\(\int \frac {(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^7} \, dx\) [173]

Optimal result

Integrand size = 26, antiderivative size = 165 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=-\frac {\sqrt {a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac {\sqrt {a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{256 c^{5/2} \left (b^2-4 a c\right )^{3/2} d^7} \] Output:

-1/64*(c*x^2+b*x+a)^(1/2)/c^2/d^7/(2*c*x+b)^4+1/128*(c*x^2+b*x+a)^(1/2)/c^ 
2/(-4*a*c+b^2)/d^7/(2*c*x+b)^2-1/12*(c*x^2+b*x+a)^(3/2)/c/d^7/(2*c*x+b)^6+ 
1/256*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(5/2)/(-4 
*a*c+b^2)^(3/2)/d^7
 

Mathematica [C] (verified)

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

Time = 10.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\frac {2 (a+x (b+c x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},4,\frac {7}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{5 \left (b^2-4 a c\right )^4 d^7} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]
 

Output:

(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, (4*c*(a + x*(b + 
 c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4*a*c)^4*d^7)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.02, 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, 1117, 1112, 218}

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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a}}{d^5 (b+2 c x)^5}dx}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a}}{(b+2 c x)^5}dx}{8 c d^7}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}\)

\(\Big \downarrow \) 1108

\(\displaystyle \frac {\frac {\int \frac {1}{(b+2 c x)^3 \sqrt {c x^2+b x+a}}dx}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c d^7}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}\)

\(\Big \downarrow \) 1117

\(\displaystyle \frac {\frac {\frac {\int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{2 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c d^7}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}\)

\(\Big \downarrow \) 1112

\(\displaystyle \frac {\frac {\frac {2 c \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{b^2-4 a c}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c d^7}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c d^7}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]
 

Output:

-1/12*(a + b*x + c*x^2)^(3/2)/(c*d^7*(b + 2*c*x)^6) + (-1/8*Sqrt[a + b*x + 
 c*x^2]/(c*(b + 2*c*x)^4) + (Sqrt[a + b*x + c*x^2]/((b^2 - 4*a*c)*(b + 2*c 
*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(2*Sq 
rt[c]*(b^2 - 4*a*c)^(3/2)))/(16*c))/(8*c*d^7)
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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]
 

rule 1112

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

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 = 1.05 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {-\frac {3 \left (2 c x +b \right )^{6} \operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {4 a \,c^{2}-b^{2} c}}\right )}{256}+\left (\frac {c^{2} x^{2}}{4}+\left (\frac {b x}{4}+a \right ) c -\frac {3 b^{2}}{16}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (\frac {3 c^{2} x^{2}}{2}+\left (\frac {3 b x}{2}+a \right ) c +\frac {b^{2}}{8}\right ) \sqrt {4 a \,c^{2}-b^{2} c}}{12 \sqrt {4 a \,c^{2}-b^{2} c}\, \left (2 c x +b \right )^{6} \left (a c -\frac {b^{2}}{4}\right ) d^{7} c^{2}}\)	\(157\)
default	\(\frac {-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}-\frac {2 c^{2} \left (-\frac {c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {c^{2} \left (-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {6 c^{2} \left (\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}}{128 d^{7} c^{7}}\)	\(415\)

Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x,method=_RETURNVERBOSE)
 

Output:

-1/12*(-3/256*(2*c*x+b)^6*arctanh(2*(c*x^2+b*x+a)^(1/2)*c/(4*a*c^2-b^2*c)^ 
(1/2))+(1/4*c^2*x^2+(1/4*b*x+a)*c-3/16*b^2)*(c*x^2+b*x+a)^(1/2)*(3/2*c^2*x 
^2+(3/2*b*x+a)*c+1/8*b^2)*(4*a*c^2-b^2*c)^(1/2))/(4*a*c^2-b^2*c)^(1/2)/(2* 
c*x+b)^6/(a*c-1/4*b^2)/d^7/c^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (141) = 282\).

Time = 2.01 (sec) , antiderivative size = 1057, normalized size of antiderivative = 6.41 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x, algorithm="fricas")
 

Output:

[1/1536*(3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 
 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x 
^2 + 4*b*c*x - b^2 + 8*a*c + 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x + a 
))/(4*c^2*x^2 + 4*b*c*x + b^2)) - 4*(3*b^6*c - 4*a*b^4*c^2 - 160*a^2*b^2*c 
^3 + 512*a^3*c^4 - 48*(b^2*c^5 - 4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*x 
^3 - 16*(b^4*c^3 + 10*a*b^2*c^4 - 56*a^2*c^5)*x^2 + 32*(b^5*c^2 - 11*a*b^3 
*c^3 + 28*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^4*c^9 - 8*a*b^2*c^10 
 + 16*a^2*c^11)*d^7*x^6 + 192*(b^5*c^8 - 8*a*b^3*c^9 + 16*a^2*b*c^10)*d^7* 
x^5 + 240*(b^6*c^7 - 8*a*b^4*c^8 + 16*a^2*b^2*c^9)*d^7*x^4 + 160*(b^7*c^6 
- 8*a*b^5*c^7 + 16*a^2*b^3*c^8)*d^7*x^3 + 60*(b^8*c^5 - 8*a*b^6*c^6 + 16*a 
^2*b^4*c^7)*d^7*x^2 + 12*(b^9*c^4 - 8*a*b^7*c^5 + 16*a^2*b^5*c^6)*d^7*x + 
(b^10*c^3 - 8*a*b^8*c^4 + 16*a^2*b^6*c^5)*d^7), -1/768*(3*(64*c^6*x^6 + 19 
2*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5* 
c*x + b^6)*sqrt(b^2*c - 4*a*c^2)*arctan(-2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^ 
2 + b*x + a)/(b^2 - 4*a*c)) + 2*(3*b^6*c - 4*a*b^4*c^2 - 160*a^2*b^2*c^3 + 
 512*a^3*c^4 - 48*(b^2*c^5 - 4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*x^3 - 
 16*(b^4*c^3 + 10*a*b^2*c^4 - 56*a^2*c^5)*x^2 + 32*(b^5*c^2 - 11*a*b^3*c^3 
 + 28*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^4*c^9 - 8*a*b^2*c^10 + 1 
6*a^2*c^11)*d^7*x^6 + 192*(b^5*c^8 - 8*a*b^3*c^9 + 16*a^2*b*c^10)*d^7*x^5 
+ 240*(b^6*c^7 - 8*a*b^4*c^8 + 16*a^2*b^2*c^9)*d^7*x^4 + 160*(b^7*c^6 -...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \] Input:

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**7,x)
 

Output:

(Integral(a*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 
 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c* 
*6*x**6 + 128*c**7*x**7), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**7 + 
 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 
 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(c* 
x**2*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280* 
b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 
 + 128*c**7*x**7), x))/d**7
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,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. 1312 vs. \(2 (141) = 282\).

Time = 0.51 (sec) , antiderivative size = 1312, normalized size of antiderivative = 7.95 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x, algorithm="giac")
 

Output:

1/128*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b 
^2*c - 4*a*c^2))/((b^2*c^2*d^7 - 4*a*c^3*d^7)*sqrt(b^2*c - 4*a*c^2)) - 1/3 
84*(96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*c^5 + 528*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^10*b*c^(9/2) + 944*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9 
*b^2*c^4 + 1504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*c^5 + 288*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^8*b^3*c^(7/2) + 6768*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^8*a*b*c^(9/2) - 1248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4* 
c^3 + 12288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^4 + 2496*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^5 - 2016*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^6*b^5*c^(5/2) + 11424*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^3* 
c^(7/2) + 8736*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b*c^(9/2) - 1488* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^6*c^2 + 5760*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^5*a*b^4*c^3 + 11232*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a 
^2*b^2*c^4 + 2496*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^5 - 648*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^4*b^7*c^(3/2) + 1632*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^4*a*b^5*c^(5/2) + 6240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
4*a^2*b^3*c^(7/2) + 6240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b*c^(9/ 
2) - 182*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^8*c + 320*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^3*a*b^6*c^2 + 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^3*a^2*b^4*c^3 + 4736*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^2*c^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^7} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x)
 

Output:

int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7, x)
 

Reduce [B] (verification not implemented)

Time = 3.19 (sec) , antiderivative size = 1512, normalized size of antiderivative = 9.16 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x)
 

Output:

( - 3*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sq 
rt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**6 - 36*sqrt(c)*sq 
rt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x 
**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*c*x - 180*sqrt(c)*sqrt(4*a*c - 
b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 
2*c*x)/sqrt(4*a*c - b**2))*b**4*c**2*x**2 - 480*sqrt(c)*sqrt(4*a*c - b**2) 
*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x 
)/sqrt(4*a*c - b**2))*b**3*c**3*x**3 - 720*sqrt(c)*sqrt(4*a*c - b**2)*log( 
( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqr 
t(4*a*c - b**2))*b**2*c**4*x**4 - 576*sqrt(c)*sqrt(4*a*c - b**2)*log(( - s 
qrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a 
*c - b**2))*b*c**5*x**5 - 192*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a* 
c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b** 
2))*c**6*x**6 + 3*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*s 
qrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**6 + 36*s 
qrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x 
 + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*c*x + 180*sqrt(c)*sqrt(4* 
a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b 
 + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c**2*x**2 + 480*sqrt(c)*sqrt(4*a*c - b* 
*2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*...