\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^6} \, dx\) [187]

Optimal result

Integrand size = 26, antiderivative size = 139 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=-\frac {\sqrt {a+b x+c x^2}}{32 c^3 d^6 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{24 c^2 d^6 (b+2 c x)^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2} d^6} \] Output:

-1/32*(c*x^2+b*x+a)^(1/2)/c^3/d^6/(2*c*x+b)-1/24*(c*x^2+b*x+a)^(3/2)/c^2/d 
^6/(2*c*x+b)^3-1/10*(c*x^2+b*x+a)^(5/2)/c/d^6/(2*c*x+b)^5+1/64*arctanh(1/2 
*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/d^6
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=-\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{320 c^3 d^6 (b+2 c x)^5 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]
 

Output:

-1/320*((b^2 - 4*a*c)^2*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, -5/2 
, -3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^3*d^6*(b + 2*c*x)^5*Sqrt[(c*(a + 
x*(b + c*x)))/(-b^2 + 4*a*c)])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1108, 27, 1108, 1108, 1092, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {\int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^4 (b+2 c x)^4}dx}{4 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^4}dx}{4 c d^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {\frac {\int \frac {\sqrt {c x^2+b x+a}}{(b+2 c x)^2}dx}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c (b+2 c x)^3}}{4 c d^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {\frac {\frac {\int \frac {1}{\sqrt {c x^2+b x+a}}dx}{4 c}-\frac {\sqrt {a+b x+c x^2}}{2 c (b+2 c x)}}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c (b+2 c x)^3}}{4 c d^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{2 c}-\frac {\sqrt {a+b x+c x^2}}{2 c (b+2 c x)}}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c (b+2 c x)^3}}{4 c d^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 c^{3/2}}-\frac {\sqrt {a+b x+c x^2}}{2 c (b+2 c x)}}{4 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c (b+2 c x)^3}}{4 c d^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{10 c d^6 (b+2 c x)^5}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x]
 

Output:

-1/10*(a + b*x + c*x^2)^(5/2)/(c*d^6*(b + 2*c*x)^5) + (-1/6*(a + b*x + c*x 
^2)^(3/2)/(c*(b + 2*c*x)^3) + (-1/2*Sqrt[a + b*x + c*x^2]/(c*(b + 2*c*x)) 
+ ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/(4*c^(3/2)))/(4*c 
))/(4*c*d^6)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1108

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[b*(p/(d*e*(m + 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] && NeQ[m + 2*p + 
3, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0] 
) && IntegerQ[2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(439\) vs. \(2(117)=234\).

Time = 2.74 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.17

method	result	size
default	\(\frac {-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5}}+\frac {8 c^{2} \left (-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {16 c^{2} \left (-\frac {4 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{5 \left (4 a c -b^{2}\right )}}{64 d^{6} c^{6}}\)	\(440\)

Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x,method=_RETURNVERBOSE)
 

Output:

1/64/d^6/c^6*(-4/5/(4*a*c-b^2)*c/(x+1/2*b/c)^5*(c*(x+1/2*b/c)^2+1/4*(4*a*c 
-b^2)/c)^(7/2)+8/5*c^2/(4*a*c-b^2)*(-4/3/(4*a*c-b^2)*c/(x+1/2*b/c)^3*(c*(x 
+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(7/2)+16/3*c^2/(4*a*c-b^2)*(-4/(4*a*c-b^2)* 
c/(x+1/2*b/c)*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(7/2)+24*c^2/(4*a*c-b^2) 
*(1/6*(x+1/2*b/c)*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(5/2)+5/24*(4*a*c-b^ 
2)/c*(1/4*(x+1/2*b/c)*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(3/2)+3/16*(4*a* 
c-b^2)/c*(1/2*(x+1/2*b/c)*(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/c)^(1/2)+1/8*(4 
*a*c-b^2)/c^(3/2)*ln(c^(1/2)*(x+1/2*b/c)+(c*(x+1/2*b/c)^2+1/4*(4*a*c-b^2)/ 
c)^(1/2))))))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (117) = 234\).

Time = 1.33 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.95 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\left [\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1920 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}, -\frac {15 \, {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (368 \, c^{5} x^{4} + 736 \, b c^{4} x^{3} + 15 \, b^{4} c + 20 \, a b^{2} c^{2} + 48 \, a^{2} c^{3} + 4 \, {\left (127 \, b^{2} c^{3} + 44 \, a c^{4}\right )} x^{2} + 4 \, {\left (35 \, b^{3} c^{2} + 44 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{960 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}}\right ] \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")
 

Output:

