\(\int \frac {(b d+2 c d x)^{11/2}}{(a+b x+c x^2)^3} \, dx\) [134]

Optimal result

Integrand size = 26, antiderivative size = 193 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=90 c^2 d^5 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^3 (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )}-45 c^2 \sqrt [4]{b^2-4 a c} d^{11/2} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-45 c^2 \sqrt [4]{b^2-4 a c} d^{11/2} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \] Output:

90*c^2*d^5*(2*c*d*x+b*d)^(1/2)-1/2*d*(2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^2-9 
*c*d^3*(2*c*d*x+b*d)^(5/2)/(2*c*x^2+2*b*x+2*a)-45*c^2*(-4*a*c+b^2)^(1/4)*d 
^(11/2)*arctan((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2))-45*c^2*(-4* 
a*c+b^2)^(1/4)*d^(11/2)*arctanh((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^( 
1/2))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.54 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=\left (\frac {1}{2}+\frac {i}{2}\right ) c^2 (d (b+2 c x))^{11/2} \left (\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (45 b^4-360 a b^2 c+720 a^2 c^2-81 b^2 (b+2 c x)^2+324 a c (b+2 c x)^2+32 (b+2 c x)^4\right )}{c^2 (b+2 c x)^5 (a+x (b+c x))^2}-\frac {45 i \sqrt [4]{b^2-4 a c} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}+\frac {45 i \sqrt [4]{b^2-4 a c} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}+\frac {45 i \sqrt [4]{b^2-4 a c} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{11/2}}\right ) \] Input:

Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^3,x]
 

Output:

(1/2 + I/2)*c^2*(d*(b + 2*c*x))^(11/2)*(((1/8 - I/8)*(45*b^4 - 360*a*b^2*c 
 + 720*a^2*c^2 - 81*b^2*(b + 2*c*x)^2 + 324*a*c*(b + 2*c*x)^2 + 32*(b + 2* 
c*x)^4))/(c^2*(b + 2*c*x)^5*(a + x*(b + c*x))^2) - ((45*I)*(b^2 - 4*a*c)^( 
1/4)*ArcTan[1 - ((1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x 
)^(11/2) + ((45*I)*(b^2 - 4*a*c)^(1/4)*ArcTan[1 + ((1 + I)*Sqrt[b + 2*c*x] 
)/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(11/2) + ((45*I)*(b^2 - 4*a*c)^(1/4)*A 
rcTanh[((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*c*x)^(11/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1110, 1110, 1116, 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.

\(\displaystyle \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1110

\(\displaystyle \frac {9}{2} c d^2 \int \frac {(b d+2 c x d)^{7/2}}{\left (c x^2+b x+a\right )^2}dx-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1110

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \int \frac {(b d+2 c x d)^{3/2}}{c x^2+b x+a}dx-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \left (c x^2+b x+a\right )}dx+4 d \sqrt {b d+2 c d x}\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1118

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (\frac {d \left (b^2-4 a c\right ) \int \frac {4 c d^2}{\sqrt {b d+2 c x d} \left (\left (4 a-\frac {b^2}{c}\right ) c d^2+(b d+2 c x d)^2\right )}d(b d+2 c x d)}{2 c}+4 d \sqrt {b d+2 c d x}\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (2 d^3 \left (b^2-4 a c\right ) \int -\frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)+4 d \sqrt {b d+2 c d x}\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (4 d \sqrt {b d+2 c d x}-2 d^3 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \left (\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2\right )}d(b d+2 c x d)\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \int \frac {1}{\left (b^2-4 a c\right ) d^2-(b d+2 c x d)^2}d\sqrt {b d+2 c x d}\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\int \frac {1}{b d+2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}\right )\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}}{2 d \sqrt {b^2-4 a c}}+\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {9}{2} c d^2 \left (5 c d^2 \left (4 d \sqrt {b d+2 c d x}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 d^{3/2} \left (b^2-4 a c\right )^{3/4}}\right )\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}\right )-\frac {d (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )^2}\)

Input:

Int[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*(d*(b*d + 2*c*d*x)^(9/2))/(a + b*x + c*x^2)^2 + (9*c*d^2*(-((d*(b*d + 
 2*c*d*x)^(5/2))/(a + b*x + c*x^2)) + 5*c*d^2*(4*d*Sqrt[b*d + 2*c*d*x] - 4 
*(b^2 - 4*a*c)*d^3*(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))))))/2
 

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 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])
 

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. \(358\) vs. \(2(167)=334\).

