\(\int \frac {1}{(b d+2 c d x)^4 (a+b x+c x^2)^{5/2}} \, dx\) [219]

Optimal result

Integrand size = 26, antiderivative size = 162 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {16 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}+\frac {256 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {512 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x)} \] Output:

-2/3/(-4*a*c+b^2)/d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(3/2)+16*c/(-4*a*c+b^2)^2/ 
d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)+256/3*c^2*(c*x^2+b*x+a)^(1/2)/(-4*a*c+ 
b^2)^3/d^4/(2*c*x+b)^3+512/3*c^2*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^4/d^4/(2 
*c*x+b)
 

Mathematica [A] (verified)

Time = 10.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-b^6+24 b^5 c x+64 b^3 c^2 x \left (9 a+28 c x^2\right )+12 b^4 c \left (3 a+34 c x^2\right )+384 b c^3 x \left (a^2+8 a c x^2+8 c^2 x^4\right )+48 b^2 c^2 \left (3 a^2+44 a c x^2+72 c^2 x^4\right )+64 c^3 \left (-a^3+6 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right )\right )}{3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x)^3 (a+x (b+c x))^{3/2}} \] Input:

Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

(2*(-b^6 + 24*b^5*c*x + 64*b^3*c^2*x*(9*a + 28*c*x^2) + 12*b^4*c*(3*a + 34 
*c*x^2) + 384*b*c^3*x*(a^2 + 8*a*c*x^2 + 8*c^2*x^4) + 48*b^2*c^2*(3*a^2 + 
44*a*c*x^2 + 72*c^2*x^4) + 64*c^3*(-a^3 + 6*a^2*c*x^2 + 24*a*c^2*x^4 + 16* 
c^3*x^6)))/(3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)^3*(a + x*(b + c*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1111, 27, 1111, 1117, 1106}

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.

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

\(\Big \downarrow \) 1111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1111

\(\displaystyle -\frac {8 c \left (-\frac {16 c \int \frac {1}{(b+2 c x)^4 \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\right )}{d^4 \left (b^2-4 a c\right )}-\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1117

\(\displaystyle -\frac {8 c \left (-\frac {16 c \left (\frac {2 \int \frac {1}{(b+2 c x)^2 \sqrt {c x^2+b x+a}}dx}{3 \left (b^2-4 a c\right )}+\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) (b+2 c x)^3}\right )}{b^2-4 a c}-\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\right )}{d^4 \left (b^2-4 a c\right )}-\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1106

\(\displaystyle -\frac {8 c \left (-\frac {16 c \left (\frac {4 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) (b+2 c x)^3}\right )}{b^2-4 a c}-\frac {2}{\left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\right )}{d^4 \left (b^2-4 a c\right )}-\frac {2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}\)

Input:

Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

-2/(3*(b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2)) - (8*c*(-2/ 
((b^2 - 4*a*c)*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2]) - (16*c*((2*Sqrt[a + b 
*x + c*x^2])/(3*(b^2 - 4*a*c)*(b + 2*c*x)^3) + (4*Sqrt[a + b*x + c*x^2])/( 
3*(b^2 - 4*a*c)^2*(b + 2*c*x))))/(b^2 - 4*a*c)))/((b^2 - 4*a*c)*d^4)
 

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 1106

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] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 
0] && EqQ[m + 2*p + 3, 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 1117

