Integrand size = 26, antiderivative size = 118 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}-\frac {32 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3}-\frac {64 c \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)} \] Output:
-2/(-4*a*c+b^2)/d^4/(2*c*x+b)^3/(c*x^2+b*x+a)^(1/2)-32/3*c*(c*x^2+b*x+a)^( 1/2)/(-4*a*c+b^2)^2/d^4/(2*c*x+b)^3-64/3*c*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2 )^3/d^4/(2*c*x+b)
Time = 10.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (3 b^4+48 b^3 c x+64 b c^2 x \left (a+4 c x^2\right )+8 b^2 c \left (3 a+22 c x^2\right )+16 c^2 \left (-a^2+4 a c x^2+8 c^2 x^4\right )\right )}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+x (b+c x)}} \] Input:
Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]
Output:
(-2*(3*b^4 + 48*b^3*c*x + 64*b*c^2*x*(a + 4*c*x^2) + 8*b^2*c*(3*a + 22*c*x ^2) + 16*c^2*(-a^2 + 4*a*c*x^2 + 8*c^2*x^4)))/(3*(b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3*Sqrt[a + x*(b + c*x)])
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1111, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^4} \, dx\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle -\frac {16 c \int \frac {1}{d^4 (b+2 c x)^4 \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {16 c \int \frac {1}{(b+2 c x)^4 \sqrt {c x^2+b x+a}}dx}{d^4 \left (b^2-4 a c\right )}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle -\frac {16 c \left (\frac {2 \int \frac {1}{(b+2 c x)^2 \sqrt {c x^2+b x+a}}dx}{3 \left (b^2-4 a c\right )}+\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) (b+2 c x)^3}\right )}{d^4 \left (b^2-4 a c\right )}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1106 |
| \(\displaystyle -\frac {16 c \left (\frac {4 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) (b+2 c x)^3}\right )}{d^4 \left (b^2-4 a c\right )}-\frac {2}{d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}\) |
Input:
Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x]
Output:
-2/((b^2 - 4*a*c)*d^4*(b + 2*c*x)^3*Sqrt[a + b*x + c*x^2]) - (16*c*((2*Sqr t[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)*d^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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])
Time = 1.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.94
| method | result | size |
| trager | \(-\frac {2 \left (-128 c^{4} x^{4}-256 b \,c^{3} x^{3}-64 a \,c^{3} x^{2}-176 b^{2} c^{2} x^{2}-64 a b \,c^{2} x -48 b^{3} c x +16 a^{2} c^{2}-24 c a \,b^{2}-3 b^{4}\right )}{3 d^{4} \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )^{3}}\) | \(111\) |
| gosper | \(-\frac {2 \left (-128 c^{4} x^{4}-256 b \,c^{3} x^{3}-64 a \,c^{3} x^{2}-176 b^{2} c^{2} x^{2}-64 a b \,c^{2} x -48 b^{3} c x +16 a^{2} c^{2}-24 c a \,b^{2}-3 b^{4}\right )}{3 \left (2 c x +b \right )^{3} d^{4} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {c \,x^{2}+b x +a}}\) | \(133\) |
| orering | \(-\frac {2 \left (-128 c^{4} x^{4}-256 b \,c^{3} x^{3}-64 a \,c^{3} x^{2}-176 b^{2} c^{2} x^{2}-64 a b \,c^{2} x -48 b^{3} c x +16 a^{2} c^{2}-24 c a \,b^{2}-3 b^{4}\right ) \left (2 c x +b \right )}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c d x +b d \right )^{4}}\) | \(139\) |
| default | \(\frac {-\frac {4 c}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3} \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}-\frac {16 c^{2} \left (-\frac {4 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right ) \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}-\frac {32 c^{3} \left (x +\frac {b}{2 c}\right )}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}\right )}{3 \left (4 a c -b^{2}\right )}}{16 d^{4} c^{4}}\) | \(193\) |
Input:
int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3/d^4*(-128*c^4*x^4-256*b*c^3*x^3-64*a*c^3*x^2-176*b^2*c^2*x^2-64*a*b*c ^2*x-48*b^3*c*x+16*a^2*c^2-24*a*b^2*c-3*b^4)/(c*x^2+b*x+a)^(1/2)/(4*a*c-b^ 2)^3/(2*c*x+b)^3
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (108) = 216\).
