Integrand size = 26, antiderivative size = 207 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^2}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{512 c^2}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}+\frac {3 \left (b^2-4 a c\right )^4 d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{5/2}} \]
1/16*d^4*(2*c*x+b)^5*(c*x^2+b*x+a)^(3/2)/c+3/2048*(-4*a*c+b^2)^4*d^4*arcta nh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)+3/1024*(-4*a*c+b^2)^ 3*d^4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^2+1/512*(-4*a*c+b^2)^2*d^4*(2*c*x+b) ^3*(c*x^2+b*x+a)^(1/2)/c^2-1/128*(-4*a*c+b^2)*d^4*(2*c*x+b)^5*(c*x^2+b*x+a )^(1/2)/c^2
Time = 3.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.07 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {d^4 \left (\sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (-3 b^6+8 b^5 c x+64 b^3 c^2 x \left (11 a+28 c x^2\right )+4 b^4 c \left (11 a+98 c x^2\right )+128 b c^3 x \left (a^2+24 a c x^2+24 c^2 x^4\right )+16 b^2 c^2 \left (11 a^2+140 a c x^2+216 c^2 x^4\right )+64 c^3 \left (-3 a^3+2 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right )\right )+3 \left (b^2-4 a c\right )^4 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right )}{1024 c^{5/2}} \]
(d^4*(Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^6 + 8*b^5*c*x + 64*b ^3*c^2*x*(11*a + 28*c*x^2) + 4*b^4*c*(11*a + 98*c*x^2) + 128*b*c^3*x*(a^2 + 24*a*c*x^2 + 24*c^2*x^4) + 16*b^2*c^2*(11*a^2 + 140*a*c*x^2 + 216*c^2*x^ 4) + 64*c^3*(-3*a^3 + 2*a^2*c*x^2 + 24*a*c^2*x^4 + 16*c^3*x^6)) + 3*(b^2 - 4*a*c)^4*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])]))/(1024* c^(5/2))
Time = 0.36 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1109, 27, 1109, 1116, 1116, 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.
| \(\displaystyle \int \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^4 \, dx\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \int d^4 (b+2 c x)^4 \sqrt {c x^2+b x+a}dx}{32 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 d^4 \left (b^2-4 a c\right ) \int (b+2 c x)^4 \sqrt {c x^2+b x+a}dx}{32 c}\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^4}{\sqrt {c x^2+b x+a}}dx}{24 c}\right )}{32 c}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{\sqrt {c x^2+b x+a}}dx+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \left (\frac {1}{2} \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+(b+2 c x) \sqrt {a+b x+c x^2}\right )+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\) |
(d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)^(3/2))/(16*c) - (3*(b^2 - 4*a*c)*d^4* (((b + 2*c*x)^5*Sqrt[a + b*x + c*x^2])/(12*c) - ((b^2 - 4*a*c)*(((b + 2*c* x)^3*Sqrt[a + b*x + c*x^2])/2 + (3*(b^2 - 4*a*c)*((b + 2*c*x)*Sqrt[a + b*x + c*x^2] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c *x^2])])/(2*Sqrt[c])))/4))/(24*c)))/(32*c)