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15

method	result	size
trager	\(-\frac {2 \left (-1024 x^{6} c^{6}-3072 x^{5} b \,c^{5}-1536 a \,c^{5} x^{4}-3456 x^{4} b^{2} c^{4}-3072 a b \,c^{4} x^{3}-1792 b^{3} c^{3} x^{3}-384 a^{2} c^{4} x^{2}-2112 a \,b^{2} c^{3} x^{2}-408 c^{2} x^{2} b^{4}-384 a^{2} b \,c^{3} x -576 x a \,b^{3} c^{2}-24 x c \,b^{5}+64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}-36 a \,b^{4} c +b^{6}\right )}{3 d^{4} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )^{4} \left (2 c x +b \right )^{3}}\)	\(187\)
gosper	\(-\frac {2 \left (-1024 x^{6} c^{6}-3072 x^{5} b \,c^{5}-1536 a \,c^{5} x^{4}-3456 x^{4} b^{2} c^{4}-3072 a b \,c^{4} x^{3}-1792 b^{3} c^{3} x^{3}-384 a^{2} c^{4} x^{2}-2112 a \,b^{2} c^{3} x^{2}-408 c^{2} x^{2} b^{4}-384 a^{2} b \,c^{3} x -576 x a \,b^{3} c^{2}-24 x c \,b^{5}+64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}-36 a \,b^{4} c +b^{6}\right )}{3 \left (2 c x +b \right )^{3} d^{4} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\)	\(218\)
orering	\(-\frac {2 \left (-1024 x^{6} c^{6}-3072 x^{5} b \,c^{5}-1536 a \,c^{5} x^{4}-3456 x^{4} b^{2} c^{4}-3072 a b \,c^{4} x^{3}-1792 b^{3} c^{3} x^{3}-384 a^{2} c^{4} x^{2}-2112 a \,b^{2} c^{3} x^{2}-408 c^{2} x^{2} b^{4}-384 a^{2} b \,c^{3} x -576 x a \,b^{3} c^{2}-24 x c \,b^{5}+64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}-36 a \,b^{4} c +b^{6}\right ) \left (2 c x +b \right )}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (2 c d x +b d \right )^{4}}\)	\(224\)
default	\(\frac {-\frac {4 c}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3} \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}-\frac {8 c^{2} \left (-\frac {4 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right ) \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}-\frac {16 c^{2} \left (\frac {4 c \left (x +\frac {b}{2 c}\right )}{3 \left (4 a c -b^{2}\right ) \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}+\frac {32 c^{2} \left (x +\frac {b}{2 c}\right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{16 d^{4} c^{4}}\)	\(264\)

Input:

int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/3/d^4*(-1024*c^6*x^6-3072*b*c^5*x^5-1536*a*c^5*x^4-3456*b^2*c^4*x^4-307 
2*a*b*c^4*x^3-1792*b^3*c^3*x^3-384*a^2*c^4*x^2-2112*a*b^2*c^3*x^2-408*b^4* 
c^2*x^2-384*a^2*b*c^3*x-576*a*b^3*c^2*x-24*b^5*c*x+64*a^3*c^3-144*a^2*b^2* 
c^2-36*a*b^4*c+b^6)/(c*x^2+b*x+a)^(3/2)/(4*a*c-b^2)^4/(2*c*x+b)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (148) = 296\).

Time = 3.68 (sec) , antiderivative size = 677, normalized size of antiderivative = 4.18 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (1024 \, c^{6} x^{6} + 3072 \, b c^{5} x^{5} - b^{6} + 36 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 384 \, {\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 256 \, {\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 24 \, {\left (17 \, b^{4} c^{2} + 88 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \, {\left (b^{5} c + 24 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{8} c^{5} - 16 \, a b^{6} c^{6} + 96 \, a^{2} b^{4} c^{7} - 256 \, a^{3} b^{2} c^{8} + 256 \, a^{4} c^{9}\right )} d^{4} x^{7} + 28 \, {\left (b^{9} c^{4} - 16 \, a b^{7} c^{5} + 96 \, a^{2} b^{5} c^{6} - 256 \, a^{3} b^{3} c^{7} + 256 \, a^{4} b c^{8}\right )} d^{4} x^{6} + 2 \, {\left (19 \, b^{10} c^{3} - 296 \, a b^{8} c^{4} + 1696 \, a^{2} b^{6} c^{5} - 4096 \, a^{3} b^{4} c^{6} + 2816 \, a^{4} b^{2} c^{7} + 2048 \, a^{5} c^{8}\right )} d^{4} x^{5} + 5 \, {\left (5 \, b^{11} c^{2} - 72 \, a b^{9} c^{3} + 352 \, a^{2} b^{7} c^{4} - 512 \, a^{3} b^{5} c^{5} - 768 \, a^{4} b^{3} c^{6} + 2048 \, a^{5} b c^{7}\right )} d^{4} x^{4} + 4 \, {\left (2 \, b^{12} c - 23 \, a b^{10} c^{2} + 50 \, a^{2} b^{8} c^{3} + 320 \, a^{3} b^{6} c^{4} - 1600 \, a^{4} b^{4} c^{5} + 1792 \, a^{5} b^{2} c^{6} + 512 \, a^{6} c^{7}\right )} d^{4} x^{3} + {\left (b^{13} - 2 \, a b^{11} c - 116 \, a^{2} b^{9} c^{2} + 896 \, a^{3} b^{7} c^{3} - 2176 \, a^{4} b^{5} c^{4} + 512 \, a^{5} b^{3} c^{5} + 3072 \, a^{6} b c^{6}\right )} d^{4} x^{2} + 2 \, {\left (a b^{12} - 13 \, a^{2} b^{10} c + 48 \, a^{3} b^{8} c^{2} + 32 \, a^{4} b^{6} c^{3} - 512 \, a^{5} b^{4} c^{4} + 768 \, a^{6} b^{2} c^{5}\right )} d^{4} x + {\left (a^{2} b^{11} - 16 \, a^{3} b^{9} c + 96 \, a^{4} b^{7} c^{2} - 256 \, a^{5} b^{5} c^{3} + 256 \, a^{6} b^{3} c^{4}\right )} d^{4}\right )}} \] Input:

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

2/3*(1024*c^6*x^6 + 3072*b*c^5*x^5 - b^6 + 36*a*b^4*c + 144*a^2*b^2*c^2 - 
64*a^3*c^3 + 384*(9*b^2*c^4 + 4*a*c^5)*x^4 + 256*(7*b^3*c^3 + 12*a*b*c^4)* 
x^3 + 24*(17*b^4*c^2 + 88*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 24*(b^5*c + 24*a*b 
^3*c^2 + 16*a^2*b*c^3)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^8*c^5 - 16*a*b^6*c^6 
 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*d^4*x^7 + 28*(b^9*c^4 - 
 16*a*b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*d^4*x^6 
+ 2*(19*b^10*c^3 - 296*a*b^8*c^4 + 1696*a^2*b^6*c^5 - 4096*a^3*b^4*c^6 + 2 
816*a^4*b^2*c^7 + 2048*a^5*c^8)*d^4*x^5 + 5*(5*b^11*c^2 - 72*a*b^9*c^3 + 3 
52*a^2*b^7*c^4 - 512*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2048*a^5*b*c^7)*d^4*x 
^4 + 4*(2*b^12*c - 23*a*b^10*c^2 + 50*a^2*b^8*c^3 + 320*a^3*b^6*c^4 - 1600 
*a^4*b^4*c^5 + 1792*a^5*b^2*c^6 + 512*a^6*c^7)*d^4*x^3 + (b^13 - 2*a*b^11* 
c - 116*a^2*b^9*c^2 + 896*a^3*b^7*c^3 - 2176*a^4*b^5*c^4 + 512*a^5*b^3*c^5 
 + 3072*a^6*b*c^6)*d^4*x^2 + 2*(a*b^12 - 13*a^2*b^10*c + 48*a^3*b^8*c^2 + 
32*a^4*b^6*c^3 - 512*a^5*b^4*c^4 + 768*a^6*b^2*c^5)*d^4*x + (a^2*b^11 - 16 
*a^3*b^9*c + 96*a^4*b^7*c^2 - 256*a^5*b^5*c^3 + 256*a^6*b^3*c^4)*d^4)
 