Time = 1.52 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.86

method	result	size
derivativedivides	\(64 c^{2} d^{5} \left (\sqrt {2 c d x +b d}-d^{2} \left (\frac {16 \left (-\frac {17 a c}{128}+\frac {17 b^{2}}{512}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {13}{32} a^{2} c^{2} d^{2}+\frac {13}{64} a \,b^{2} c \,d^{2}-\frac {13}{512} b^{4} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {45 \left (\frac {a c}{4}-\frac {b^{2}}{16}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\)	\(359\)
default	\(64 c^{2} d^{5} \left (\sqrt {2 c d x +b d}-d^{2} \left (\frac {16 \left (-\frac {17 a c}{128}+\frac {17 b^{2}}{512}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}+16 \left (-\frac {13}{32} a^{2} c^{2} d^{2}+\frac {13}{64} a \,b^{2} c \,d^{2}-\frac {13}{512} b^{4} d^{2}\right ) \sqrt {2 c d x +b d}}{\left (\left (2 c d x +b d \right )^{2}+4 a \,d^{2} c -b^{2} d^{2}\right )^{2}}+\frac {45 \left (\frac {a c}{4}-\frac {b^{2}}{16}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\)	\(359\)
pseudoelliptic	\(\frac {45 d^{3} \left (\frac {17 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (4 a c -b^{2}\right ) \left (d \left (2 c x +b \right )\right )^{\frac {5}{2}}}{90}+d^{2} \left (8 \left (\frac {32 c^{4} x^{4}}{45}+\frac {64 x^{2} \left (b x +a \right ) c^{3}}{45}+\left (\frac {64}{45} a b x +\frac {32}{45} b^{2} x^{2}+a^{2}\right ) c^{2}-\frac {13 c a \,b^{2}}{90}+\frac {13 b^{4}}{720}\right ) \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}-c^{2} d^{2} \sqrt {2}\, \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2} \left (\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )\right )\right )\right )}{4 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (c \,x^{2}+b x +a \right )^{2}}\)	\(380\)

Input:

int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

64*c^2*d^5*((2*c*d*x+b*d)^(1/2)-d^2*(16*((-17/128*a*c+17/512*b^2)*(2*c*d*x 
+b*d)^(5/2)+(-13/32*a^2*c^2*d^2+13/64*a*b^2*c*d^2-13/512*b^4*d^2)*(2*c*d*x 
+b*d)^(1/2))/((2*c*d*x+b*d)^2+4*a*d^2*c-b^2*d^2)^2+45/16*(1/4*a*c-1/16*b^2 
)/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*(ln((2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^( 
1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x+b*d-( 
4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^( 
1/2)))+2*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)-2 
*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1))))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.86 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (45 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} + 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (-i \, c^{2} x^{4} - 2 i \, b c x^{3} - 2 i \, a b x - i \, {\left (b^{2} + 2 \, a c\right )} x^{2} - i \, a^{2}\right )} \log \left (45 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} + 45 i \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (i \, c^{2} x^{4} + 2 i \, b c x^{3} + 2 i \, a b x + i \, {\left (b^{2} + 2 \, a c\right )} x^{2} + i \, a^{2}\right )} \log \left (45 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} - 45 i \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (45 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} - 45 \, \left ({\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{22}\right )^{\frac {1}{4}}\right ) - {\left (128 \, c^{4} d^{5} x^{4} + 256 \, b c^{3} d^{5} x^{3} + 3 \, {\left (37 \, b^{2} c^{2} + 108 \, a c^{3}\right )} d^{5} x^{2} - {\left (17 \, b^{3} c - 324 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 9 \, a b^{2} c - 180 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \] Input:

integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

-1/2*(45*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + 
 (b^2 + 2*a*c)*x^2 + a^2)*log(45*sqrt(2*c*d*x + b*d)*c^2*d^5 + 45*((b^2*c^ 
8 - 4*a*c^9)*d^22)^(1/4)) - 45*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)*(-I*c^2*x^ 
4 - 2*I*b*c*x^3 - 2*I*a*b*x - I*(b^2 + 2*a*c)*x^2 - I*a^2)*log(45*sqrt(2*c 
*d*x + b*d)*c^2*d^5 + 45*I*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)) - 45*((b^2*c^ 
8 - 4*a*c^9)*d^22)^(1/4)*(I*c^2*x^4 + 2*I*b*c*x^3 + 2*I*a*b*x + I*(b^2 + 2 
*a*c)*x^2 + I*a^2)*log(45*sqrt(2*c*d*x + b*d)*c^2*d^5 - 45*I*((b^2*c^8 - 4 
*a*c^9)*d^22)^(1/4)) - 45*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)*(c^2*x^4 + 2*b* 
c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*log(45*sqrt(2*c*d*x + b*d)*c^2* 
d^5 - 45*((b^2*c^8 - 4*a*c^9)*d^22)^(1/4)) - (128*c^4*d^5*x^4 + 256*b*c^3* 
d^5*x^3 + 3*(37*b^2*c^2 + 108*a*c^3)*d^5*x^2 - (17*b^3*c - 324*a*b*c^2)*d^ 
5*x - (b^4 + 9*a*b^2*c - 180*a^2*c^2)*d^5)*sqrt(2*c*d*x + b*d))/(c^2*x^4 + 
 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((2*c*d*x+b*d)**(11/2)/(c*x**2+b*x+a)**3,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^(11/2)/(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. 521 vs. \(2 (163) = 326\).

Time = 0.16 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.70 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {45}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {45}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {45}{4} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {45}{4} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{5} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 64 \, \sqrt {2 \, c d x + b d} c^{2} d^{5} + \frac {2 \, {\left (13 \, \sqrt {2 \, c d x + b d} b^{4} c^{2} d^{9} - 104 \, \sqrt {2 \, c d x + b d} a b^{2} c^{3} d^{9} + 208 \, \sqrt {2 \, c d x + b d} a^{2} c^{4} d^{9} - 17 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} c^{2} d^{7} + 68 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c^{3} d^{7}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \] Input:

integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

-45/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d^5*arctan(1/2*sqrt(2)*(sqr 
t(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)) - 45/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d^5*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)) - 45/4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4 
)*c^2*d^5*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)) + 45/4*sqrt(2)*(-b^2*d^2 + 4*a* 
c*d^2)^(1/4)*c^2*d^5*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)) + 64*sqrt(2*c*d*x + 
b*d)*c^2*d^5 + 2*(13*sqrt(2*c*d*x + b*d)*b^4*c^2*d^9 - 104*sqrt(2*c*d*x + 
b*d)*a*b^2*c^3*d^9 + 208*sqrt(2*c*d*x + b*d)*a^2*c^4*d^9 - 17*(2*c*d*x + b 
*d)^(5/2)*b^2*c^2*d^7 + 68*(2*c*d*x + b*d)^(5/2)*a*c^3*d^7)/(b^2*d^2 - 4*a 
*c*d^2 - (2*c*d*x + b*d)^2)^2
 

Mupad [B] (verification not implemented)