Time = 1.16 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.23 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 3 \, b^{4} + 24 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \, {\left (11 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \, {\left (3 \, b^{3} c + 4 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (8 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \, {\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} + {\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} + {\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x + {\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\right )} d^{4}\right )}} \] Input:
integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
-2/3*(128*c^4*x^4 + 256*b*c^3*x^3 + 3*b^4 + 24*a*b^2*c - 16*a^2*c^2 + 16*( 11*b^2*c^2 + 4*a*c^3)*x^2 + 16*(3*b^3*c + 4*a*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(8*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*x^5 + 20 *(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^4*x^4 + 2*(9*b ^8*c^2 - 104*a*b^6*c^3 + 384*a^2*b^4*c^4 - 384*a^3*b^2*c^5 - 256*a^4*c^6)* d^4*x^3 + (7*b^9*c - 72*a*b^7*c^2 + 192*a^2*b^5*c^3 + 128*a^3*b^3*c^4 - 76 8*a^4*b*c^5)*d^4*x^2 + (b^10 - 6*a*b^8*c - 24*a^2*b^6*c^2 + 224*a^3*b^4*c^ 3 - 384*a^4*b^2*c^4)*d^4*x + (a*b^9 - 12*a^2*b^7*c + 48*a^3*b^5*c^2 - 64*a ^4*b^3*c^3)*d^4)
\[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\int \frac {1}{a b^{4} \sqrt {a + b x + c x^{2}} + 8 a b^{3} c x \sqrt {a + b x + c x^{2}} + 24 a b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 32 a b c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 16 a c^{4} x^{4} \sqrt {a + b x + c x^{2}} + b^{5} x \sqrt {a + b x + c x^{2}} + 9 b^{4} c x^{2} \sqrt {a + b x + c x^{2}} + 32 b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 56 b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 48 b c^{4} x^{5} \sqrt {a + b x + c x^{2}} + 16 c^{5} x^{6} \sqrt {a + b x + c x^{2}}}\, dx}{d^{4}} \] Input:
integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral(1/(a*b**4*sqrt(a + b*x + c*x**2) + 8*a*b**3*c*x*sqrt(a + b*x + c* x**2) + 24*a*b**2*c**2*x**2*sqrt(a + b*x + c*x**2) + 32*a*b*c**3*x**3*sqrt (a + b*x + c*x**2) + 16*a*c**4*x**4*sqrt(a + b*x + c*x**2) + b**5*x*sqrt(a + b*x + c*x**2) + 9*b**4*c*x**2*sqrt(a + b*x + c*x**2) + 32*b**3*c**2*x** 3*sqrt(a + b*x + c*x**2) + 56*b**2*c**3*x**4*sqrt(a + b*x + c*x**2) + 48*b *c**4*x**5*sqrt(a + b*x + c*x**2) + 16*c**5*x**6*sqrt(a + b*x + c*x**2)), x)/d**4
Exception generated. \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),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
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (108) = 216\).