[1/1920*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 40*b^3*c^2*x^2 + 
 10*b^4*c*x + b^5)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + 
 b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(368*c^5*x^4 + 736*b*c^4*x^3 + 
15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4)*x^2 + 4* 
(35*b^3*c^2 + 44*a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/(32*c^9*d^6*x^5 + 80*b 
*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^6*x 
+ b^5*c^4*d^6), -1/960*(15*(32*c^5*x^5 + 80*b*c^4*x^4 + 80*b^2*c^3*x^3 + 4 
0*b^3*c^2*x^2 + 10*b^4*c*x + b^5)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a 
)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(368*c^5*x^4 + 736*b*c 
^4*x^3 + 15*b^4*c + 20*a*b^2*c^2 + 48*a^2*c^3 + 4*(127*b^2*c^3 + 44*a*c^4) 
*x^2 + 4*(35*b^3*c^2 + 44*a*b*c^3)*x)*sqrt(c*x^2 + b*x + a))/(32*c^9*d^6*x 
^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c 
^5*d^6*x + b^5*c^4*d^6)]
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**6,x)
 

Output:

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x 
**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6* 
x**6), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x 
+ 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5 
*x**5 + 64*c**6*x**6), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b** 
6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x 
**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*a*b*x*sqrt(a + b*x 
+ c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 2 
40*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(2*a*c*x 
**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b 
**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + 
 Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4* 
c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64 
*c**6*x**6), x))/d**6
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,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. 1013 vs. \(2 (117) = 234\).

Time = 0.68 (sec) , antiderivative size = 1013, normalized size of antiderivative = 7.29 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")
 

Output:

-1/64*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/(c^(7/2) 
*d^6) - 1/960*(720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^2*c^4 - 2880*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^5 + 2880*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^7*b^3*c^(7/2) - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b 
*c^(9/2) + 5400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^3 - 23040*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^2*c^4 + 5760*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^6*a^2*c^5 + 6120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*c^ 
(5/2) - 28800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^(7/2) + 17280* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b*c^(9/2) + 4640*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^4*b^6*c^2 - 25080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^4*a*b^4*c^3 + 28320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^4 - 8 
960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^5 + 2440*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^3*b^7*c^(3/2) - 15600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a 
))^3*a*b^5*c^(5/2) + 27840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^3*c 
^(7/2) - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^(9/2) + 880*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*c - 6760*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^2*a*b^6*c^2 + 17160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2* 
b^4*c^3 - 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^4 + 4480*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*c^5 + 200*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))*b^9*sqrt(c) - 1840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^6} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6,x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.17 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^6} \, dx=\frac {-96 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3}-40 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{2}-352 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} x -352 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} x^{2}-30 \sqrt {c \,x^{2}+b x +a}\, b^{4} c -280 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{2} x -1016 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} x^{2}-1472 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} x^{3}-736 \sqrt {c \,x^{2}+b x +a}\, c^{5} x^{4}+15 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{5}+150 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{4} c x +600 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c^{2} x^{2}+1200 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{3} x^{3}+1200 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b \,c^{4} x^{4}+480 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{5} x^{5}+5 \sqrt {c}\, b^{5}+50 \sqrt {c}\, b^{4} c x +200 \sqrt {c}\, b^{3} c^{2} x^{2}+400 \sqrt {c}\, b^{2} c^{3} x^{3}+400 \sqrt {c}\, b \,c^{4} x^{4}+160 \sqrt {c}\, c^{5} x^{5}}{960 c^{4} d^{6} \left (32 c^{5} x^{5}+80 b \,c^{4} x^{4}+80 b^{2} c^{3} x^{3}+40 b^{3} c^{2} x^{2}+10 b^{4} c x +b^{5}\right )} \] Input:

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^6,x)
 

Output:

( - 96*sqrt(a + b*x + c*x**2)*a**2*c**3 - 40*sqrt(a + b*x + c*x**2)*a*b**2 
*c**2 - 352*sqrt(a + b*x + c*x**2)*a*b*c**3*x - 352*sqrt(a + b*x + c*x**2) 
*a*c**4*x**2 - 30*sqrt(a + b*x + c*x**2)*b**4*c - 280*sqrt(a + b*x + c*x** 
2)*b**3*c**2*x - 1016*sqrt(a + b*x + c*x**2)*b**2*c**3*x**2 - 1472*sqrt(a 
+ b*x + c*x**2)*b*c**4*x**3 - 736*sqrt(a + b*x + c*x**2)*c**5*x**4 + 15*sq 
rt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2 
))*b**5 + 150*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s 
qrt(4*a*c - b**2))*b**4*c*x + 600*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c* 
x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**2*x**2 + 1200*sqrt(c)*log(( 
2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c** 
3*x**3 + 1200*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/s 
qrt(4*a*c - b**2))*b*c**4*x**4 + 480*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + 
 c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*c**5*x**5 + 5*sqrt(c)*b**5 + 50* 
sqrt(c)*b**4*c*x + 200*sqrt(c)*b**3*c**2*x**2 + 400*sqrt(c)*b**2*c**3*x**3 
 + 400*sqrt(c)*b*c**4*x**4 + 160*sqrt(c)*c**5*x**5)/(960*c**4*d**6*(b**5 + 
 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b**2*c**3*x**3 + 80*b*c**4*x**4 + 32 
*c**5*x**5))