Sympy [F]

\[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\int \frac {1}{a^{2} b^{4} \sqrt {a + b x + c x^{2}} + 8 a^{2} b^{3} c x \sqrt {a + b x + c x^{2}} + 24 a^{2} b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 a^{2} b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 a^{2} c^{4} x^{4} \sqrt {a + b x + c x^{2}} + 2 a b^{5} x \sqrt {a + b x + c x^{2}} + 18 a b^{4} c x^{2} \sqrt {a + b x + c x^{2}} + 64 a b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 112 a b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 96 a b c^{4} x^{5} \sqrt {a + b x + c x^{2}} + 32 a c^{5} x^{6} \sqrt {a + b x + c x^{2}} + b^{6} x^{2} \sqrt {a + b x + c x^{2}} + 10 b^{5} c x^{3} \sqrt {a + b x + c x^{2}} + 41 b^{4} c^{2} x^{4} \sqrt {a + b x + c x^{2}} + 88 b^{3} c^{3} x^{5} \sqrt {a + b x + c x^{2}} + 104 b^{2} c^{4} x^{6} \sqrt {a + b x + c x^{2}} + 64 b c^{5} x^{7} \sqrt {a + b x + c x^{2}} + 16 c^{6} x^{8} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \] Input:

integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Integral(1/(a**2*b**4*sqrt(a + b*x + c*x**2) + 8*a**2*b**3*c*x*sqrt(a + b* 
x + c*x**2) + 24*a**2*b**2*c**2*x**2*sqrt(a + b*x + c*x**2) + 32*a**2*b*c* 
*3*x**3*sqrt(a + b*x + c*x**2) + 16*a**2*c**4*x**4*sqrt(a + b*x + c*x**2) 
+ 2*a*b**5*x*sqrt(a + b*x + c*x**2) + 18*a*b**4*c*x**2*sqrt(a + b*x + c*x* 
*2) + 64*a*b**3*c**2*x**3*sqrt(a + b*x + c*x**2) + 112*a*b**2*c**3*x**4*sq 
rt(a + b*x + c*x**2) + 96*a*b*c**4*x**5*sqrt(a + b*x + c*x**2) + 32*a*c**5 
*x**6*sqrt(a + b*x + c*x**2) + b**6*x**2*sqrt(a + b*x + c*x**2) + 10*b**5* 
c*x**3*sqrt(a + b*x + c*x**2) + 41*b**4*c**2*x**4*sqrt(a + b*x + c*x**2) + 
 88*b**3*c**3*x**5*sqrt(a + b*x + c*x**2) + 104*b**2*c**4*x**6*sqrt(a + b* 
x + c*x**2) + 64*b*c**5*x**7*sqrt(a + b*x + c*x**2) + 16*c**6*x**8*sqrt(a 
+ b*x + c*x**2)), x)/d**4
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(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. 1389 vs. \(2 (148) = 296\).

Time = 0.46 (sec) , antiderivative size = 1389, normalized size of antiderivative = 8.57 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

2/3*(2*(16*(2*(b^14*c^3*d^8 - 28*a*b^12*c^4*d^8 + 336*a^2*b^10*c^5*d^8 - 2 
240*a^3*b^8*c^6*d^8 + 8960*a^4*b^6*c^7*d^8 - 21504*a^5*b^4*c^8*d^8 + 28672 
*a^6*b^2*c^9*d^8 - 16384*a^7*c^10*d^8)*x/(b^22*d^12 - 44*a*b^20*c*d^12 + 8 
80*a^2*b^18*c^2*d^12 - 10560*a^3*b^16*c^3*d^12 + 84480*a^4*b^14*c^4*d^12 - 
 473088*a^5*b^12*c^5*d^12 + 1892352*a^6*b^10*c^6*d^12 - 5406720*a^7*b^8*c^ 
7*d^12 + 10813440*a^8*b^6*c^8*d^12 - 14417920*a^9*b^4*c^9*d^12 + 11534336* 
a^10*b^2*c^10*d^12 - 4194304*a^11*c^11*d^12) + 3*(b^15*c^2*d^8 - 28*a*b^13 
*c^3*d^8 + 336*a^2*b^11*c^4*d^8 - 2240*a^3*b^9*c^5*d^8 + 8960*a^4*b^7*c^6* 
d^8 - 21504*a^5*b^5*c^7*d^8 + 28672*a^6*b^3*c^8*d^8 - 16384*a^7*b*c^9*d^8) 
/(b^22*d^12 - 44*a*b^20*c*d^12 + 880*a^2*b^18*c^2*d^12 - 10560*a^3*b^16*c^ 
3*d^12 + 84480*a^4*b^14*c^4*d^12 - 473088*a^5*b^12*c^5*d^12 + 1892352*a^6* 
b^10*c^6*d^12 - 5406720*a^7*b^8*c^7*d^12 + 10813440*a^8*b^6*c^8*d^12 - 144 
17920*a^9*b^4*c^9*d^12 + 11534336*a^10*b^2*c^10*d^12 - 4194304*a^11*c^11*d 
^12))*x + 3*(5*b^16*c*d^8 - 128*a*b^14*c^2*d^8 + 1344*a^2*b^12*c^3*d^8 - 7 
168*a^3*b^10*c^4*d^8 + 17920*a^4*b^8*c^5*d^8 - 114688*a^6*b^4*c^7*d^8 + 26 
2144*a^7*b^2*c^8*d^8 - 196608*a^8*c^9*d^8)/(b^22*d^12 - 44*a*b^20*c*d^12 + 
 880*a^2*b^18*c^2*d^12 - 10560*a^3*b^16*c^3*d^12 + 84480*a^4*b^14*c^4*d^12 
 - 473088*a^5*b^12*c^5*d^12 + 1892352*a^6*b^10*c^6*d^12 - 5406720*a^7*b^8* 
c^7*d^12 + 10813440*a^8*b^6*c^8*d^12 - 14417920*a^9*b^4*c^9*d^12 + 1153433 
6*a^10*b^2*c^10*d^12 - 4194304*a^11*c^11*d^12))*x - (b^17*d^8 - 64*a*b^...
 