Time = 5.19 (sec) , antiderivative size = 720, normalized size of antiderivative = 3.73 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (136\,a\,c^3\,d^7-34\,b^2\,c^2\,d^7\right )+\sqrt {b\,d+2\,c\,d\,x}\,\left (416\,a^2\,c^4\,d^9-208\,a\,b^2\,c^3\,d^9+26\,b^4\,c^2\,d^9\right )}{{\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}+64\,c^2\,d^5\,\sqrt {b\,d+2\,c\,d\,x}-45\,c^2\,d^{11/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}-c^2\,d^{11/2}\,\mathrm {atan}\left (\frac {\frac {c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )-\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,45{}\mathrm {i}}{2}+\frac {c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )+\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,45{}\mathrm {i}}{2}}{\frac {45\,c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )-\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}}{2}-\frac {45\,c^2\,d^{11/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (518400\,a^2\,c^6\,d^{14}-259200\,a\,b^2\,c^5\,d^{14}+32400\,b^4\,c^4\,d^{14}\right )+\frac {45\,c^2\,d^{11/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (23040\,a^2\,c^4\,d^9-11520\,a\,b^2\,c^3\,d^9+1440\,b^4\,c^2\,d^9\right )}{2}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}}{2}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,45{}\mathrm {i} \] Input:

int((b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^3,x)
 

Output:

((b*d + 2*c*d*x)^(5/2)*(136*a*c^3*d^7 - 34*b^2*c^2*d^7) + (b*d + 2*c*d*x)^ 
(1/2)*(416*a^2*c^4*d^9 + 26*b^4*c^2*d^9 - 208*a*b^2*c^3*d^9))/((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) + 64*c^2*d^5*(b*d + 2*c*d*x)^(1/2) - 45*c^2*d^(11/2)* 
atan((b*d + 2*c*d*x)^(1/2)/(d^(1/2)*(b^2 - 4*a*c)^(1/4)))*(b^2 - 4*a*c)^(1 
/4) - c^2*d^(11/2)*atan(((c^2*d^(11/2)*((b*d + 2*c*d*x)^(1/2)*(518400*a^2* 
c^6*d^14 + 32400*b^4*c^4*d^14 - 259200*a*b^2*c^5*d^14) - (45*c^2*d^(11/2)* 
(b^2 - 4*a*c)^(1/4)*(23040*a^2*c^4*d^9 + 1440*b^4*c^2*d^9 - 11520*a*b^2*c^ 
3*d^9))/2)*(b^2 - 4*a*c)^(1/4)*45i)/2 + (c^2*d^(11/2)*((b*d + 2*c*d*x)^(1/ 
2)*(518400*a^2*c^6*d^14 + 32400*b^4*c^4*d^14 - 259200*a*b^2*c^5*d^14) + (4 
5*c^2*d^(11/2)*(b^2 - 4*a*c)^(1/4)*(23040*a^2*c^4*d^9 + 1440*b^4*c^2*d^9 - 
 11520*a*b^2*c^3*d^9))/2)*(b^2 - 4*a*c)^(1/4)*45i)/2)/((45*c^2*d^(11/2)*(( 
b*d + 2*c*d*x)^(1/2)*(518400*a^2*c^6*d^14 + 32400*b^4*c^4*d^14 - 259200*a* 
b^2*c^5*d^14) - (45*c^2*d^(11/2)*(b^2 - 4*a*c)^(1/4)*(23040*a^2*c^4*d^9 + 
1440*b^4*c^2*d^9 - 11520*a*b^2*c^3*d^9))/2)*(b^2 - 4*a*c)^(1/4))/2 - (45*c 
^2*d^(11/2)*((b*d + 2*c*d*x)^(1/2)*(518400*a^2*c^6*d^14 + 32400*b^4*c^4*d^ 
14 - 259200*a*b^2*c^5*d^14) + (45*c^2*d^(11/2)*(b^2 - 4*a*c)^(1/4)*(23040* 
a^2*c^4*d^9 + 1440*b^4*c^2*d^9 - 11520*a*b^2*c^3*d^9))/2)*(b^2 - 4*a*c)^(1 
/4))/2))*(b^2 - 4*a*c)^(1/4)*45i
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1735, normalized size of antiderivative = 8.99 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^3,x)
 

Output:

(sqrt(d)*d**5*(90*(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 
+ 180*(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 + 180*(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 + 90*(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 + 180*(4*a*c - b**2)**(1/4)*s 
qrt(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 + 90*(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)))*c**4*x**4 - 90*(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 - 180*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*s 
qrt(2) + 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*b*c**2*x - 
180*(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 - 90*(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 - 180*(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)...