Time = 0.41 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.37 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, c x}{b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}} + \frac {b}{b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {4 \, {\left (12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} c^{\frac {5}{2}} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c^{2} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c^{\frac {3}{2}} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {5}{2}} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{2} + 5 \, b^{4} \sqrt {c} - 22 \, a b^{2} c^{\frac {3}{2}} + 20 \, a^{2} c^{\frac {5}{2}}\right )}}{3 \, {\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{3}} \] Input:
integrate(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
-2*(2*c*x/(b^6*d^4 - 12*a*b^4*c*d^4 + 48*a^2*b^2*c^2*d^4 - 64*a^3*c^3*d^4) + b/(b^6*d^4 - 12*a*b^4*c*d^4 + 48*a^2*b^2*c^2*d^4 - 64*a^3*c^3*d^4))/sqr t(c*x^2 + b*x + a) - 4/3*(12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(5/2) + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^2 + 30*(sqrt(c)*x - sqrt(c *x^2 + b*x + a))^2*b^2*c^(3/2) - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2* a*c^(5/2) + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^2 + 5*b^4*sqrt(c) - 22*a*b^2*c^(3/2) + 20*a^ 2*c^(5/2))/((b^4*d^4 - 8*a*b^2*c*d^4 + 16*a^2*c^2*d^4)*(2*(sqrt(c)*x - sqr t(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^3)
Time = 6.77 (sec) , antiderivative size = 4588, normalized size of antiderivative = 38.88 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int(1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(3/2)),x)
Output:
(8*b^6*c^2)/((a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c ^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4 *x)) - (128*b^8*c^5)/((a + b*x + c*x^2)^(1/2)*(32*b^13*c^5*d^4 - 768*a*b^1 1*c^6*d^4 + 131072*a^6*b*c^11*d^4 + 262144*a^6*c^12*d^4*x + 64*b^12*c^6*d^ 4*x + 7680*a^2*b^9*c^7*d^4 - 40960*a^3*b^7*c^8*d^4 + 122880*a^4*b^5*c^9*d^ 4 - 196608*a^5*b^3*c^10*d^4 - 1536*a*b^10*c^7*d^4*x + 15360*a^2*b^8*c^8*d^ 4*x - 81920*a^3*b^6*c^9*d^4*x + 245760*a^4*b^4*c^10*d^4*x - 393216*a^5*b^2 *c^11*d^4*x)) - (2560*a^3*c^5)/(3*(a + b*x + c*x^2)^(1/2)*(4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4*x + 8*b^10*c^3* d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 5120*a^4*b^3*c^6*d^4 - 160*a*b^8*c^4*d^4*x + 1280*a^2*b^6*c^5*d^4*x - 5120*a^3*b^4*c^6*d^4*x + 10240*a^4*b^2*c^7*d^4*x)) - (32*c^4*(a + b*x + c*x^2)^(1/2))/(12*b^7*c^3*d ^4 - 96*a*b^5*c^4*d^4 + 72*b^6*c^4*d^4*x + 192*a^2*b^3*c^5*d^4 + 1536*a^2* c^8*d^4*x^3 + 144*b^5*c^5*d^4*x^2 + 96*b^4*c^6*d^4*x^3 - 576*a*b^4*c^5*d^4 *x + 1152*a^2*b^2*c^6*d^4*x - 1152*a*b^3*c^6*d^4*x^2 + 2304*a^2*b*c^7*d^4* x^2 - 768*a*b^2*c^7*d^4*x^3) - (352*a*b^4*c^3)/(3*(a + b*x + c*x^2)^(1/2)* (4*b^11*c^2*d^4 - 80*a*b^9*c^3*d^4 - 4096*a^5*b*c^7*d^4 - 8192*a^5*c^8*d^4 *x + 8*b^10*c^3*d^4*x + 640*a^2*b^7*c^4*d^4 - 2560*a^3*b^5*c^5*d^4 + 51...