Mupad [B] (verification not implemented)

Time = 7.30 (sec) , antiderivative size = 5604, normalized size of antiderivative = 34.59 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

int(1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2)),x)
 

Output:

(128*c^6*(a + b*x + c*x^2)^(1/2))/(12*b^9*c^4*d^4 - 144*a*b^7*c^5*d^4 + 72 
*b^8*c^5*d^4*x + 576*a^2*b^5*c^6*d^4 - 768*a^3*b^3*c^7*d^4 - 6144*a^3*c^10 
*d^4*x^3 + 144*b^7*c^6*d^4*x^2 + 96*b^6*c^7*d^4*x^3 + 6912*a^2*b^3*c^8*d^4 
*x^2 + 4608*a^2*b^2*c^9*d^4*x^3 - 864*a*b^6*c^6*d^4*x + 3456*a^2*b^4*c^7*d 
^4*x - 4608*a^3*b^2*c^8*d^4*x - 1728*a*b^5*c^7*d^4*x^2 - 9216*a^3*b*c^9*d^ 
4*x^2 - 1152*a*b^4*c^8*d^4*x^3) - (800*b^7*c^5)/((a + b*x + c*x^2)^(3/2)*( 
480*b^12*c^5*d^4 - 9600*a*b^10*c^6*d^4 + 1920*b^11*c^6*d^4*x + 76800*a^2*b 
^8*c^7*d^4 - 307200*a^3*b^6*c^8*d^4 + 614400*a^4*b^4*c^9*d^4 - 491520*a^5* 
b^2*c^10*d^4 - 1966080*a^5*c^12*d^4*x^2 + 1920*b^10*c^7*d^4*x^2 + 307200*a 
^2*b^6*c^9*d^4*x^2 - 1228800*a^3*b^4*c^10*d^4*x^2 + 2457600*a^4*b^2*c^11*d 
^4*x^2 - 38400*a*b^9*c^7*d^4*x - 1966080*a^5*b*c^11*d^4*x + 307200*a^2*b^7 
*c^8*d^4*x - 1228800*a^3*b^5*c^9*d^4*x + 2457600*a^4*b^3*c^10*d^4*x - 3840 
0*a*b^8*c^8*d^4*x^2)) + (384*a*c^4)/((a + b*x + c*x^2)^(1/2)*(4*b^9*c^2*d^ 
4 - 64*a*b^7*c^3*d^4 + 1024*a^4*b*c^6*d^4 + 2048*a^4*c^7*d^4*x + 8*b^8*c^3 
*d^4*x + 384*a^2*b^5*c^4*d^4 - 1024*a^3*b^3*c^5*d^4 - 128*a*b^6*c^4*d^4*x 
+ 768*a^2*b^4*c^5*d^4*x - 2048*a^3*b^2*c^6*d^4*x)) - (224*b^2*c^3)/(3*(a + 
 b*x + c*x^2)^(1/2)*(4*b^9*c^2*d^4 - 64*a*b^7*c^3*d^4 + 1024*a^4*b*c^6*d^4 
 + 2048*a^4*c^7*d^4*x + 8*b^8*c^3*d^4*x + 384*a^2*b^5*c^4*d^4 - 1024*a^3*b 
^3*c^5*d^4 - 128*a*b^6*c^4*d^4*x + 768*a^2*b^4*c^5*d^4*x - 2048*a^3*b^2*c^ 
6*d^4*x)) + (256*c^5*x^2)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^9*c^2*d^4 - 6...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1084, normalized size of antiderivative = 6.69 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(5/2),x)
 

