\(377\)

Input:

int((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/8*(d*(2*c*x+b))^(3/2)/(a+x*(c*x+b))^2/(a*c-1/4*b^2)/d+5/32*c*(d*(2*c*x+b 
))^(3/2)/(c*x^2+b*x+a)/(a*c-1/4*b^2)^2/d+5/4*c^2*d^5/(d^2*(4*a*c-b^2))^(9/ 
4)*2^(1/2)*ln(((d^2*(4*a*c-b^2))^(1/2)-(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b 
))^(1/2)*2^(1/2)+d*(2*c*x+b))/((d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^(1/2) 
*2^(1/2)+(d^2*(4*a*c-b^2))^(1/2)+d*(2*c*x+b)))+5/2*c^2*d^5/(d^2*(4*a*c-b^2 
))^(9/4)*2^(1/2)*arctan(2^(1/2)/(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^(1/2 
)+1)+5/2*c^2*d^5/(d^2*(4*a*c-b^2))^(9/4)*2^(1/2)*arctan(2^(1/2)/(d^2*(4*a* 
c-b^2))^(1/4)*(d*(2*c*x+b))^(1/2)-1)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 1980, normalized size of antiderivative = 10.21 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

-1/2*(5*(c^8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^ 
3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824* 
a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9))^(1/4)*(a^2*b^4 - 8*a^3 
*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c 
- 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2 
*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*log(125*sqrt(2*c*d*x + b*d)*c^6*d 
 + 125*(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^ 
4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*(c^8*d^ 
2/(b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b 
^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 5 
89824*a^8*b^2*c^8 - 262144*a^9*c^9))^(3/4)) - 5*(c^8*d^2/(b^18 - 36*a*b^16 
*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^ 
5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 
 262144*a^9*c^9))^(1/4)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8 
*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 
+ (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c 
^2)*x)*log(125*sqrt(2*c*d*x + b*d)*c^6*d - 125*(b^14 - 28*a*b^12*c + 336*a 
^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28 
672*a^6*b^2*c^6 - 16384*a^7*c^7)*(c^8*d^2/(b^18 - 36*a*b^16*c + 576*a^2*b^ 
14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + ...
 

Sympy [F]

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

integrate((2*c*d*x+b*d)**(1/2)/(c*x**2+b*x+a)**3,x)
 

Output:

Integral(sqrt(d*(b + 2*c*x))/(a + b*x + c*x**2)**3, x)
 

Maxima [F(-2)]

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

integrate((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^3,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. 645 vs. \(2 (166) = 332\).

Time = 0.19 (sec) , antiderivative size = 645, normalized size of antiderivative = 3.32 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{6} d - 12 \, \sqrt {2} a b^{4} c d + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d - 64 \, \sqrt {2} a^{3} c^{3} d} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{6} d - 12 \, \sqrt {2} a b^{4} c d + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d - 64 \, \sqrt {2} a^{3} c^{3} d} + \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{6} d - 12 \, \sqrt {2} a b^{4} c d + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d - 64 \, \sqrt {2} a^{3} c^{3} d\right )}} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} c^{2} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{6} d - 12 \, \sqrt {2} a b^{4} c d + 48 \, \sqrt {2} a^{2} b^{2} c^{2} d - 64 \, \sqrt {2} a^{3} c^{3} d\right )}} - \frac {2 \, {\left (9 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{3} - 36 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{3} d^{3} - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{2} d\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \] Input:

integrate((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

-5*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 
+ 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))/ 
(sqrt(2)*b^6*d - 12*sqrt(2)*a*b^4*c*d + 48*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt 
(2)*a^3*c^3*d) - 5*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*arctan(-1/2*sqrt(2)*(s 
qrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4 
*a*c*d^2)^(1/4))/(sqrt(2)*b^6*d - 12*sqrt(2)*a*b^4*c*d + 48*sqrt(2)*a^2*b^ 
2*c^2*d - 64*sqrt(2)*a^3*c^3*d) + 5/2*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*log 
(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) 
+ sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^6*d - 12*sqrt(2)*a*b^4*c*d + 48*s 
qrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) - 5/2*(-b^2*d^2 + 4*a*c*d^2)^ 
(3/4)*c^2*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2* 
c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))/(sqrt(2)*b^6*d - 12*sqrt(2)*a*b 
^4*c*d + 48*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) - 2*(9*(2*c*d*x 
+ b*d)^(3/2)*b^2*c^2*d^3 - 36*(2*c*d*x + b*d)^(3/2)*a*c^3*d^3 - 5*(2*c*d*x 
 + b*d)^(7/2)*c^2*d)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*(b^2*d^2 - 4*a*c*d^2 
- (2*c*d*x + b*d)^2)^2)
 

Mupad [B] (verification not implemented)

Time = 5.35 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {10\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {18\,c^2\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}{4\,a\,c-b^2}}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}+\frac {5\,c^2\,\sqrt {d}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{9/4}}+\frac {c^2\,\sqrt {d}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,5{}\mathrm {i}}{{\left (b^2-4\,a\,c\right )}^{9/4}} \] Input:

int((b*d + 2*c*d*x)^(1/2)/(a + b*x + c*x^2)^3,x)
 

Output:

((10*c^2*d*(b*d + 2*c*d*x)^(7/2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (18*c^2 
*d^3*(b*d + 2*c*d*x)^(3/2))/(4*a*c - b^2))/((b*d + 2*c*d*x)^4 - (b*d + 2*c 
*d*x)^2*(2*b^2*d^2 - 8*a*c*d^2) + b^4*d^4 + 16*a^2*c^2*d^4 - 8*a*b^2*c*d^4 
) + (5*c^2*d^(1/2)*atan((b^4*(b*d + 2*c*d*x)^(1/2) + 16*a^2*c^2*(b*d + 2*c 
*d*x)^(1/2) - 8*a*b^2*c*(b*d + 2*c*d*x)^(1/2))/(d^(1/2)*(b^2 - 4*a*c)^(9/4 
))))/(b^2 - 4*a*c)^(9/4) + (c^2*d^(1/2)*atan((b^4*(b*d + 2*c*d*x)^(1/2)*1i 
 + a^2*c^2*(b*d + 2*c*d*x)^(1/2)*16i - a*b^2*c*(b*d + 2*c*d*x)^(1/2)*8i)/( 
d^(1/2)*(b^2 - 4*a*c)^(9/4)))*5i)/(b^2 - 4*a*c)^(9/4)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1942, normalized size of antiderivative = 10.01 \[ \int \frac {\sqrt {b d+2 c d x}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^3,x)
 

Output:

(sqrt(d)*( - 10*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)* 
sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**2*c**2 - 
20*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*s 
qrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*b*c**2*x - 20*(4*a*c - 
b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c* 
x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*c**3*x**2 - 10*(4*a*c - b**2)**(3/4 
)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c 
 - b**2)**(1/4)*sqrt(2)))*b**2*c**2*x**2 - 20*(4*a*c - b**2)**(3/4)*sqrt(2 
)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2) 
**(1/4)*sqrt(2)))*b*c**3*x**3 - 10*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4* 
a*c - b**2)**(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqr 
t(2)))*c**4*x**4 + 10*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)** 
(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a**2*c 
**2 + 20*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) 
 + 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*b*c**2*x + 20*(4* 
a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b 
+ 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*c**3*x**2 + 10*(4*a*c - b**2) 
**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/( 
(4*a*c - b**2)**(1/4)*sqrt(2)))*b**2*c**2*x**2 + 20*(4*a*c - b**2)**(3/4)* 
sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/((4*a*...