3.13.3.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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 + 2*p + 1))), x ] - Simp[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))) 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] && GtQ[p, 0] && !LtQ[m, -1] && !(IGtQ[(m - 1 )/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
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])
Time = 2.62 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(-\frac {\left (-2048 c^{7} x^{7}-7168 b \,c^{6} x^{6}-3072 a \,c^{6} x^{5}-9984 b^{2} c^{5} x^{5}-7680 a b \,c^{5} x^{4}-7040 b^{3} c^{4} x^{4}-256 a^{2} c^{5} x^{3}-7552 a \,b^{2} c^{4} x^{3}-2576 b^{4} c^{3} x^{3}-384 a^{2} b \,c^{4} x^{2}-3648 a \,b^{3} c^{3} x^{2}-408 b^{5} c^{2} x^{2}+384 a^{3} c^{4} x -480 a^{2} b^{2} c^{3} x -792 c^{2} a \,b^{4} x -2 b^{6} c x +192 a^{3} c^{3} b -176 a^{2} c^{2} b^{3}-44 a \,b^{5} c +3 b^{7}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}}{1024 c^{2}}+\frac {3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) d^{4}}{2048 c^{\frac {5}{2}}}\) | \(294\) |
| default | \(\text {Expression too large to display}\) | \(1662\) |
-1/1024/c^2*(-2048*c^7*x^7-7168*b*c^6*x^6-3072*a*c^6*x^5-9984*b^2*c^5*x^5- 7680*a*b*c^5*x^4-7040*b^3*c^4*x^4-256*a^2*c^5*x^3-7552*a*b^2*c^4*x^3-2576* b^4*c^3*x^3-384*a^2*b*c^4*x^2-3648*a*b^3*c^3*x^2-408*b^5*c^2*x^2+384*a^3*c ^4*x-480*a^2*b^2*c^3*x-792*a*b^4*c^2*x-2*b^6*c*x+192*a^3*b*c^3-176*a^2*b^3 *c^2-44*a*b^5*c+3*b^7)*(c*x^2+b*x+a)^(1/2)*d^4+3/2048*(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)/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c* x^2+b*x+a)^(1/2))*d^4
Time = 0.44 (sec) , antiderivative size = 677, normalized size of antiderivative = 3.27 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\left [\frac {3 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c} d^{4} \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 (2048 \, c^{8} d^{4} x^{7} + 7168 \, b c^{7} d^{4} x^{6} + 768 \, {\left (13 \, b^{2} c^{6} + 4 \, a c^{7}\right )} d^{4} x^{5} + 640 \, {\left (11 \, b^{3} c^{5} + 12 \, a b c^{6}\right )} d^{4} x^{4} + 16 \, {\left (161 \, b^{4} c^{4} + 472 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 24 \, {\left (17 \, b^{5} c^{3} + 152 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 2 \, {\left (b^{6} c^{2} + 396 \, a b^{4} c^{3} + 240 \, a^{2} b^{2} c^{4} - 192 \, a^{3} c^{5}\right )} d^{4} x - {\left (3 \, b^{7} c - 44 \, a b^{5} c^{2} - 176 \, a^{2} b^{3} c^{3} + 192 \, a^{3} b c^{4}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{4096 \, c^{3}}, -\frac {3 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-c} d^{4} \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 (2048 \, c^{8} d^{4} x^{7} + 7168 \, b c^{7} d^{4} x^{6} + 768 \, {\left (13 \, b^{2} c^{6} + 4 \, a c^{7}\right )} d^{4} x^{5} + 640 \, {\left (11 \, b^{3} c^{5} + 12 \, a b c^{6}\right )} d^{4} x^{4} + 16 \, {\left (161 \, b^{4} c^{4} + 472 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{4} x^{3} + 24 \, {\left (17 \, b^{5} c^{3} + 152 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{4} x^{2} + 2 \, {\left (b^{6} c^{2} + 396 \, a b^{4} c^{3} + 240 \, a^{2} b^{2} c^{4} - 192 \, a^{3} c^{5}\right )} d^{4} x - {\left (3 \, b^{7} c - 44 \, a b^{5} c^{2} - 176 \, a^{2} b^{3} c^{3} + 192 \, a^{3} b c^{4}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{2048 \, c^{3}}\right ] \]