Output:

(2*( - 64*sqrt(a + b*x + c*x**2)*a**3*c**3 + 144*sqrt(a + b*x + c*x**2)*a* 
*2*b**2*c**2 + 384*sqrt(a + b*x + c*x**2)*a**2*b*c**3*x + 384*sqrt(a + b*x 
 + c*x**2)*a**2*c**4*x**2 + 36*sqrt(a + b*x + c*x**2)*a*b**4*c + 576*sqrt( 
a + b*x + c*x**2)*a*b**3*c**2*x + 2112*sqrt(a + b*x + c*x**2)*a*b**2*c**3* 
x**2 + 3072*sqrt(a + b*x + c*x**2)*a*b*c**4*x**3 + 1536*sqrt(a + b*x + c*x 
**2)*a*c**5*x**4 - sqrt(a + b*x + c*x**2)*b**6 + 24*sqrt(a + b*x + c*x**2) 
*b**5*c*x + 408*sqrt(a + b*x + c*x**2)*b**4*c**2*x**2 + 1792*sqrt(a + b*x 
+ c*x**2)*b**3*c**3*x**3 + 3456*sqrt(a + b*x + c*x**2)*b**2*c**4*x**4 + 30 
72*sqrt(a + b*x + c*x**2)*b*c**5*x**5 + 1024*sqrt(a + b*x + c*x**2)*c**6*x 
**6 - 128*sqrt(c)*a**2*b**3*c - 768*sqrt(c)*a**2*b**2*c**2*x - 1536*sqrt(c 
)*a**2*b*c**3*x**2 - 1024*sqrt(c)*a**2*c**4*x**3 - 256*sqrt(c)*a*b**4*c*x 
- 1792*sqrt(c)*a*b**3*c**2*x**2 - 4608*sqrt(c)*a*b**2*c**3*x**3 - 5120*sqr 
t(c)*a*b*c**4*x**4 - 2048*sqrt(c)*a*c**5*x**5 - 128*sqrt(c)*b**5*c*x**2 - 
1024*sqrt(c)*b**4*c**2*x**3 - 3200*sqrt(c)*b**3*c**3*x**4 - 4864*sqrt(c)*b 
**2*c**4*x**5 - 3584*sqrt(c)*b*c**5*x**6 - 1024*sqrt(c)*c**6*x**7))/(3*d** 
4*(256*a**6*b**3*c**4 + 1536*a**6*b**2*c**5*x + 3072*a**6*b*c**6*x**2 + 20 
48*a**6*c**7*x**3 - 256*a**5*b**5*c**3 - 1024*a**5*b**4*c**4*x + 512*a**5* 
b**3*c**5*x**2 + 7168*a**5*b**2*c**6*x**3 + 10240*a**5*b*c**7*x**4 + 4096* 
a**5*c**8*x**5 + 96*a**4*b**7*c**2 + 64*a**4*b**6*c**3*x - 2176*a**4*b**5* 
c**4*x**2 - 6400*a**4*b**4*c**5*x**3 - 3840*a**4*b**3*c**6*x**4 + 5632*...