Time = 0.22 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.92 \[ \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-\frac {32 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{2}}{3}+16 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c +\frac {128 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} x}{3}+\frac {128 \sqrt {c \,x^{2}+b x +a}\, a \,c^{3} x^{2}}{3}+2 \sqrt {c \,x^{2}+b x +a}\, b^{4}+32 \sqrt {c \,x^{2}+b x +a}\, b^{3} c x +\frac {352 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} x^{2}}{3}+\frac {512 \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} x^{3}}{3}+\frac {256 \sqrt {c \,x^{2}+b x +a}\, c^{4} x^{4}}{3}-\frac {32 \sqrt {c}\, a \,b^{3}}{3}-64 \sqrt {c}\, a \,b^{2} c x -128 \sqrt {c}\, a b \,c^{2} x^{2}-\frac {256 \sqrt {c}\, a \,c^{3} x^{3}}{3}-\frac {32 \sqrt {c}\, b^{4} x}{3}-\frac {224 \sqrt {c}\, b^{3} c \,x^{2}}{3}-192 \sqrt {c}\, b^{2} c^{2} x^{3}-\frac {640 \sqrt {c}\, b \,c^{3} x^{4}}{3}-\frac {256 \sqrt {c}\, c^{4} x^{5}}{3}}{d^{4} \left (512 a^{3} c^{7} x^{5}-384 a^{2} b^{2} c^{6} x^{5}+96 a \,b^{4} c^{5} x^{5}-8 b^{6} c^{4} x^{5}+1280 a^{3} b \,c^{6} x^{4}-960 a^{2} b^{3} c^{5} x^{4}+240 a \,b^{5} c^{4} x^{4}-20 b^{7} c^{3} x^{4}+512 a^{4} c^{6} x^{3}+768 a^{3} b^{2} c^{5} x^{3}-768 a^{2} b^{4} c^{4} x^{3}+208 a \,b^{6} c^{3} x^{3}-18 b^{8} c^{2} x^{3}+768 a^{4} b \,c^{5} x^{2}-128 a^{3} b^{3} c^{4} x^{2}-192 a^{2} b^{5} c^{3} x^{2}+72 a \,b^{7} c^{2} x^{2}-7 b^{9} c \,x^{2}+384 a^{4} b^{2} c^{4} x -224 a^{3} b^{4} c^{3} x +24 a^{2} b^{6} c^{2} x +6 a \,b^{8} c x -b^{10} x +64 a^{4} b^{3} c^{3}-48 a^{3} b^{5} c^{2}+12 a^{2} b^{7} c -a \,b^{9}\right )} \] Input:
int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a)^(3/2),x)
Output:
(2*( - 16*sqrt(a + b*x + c*x**2)*a**2*c**2 + 24*sqrt(a + b*x + c*x**2)*a*b **2*c + 64*sqrt(a + b*x + c*x**2)*a*b*c**2*x + 64*sqrt(a + b*x + c*x**2)*a *c**3*x**2 + 3*sqrt(a + b*x + c*x**2)*b**4 + 48*sqrt(a + b*x + c*x**2)*b** 3*c*x + 176*sqrt(a + b*x + c*x**2)*b**2*c**2*x**2 + 256*sqrt(a + b*x + c*x **2)*b*c**3*x**3 + 128*sqrt(a + b*x + c*x**2)*c**4*x**4 - 16*sqrt(c)*a*b** 3 - 96*sqrt(c)*a*b**2*c*x - 192*sqrt(c)*a*b*c**2*x**2 - 128*sqrt(c)*a*c**3 *x**3 - 16*sqrt(c)*b**4*x - 112*sqrt(c)*b**3*c*x**2 - 288*sqrt(c)*b**2*c** 2*x**3 - 320*sqrt(c)*b*c**3*x**4 - 128*sqrt(c)*c**4*x**5))/(3*d**4*(64*a** 4*b**3*c**3 + 384*a**4*b**2*c**4*x + 768*a**4*b*c**5*x**2 + 512*a**4*c**6* x**3 - 48*a**3*b**5*c**2 - 224*a**3*b**4*c**3*x - 128*a**3*b**3*c**4*x**2 + 768*a**3*b**2*c**5*x**3 + 1280*a**3*b*c**6*x**4 + 512*a**3*c**7*x**5 + 1 2*a**2*b**7*c + 24*a**2*b**6*c**2*x - 192*a**2*b**5*c**3*x**2 - 768*a**2*b **4*c**4*x**3 - 960*a**2*b**3*c**5*x**4 - 384*a**2*b**2*c**6*x**5 - a*b**9 + 6*a*b**8*c*x + 72*a*b**7*c**2*x**2 + 208*a*b**6*c**3*x**3 + 240*a*b**5* c**4*x**4 + 96*a*b**4*c**5*x**5 - b**10*x - 7*b**9*c*x**2 - 18*b**8*c**2*x **3 - 20*b**7*c**3*x**4 - 8*b**6*c**4*x**5))