[1/4096*(3*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4* c^4)*sqrt(c)*d^4*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*(2048*c^8*d^4*x^7 + 7168*b*c^7*d^4*x^6 + 768*(13*b^2*c^6 + 4*a*c^7)*d^4*x^5 + 640*(11*b^3*c^5 + 12*a*b*c^6)*d^4*x^4 + 16*(161*b^4*c^4 + 472*a*b^2*c^5 + 16*a^2*c^6)*d^4*x^3 + 24*(17*b^5*c^3 + 152*a*b^3*c^4 + 16*a^2*b*c^5)*d^4*x^2 + 2*(b^6*c^2 + 396*a*b^4*c^3 + 240 *a^2*b^2*c^4 - 192*a^3*c^5)*d^4*x - (3*b^7*c - 44*a*b^5*c^2 - 176*a^2*b^3* c^3 + 192*a^3*b*c^4)*d^4)*sqrt(c*x^2 + b*x + a))/c^3, -1/2048*(3*(b^8 - 16 *a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-c)*d^4*ar ctan(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*(2048*c^8*d^4*x^7 + 7168*b*c^7*d^4*x^6 + 768*(13*b^2*c^6 + 4*a*c^7) *d^4*x^5 + 640*(11*b^3*c^5 + 12*a*b*c^6)*d^4*x^4 + 16*(161*b^4*c^4 + 472*a *b^2*c^5 + 16*a^2*c^6)*d^4*x^3 + 24*(17*b^5*c^3 + 152*a*b^3*c^4 + 16*a^2*b *c^5)*d^4*x^2 + 2*(b^6*c^2 + 396*a*b^4*c^3 + 240*a^2*b^2*c^4 - 192*a^3*c^5 )*d^4*x - (3*b^7*c - 44*a*b^5*c^2 - 176*a^2*b^3*c^3 + 192*a^3*b*c^4)*d^4)* sqrt(c*x^2 + b*x + a))/c^3]
Leaf count of result is larger than twice the leaf count of optimal. 3067 vs. \(2 (197) = 394\).
Time = 1.01 (sec) , antiderivative size = 3067, normalized size of antiderivative = 14.82 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a + b*x + c*x**2)*(7*b*c**4*d**4*x**6 + 2*c**5*d**4*x**7 + x**5*(18*a*c**5*d**4 + 117*b**2*c**4*d**4/2)/(6*c) + x**4*(54*a*b*c**4*d* *4 + 88*b**3*c**3*d**4 - 11*b*(18*a*c**5*d**4 + 117*b**2*c**4*d**4/2)/(12* c))/(5*c) + x**3*(16*a**2*c**4*d**4 + 112*a*b**2*c**3*d**4 - 5*a*(18*a*c** 5*d**4 + 117*b**2*c**4*d**4/2)/(6*c) + 41*b**4*c**2*d**4 - 9*b*(54*a*b*c** 4*d**4 + 88*b**3*c**3*d**4 - 11*b*(18*a*c**5*d**4 + 117*b**2*c**4*d**4/2)/ (12*c))/(10*c))/(4*c) + x**2*(32*a**2*b*c**3*d**4 + 64*a*b**3*c**2*d**4 - 4*a*(54*a*b*c**4*d**4 + 88*b**3*c**3*d**4 - 11*b*(18*a*c**5*d**4 + 117*b** 2*c**4*d**4/2)/(12*c))/(5*c) + 10*b**5*c*d**4 - 7*b*(16*a**2*c**4*d**4 + 1 12*a*b**2*c**3*d**4 - 5*a*(18*a*c**5*d**4 + 117*b**2*c**4*d**4/2)/(6*c) + 41*b**4*c**2*d**4 - 9*b*(54*a*b*c**4*d**4 + 88*b**3*c**3*d**4 - 11*b*(18*a *c**5*d**4 + 117*b**2*c**4*d**4/2)/(12*c))/(10*c))/(8*c))/(3*c) + x*(24*a* *2*b**2*c**2*d**4 + 18*a*b**4*c*d**4 - 3*a*(16*a**2*c**4*d**4 + 112*a*b**2 *c**3*d**4 - 5*a*(18*a*c**5*d**4 + 117*b**2*c**4*d**4/2)/(6*c) + 41*b**4*c **2*d**4 - 9*b*(54*a*b*c**4*d**4 + 88*b**3*c**3*d**4 - 11*b*(18*a*c**5*d** 4 + 117*b**2*c**4*d**4/2)/(12*c))/(10*c))/(4*c) + b**6*d**4 - 5*b*(32*a**2 *b*c**3*d**4 + 64*a*b**3*c**2*d**4 - 4*a*(54*a*b*c**4*d**4 + 88*b**3*c**3* d**4 - 11*b*(18*a*c**5*d**4 + 117*b**2*c**4*d**4/2)/(12*c))/(5*c) + 10*b** 5*c*d**4 - 7*b*(16*a**2*c**4*d**4 + 112*a*b**2*c**3*d**4 - 5*a*(18*a*c**5* d**4 + 117*b**2*c**4*d**4/2)/(6*c) + 41*b**4*c**2*d**4 - 9*b*(54*a*b*c*...
