\(\int \frac {1}{\sqrt {b d+2 c d x} (a+b x+c x^2)^3} \, dx\) [140]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac {7 c \sqrt {b d+2 c d x}}{2 \left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}-\frac {21 c^2 \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{11/4} \sqrt {d}}-\frac {21 c^2 \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{11/4} \sqrt {d}} \] Output:

-1/2*(2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)/d/(c*x^2+b*x+a)^2+7/2*c*(2*c*d*x+b*d 
)^(1/2)/(-4*a*c+b^2)^2/d/(c*x^2+b*x+a)-21*c^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)^(11/4)/d^(1/2)-21*c^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)^(11/4)/d^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) c^2 \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x) \left (b^2-7 b c x-c \left (11 a+7 c x^2\right )\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {21 i \sqrt {b+2 c x} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{11/4}}+\frac {21 i \sqrt {b+2 c x} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{11/4}}+\frac {21 i \sqrt {b+2 c x} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{\left (b^2-4 a c\right )^{11/4}}\right )}{\sqrt {d (b+2 c x)}} \] Input:

Integrate[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^3),x]
 

Output:

((1/2 + I/2)*c^2*(((-1/2 + I/2)*(b + 2*c*x)*(b^2 - 7*b*c*x - c*(11*a + 7*c 
*x^2)))/(c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) - ((21*I)*Sqrt[b + 2*c*x 
]*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c) 
^(11/4) + ((21*I)*Sqrt[b + 2*c*x]*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x])/(b^ 
2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(11/4) + ((21*I)*Sqrt[b + 2*c*x]*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)^(11/4)))/Sqrt[d*(b + 2*c*x)]
 

Rubi [A] (verified)

Time = 0.39 (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, 756, 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.

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

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {7 c \int \frac {1}{\sqrt {b d+2 c x d} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {7 c \left (-\frac {3 c \int \frac {1}{\sqrt {b d+2 c x d} \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1118

\(\displaystyle -\frac {7 c \left (-\frac {3 \int \frac {4 c d^2}{\sqrt {b d+2 c x d} \left (\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2\right )}d(b d+2 c x d)}{2 d \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {7 c \left (-\frac {6 c d \int -\frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {7 c \left (\frac {6 c d \int \frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {7 c \left (\frac {12 c d \int \frac {1}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 756

\(\displaystyle -\frac {7 c \left (\frac {12 c d \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}\right )}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {7 c \left (\frac {12 c d \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {7 c \left (\frac {12 c d \left (\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )}{b^2-4 a c}-\frac {\sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\right )}{2 \left (b^2-4 a c\right )}-\frac {\sqrt {b d+2 c d x}}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

Int[1/(Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2)^3),x]
 

Output:

-1/2*Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2) - (7*c*(-(S 
qrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)*d*(a + b*x + c*x^2))) + (12*c*d*(ArcTan[ 
Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]/(2*(b^2 - 4*a*c)^(3/4)* 
d^(3/2)) + ArcTanh[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]/(2*( 
b^2 - 4*a*c)^(3/4)*d^(3/2))))/(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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   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 = 1.29 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.75

method	result	size
pseudoelliptic	\(\frac {\sqrt {d \left (2 c x +b \right )}}{8 d \left (a +x \left (c x +b \right )\right )^{2} \left (a c -\frac {b^{2}}{4}\right )}+\frac {7 c \sqrt {d \left (2 c x +b \right )}}{32 d \left (c \,x^{2}+b x +a \right ) \left (a c -\frac {b^{2}}{4}\right )^{2}}+\frac {21 c^{2} d^{5} \sqrt {2}\, \ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )}{4 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {11}{4}}}+\frac {21 c^{2} d^{5} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {11}{4}}}+\frac {21 c^{2} d^{5} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )}{2 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {11}{4}}}\)	\(340\)
derivativedivides	\(64 d^{5} c^{2} \left (\frac {\sqrt {2 c d x +b d}}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {\frac {7 \sqrt {2 c d x +b d}}{32 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {7}{4}}}}{4 a \,d^{2} c -b^{2} d^{2}}\right )\)	\(377\)
default	\(64 d^{5} c^{2} \left (\frac {\sqrt {2 c d x +b d}}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {\frac {7 \sqrt {2 c d x +b d}}{32 \left (4 a \,d^{2} c -b^{2} d^{2}\right ) \left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )}+\frac {21 \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{256 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {7}{4}}}}{4 a \,d^{2} c -b^{2} d^{2}}\right )\)	\(377\)