Exception generated. \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \]
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. 389 vs. \(2 (181) = 362\).
Time = 0.29 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.88 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{1024} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (2 \, c^{5} d^{4} x + 7 \, b c^{4} d^{4}\right )} x + \frac {3 \, {\left (13 \, b^{2} c^{10} d^{4} + 4 \, a c^{11} d^{4}\right )}}{c^{7}}\right )} x + \frac {5 \, {\left (11 \, b^{3} c^{9} d^{4} + 12 \, a b c^{10} d^{4}\right )}}{c^{7}}\right )} x + \frac {161 \, b^{4} c^{8} d^{4} + 472 \, a b^{2} c^{9} d^{4} + 16 \, a^{2} c^{10} d^{4}}{c^{7}}\right )} x + \frac {3 \, {\left (17 \, b^{5} c^{7} d^{4} + 152 \, a b^{3} c^{8} d^{4} + 16 \, a^{2} b c^{9} d^{4}\right )}}{c^{7}}\right )} x + \frac {b^{6} c^{6} d^{4} + 396 \, a b^{4} c^{7} d^{4} + 240 \, a^{2} b^{2} c^{8} d^{4} - 192 \, a^{3} c^{9} d^{4}}{c^{7}}\right )} x - \frac {3 \, b^{7} c^{5} d^{4} - 44 \, a b^{5} c^{6} d^{4} - 176 \, a^{2} b^{3} c^{7} d^{4} + 192 \, a^{3} b c^{8} d^{4}}{c^{7}}\right )} - \frac {3 \, {\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {5}{2}}} \]
1/1024*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(2*c^5*d^4*x + 7*b*c^4*d^4) *x + 3*(13*b^2*c^10*d^4 + 4*a*c^11*d^4)/c^7)*x + 5*(11*b^3*c^9*d^4 + 12*a* b*c^10*d^4)/c^7)*x + (161*b^4*c^8*d^4 + 472*a*b^2*c^9*d^4 + 16*a^2*c^10*d^ 4)/c^7)*x + 3*(17*b^5*c^7*d^4 + 152*a*b^3*c^8*d^4 + 16*a^2*b*c^9*d^4)/c^7) *x + (b^6*c^6*d^4 + 396*a*b^4*c^7*d^4 + 240*a^2*b^2*c^8*d^4 - 192*a^3*c^9* d^4)/c^7)*x - (3*b^7*c^5*d^4 - 44*a*b^5*c^6*d^4 - 176*a^2*b^3*c^7*d^4 + 19 2*a^3*b*c^8*d^4)/c^7) - 3/2048*(b^8*d^4 - 16*a*b^6*c*d^4 + 96*a^2*b^4*c^2* d^4 - 256*a^3*b^2*c^3*d^4 + 256*a^4*c^4*d^4)*log(abs(2*(sqrt(c)*x - sqrt(c *x^2 + b*x + a))*sqrt(c) + b))/c^(5/2)
Timed out. \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]