Input:

int(1/(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))^(1/2)/d/(a+x*(c*x+b))^2/(a*c-1/4*b^2)+7/32*c*(d*(2*c*x+b 
))^(1/2)/d/(c*x^2+b*x+a)/(a*c-1/4*b^2)^2+21/4*c^2*d^5/(d^2*(4*a*c-b^2))^(1 
1/4)*2^(1/2)*ln(((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))/((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)))+21/2*c^2*d^5/(d^2*(4*a*c- 
b^2))^(11/4)*2^(1/2)*arctan(2^(1/2)/(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^ 
(1/2)+1)+21/2*c^2*d^5/(d^2*(4*a*c-b^2))^(11/4)*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 = 1991, normalized size of antiderivative = 10.26 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(21*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3* 
c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^ 
5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2) 
*d)*(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84 
480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^ 
7*b^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^ 
2*c^10 - 4194304*a^11*c^11)*d^2))^(1/4)*log(21*sqrt(2*c*d*x + b*d)*c^2 + 2 
1*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c^8/((b^22 - 44*a*b^20 
*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a 
^5*b^12*c^5 + 1892352*a^6*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^ 
6*c^8 - 14417920*a^9*b^4*c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11) 
*d^2))^(1/4)*d) + 21*(I*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^4 + 2*I*( 
b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + I*(b^6 - 6*a*b^4*c + 32*a^3*c^ 
3)*d*x^2 + 2*I*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + I*(a^2*b^4 - 8*a 
^3*b^2*c + 16*a^4*c^2)*d)*(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 1 
0560*a^3*b^16*c^3 + 84480*a^4*b^14*c^4 - 473088*a^5*b^12*c^5 + 1892352*a^6 
*b^10*c^6 - 5406720*a^7*b^8*c^7 + 10813440*a^8*b^6*c^8 - 14417920*a^9*b^4* 
c^9 + 11534336*a^10*b^2*c^10 - 4194304*a^11*c^11)*d^2))^(1/4)*log(21*sqrt( 
2*c*d*x + b*d)*c^2 + 21*I*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3) 
*(c^8/((b^22 - 44*a*b^20*c + 880*a^2*b^18*c^2 - 10560*a^3*b^16*c^3 + 84...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(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 = 0.15 (sec) , antiderivative size = 645, normalized size of antiderivative = 3.32 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=-\frac {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{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 {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{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 {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{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 {21 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{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 (11 \, \sqrt {2 \, c d x + b d} b^{2} c^{2} d^{3} - 44 \, \sqrt {2 \, c d x + b d} a c^{3} d^{3} - 7 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{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(1/(2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

-21*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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*sqr 
t(2)*a^3*c^3*d) - 21*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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) - 21/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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 + 4 
8*sqrt(2)*a^2*b^2*c^2*d - 64*sqrt(2)*a^3*c^3*d) + 21/2*(-b^2*d^2 + 4*a*c*d 
^2)^(1/4)*c^2*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqr 
t(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*(11*sqrt 
(2*c*d*x + b*d)*b^2*c^2*d^3 - 44*sqrt(2*c*d*x + b*d)*a*c^3*d^3 - 7*(2*c*d* 
x + b*d)^(5/2)*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 = 5.27 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {14\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {22\,c^2\,d^3\,\sqrt {b\,d+2\,c\,d\,x}}{4\,a\,c-b^2}}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}-\frac {21\,c^2\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{15/4}}{\sqrt {d}\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}\right )}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{11/4}}-\frac {21\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{15/4}}{\sqrt {d}\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}\right )}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{11/4}} \] Input:

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

Output:

((14*c^2*d*(b*d + 2*c*d*x)^(5/2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (22*c^2 
*d^3*(b*d + 2*c*d*x)^(1/2))/(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 
) - (21*c^2*atan(((b*d + 2*c*d*x)^(1/2)*(b^2 - 4*a*c)^(15/4))/(d^(1/2)*(b^ 
8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c))))/(d^(1/ 
2)*(b^2 - 4*a*c)^(11/4)) - (21*c^2*atanh(((b*d + 2*c*d*x)^(1/2)*(b^2 - 4*a 
*c)^(15/4))/(d^(1/2)*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 
 - 16*a*b^6*c))))/(d^(1/2)*(b^2 - 4*a*c)^(11/4))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1892, normalized size of antiderivative = 9.75 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(d)*( - 42*(4*a*c - b**2)**(1/4)*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 - 
84*(4*a*c - b**2)**(1/4)*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 - 84*(4*a*c - 
b**2)**(1/4)*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 - 42*(4*a*c - b**2)**(1/4 
)*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 - 84*(4*a*c - b**2)**(1/4)*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 - 42*(4*a*c - b**2)**(1/4)*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 + 42*(4*a*c - b**2)**(1/4)*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 + 84*(4*a*c - b**2)**(1/4)*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 + 84*(4* 
a*c - b**2)**(1/4)*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 + 42*(4*a*c - b**2) 
**(1/4)*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 + 84*(4*a*c - b**2)**(1/4)* 
sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/((4*a*...