Integrand size = 26, antiderivative size = 160 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right )^2 e (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3}+\frac {\left (b^2-4 a c\right ) e (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^2}+\frac {(12 c d-b e+10 c e x) \left (a+b x+c x^2\right )^{5/2}}{30 c}+\frac {\left (b^2-4 a c\right )^3 e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2}} \] Output:
-1/256*(-4*a*c+b^2)^2*e*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^3+1/96*(-4*a*c+b^2 )*e*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^2+1/30*(10*c*e*x-b*e+12*c*d)*(c*x^2+b* x+a)^(5/2)/c+1/512*(-4*a*c+b^2)^3*e*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b *x+a)^(1/2))/c^(7/2)
Time = 3.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.40 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-15 b^5 e+10 b^4 c e x-8 b^3 c^2 e x^2+256 c^5 x^4 (6 d+5 e x)+128 b c^4 x^3 (24 d+19 e x)+48 b^2 c^3 x^2 (32 d+23 e x)+48 a^2 c^2 (32 c d-11 b e+10 c e x)+32 a c \left (5 b^3 e-3 b^2 c e x+3 b c^2 x (32 d+19 e x)+2 c^3 x^2 (48 d+35 e x)\right )\right )+15 \left (b^2-4 a c\right )^3 e \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{3840 c^{7/2}} \] Input:
Integrate[(b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^(3/2),x]
Output:
(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-15*b^5*e + 10*b^4*c*e*x - 8*b^3*c^2*e*x^2 + 256*c^5*x^4*(6*d + 5*e*x) + 128*b*c^4*x^3*(24*d + 19*e*x) + 48*b^2*c^3* x^2*(32*d + 23*e*x) + 48*a^2*c^2*(32*c*d - 11*b*e + 10*c*e*x) + 32*a*c*(5* b^3*e - 3*b^2*c*e*x + 3*b*c^2*x*(32*d + 19*e*x) + 2*c^3*x^2*(48*d + 35*e*x ))) + 15*(b^2 - 4*a*c)^3*e*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(3840*c^(7/2))
Time = 0.52 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1225, 1087, 1087, 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.
| \(\displaystyle \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {e \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{12 c}+\frac {\left (a+b x+c x^2\right )^{5/2} (-b e+12 c d+10 c e x)}{30 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {e \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{12 c}+\frac {\left (a+b x+c x^2\right )^{5/2} (-b e+12 c d+10 c e x)}{30 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {e \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{12 c}+\frac {\left (a+b x+c x^2\right )^{5/2} (-b e+12 c d+10 c e x)}{30 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {e \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{12 c}+\frac {\left (a+b x+c x^2\right )^{5/2} (-b e+12 c d+10 c e x)}{30 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {e \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{12 c}+\frac {\left (a+b x+c x^2\right )^{5/2} (-b e+12 c d+10 c e x)}{30 c}\) |
Input:
Int[(b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^(3/2),x]
Output:
((12*c*d - b*e + 10*c*e*x)*(a + b*x + c*x^2)^(5/2))/(30*c) + ((b^2 - 4*a*c )*e*(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/ (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c)))/(12*c)
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
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_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Time = 1.51 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(-\frac {\left (-1280 c^{5} e \,x^{5}-2432 b \,c^{4} e \,x^{4}-1536 c^{5} d \,x^{4}-2240 a \,c^{4} e \,x^{3}-1104 b^{2} c^{3} e \,x^{3}-3072 b \,c^{4} d \,x^{3}-1824 b \,c^{3} a e \,x^{2}-3072 a \,c^{4} d \,x^{2}+8 b^{3} c^{2} e \,x^{2}-1536 b^{2} c^{3} d \,x^{2}-480 a^{2} c^{3} e x +96 a \,b^{2} c^{2} e x -3072 a b \,c^{3} d x -10 b^{4} c e x +528 a^{2} b \,c^{2} e -1536 a^{2} c^{3} d -160 a \,b^{3} c e +15 b^{5} e \right ) \sqrt {c \,x^{2}+b x +a}}{3840 c^{3}}-\frac {e \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {7}{2}}}\) | \(259\) |
| default | \(b d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+\left (b e +2 c d \right ) \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+2 c e \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )\) | \(509\) |
Input:
int((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3840/c^3*(-1280*c^5*e*x^5-2432*b*c^4*e*x^4-1536*c^5*d*x^4-2240*a*c^4*e* x^3-1104*b^2*c^3*e*x^3-3072*b*c^4*d*x^3-1824*a*b*c^3*e*x^2-3072*a*c^4*d*x^ 2+8*b^3*c^2*e*x^2-1536*b^2*c^3*d*x^2-480*a^2*c^3*e*x+96*a*b^2*c^2*e*x-3072 *a*b*c^3*d*x-10*b^4*c*e*x+528*a^2*b*c^2*e-1536*a^2*c^3*d-160*a*b^3*c*e+15* b^5*e)*(c*x^2+b*x+a)^(1/2)-1/512*e*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b ^6)/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (138) = 276\).
Time = 0.11 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.49 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} e \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (1280 \, c^{6} e x^{5} + 1536 \, a^{2} c^{4} d + 128 \, {\left (12 \, c^{6} d + 19 \, b c^{5} e\right )} x^{4} + 16 \, {\left (192 \, b c^{5} d + {\left (69 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e\right )} x^{3} + 8 \, {\left (192 \, {\left (b^{2} c^{4} + 2 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 228 \, a b c^{4}\right )} e\right )} x^{2} - {\left (15 \, b^{5} c - 160 \, a b^{3} c^{2} + 528 \, a^{2} b c^{3}\right )} e + 2 \, {\left (1536 \, a b c^{4} d + {\left (5 \, b^{4} c^{2} - 48 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{4}}, -\frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} e \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (1280 \, c^{6} e x^{5} + 1536 \, a^{2} c^{4} d + 128 \, {\left (12 \, c^{6} d + 19 \, b c^{5} e\right )} x^{4} + 16 \, {\left (192 \, b c^{5} d + {\left (69 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e\right )} x^{3} + 8 \, {\left (192 \, {\left (b^{2} c^{4} + 2 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 228 \, a b c^{4}\right )} e\right )} x^{2} - {\left (15 \, b^{5} c - 160 \, a b^{3} c^{2} + 528 \, a^{2} b c^{3}\right )} e + 2 \, {\left (1536 \, a b c^{4} d + {\left (5 \, b^{4} c^{2} - 48 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{4}}\right ] \] Input:
integrate((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
[-1/15360*(15*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c)*e*l og(-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*(1280*c^6*e*x^5 + 1536*a^2*c^4*d + 128*(12*c^6*d + 19*b*c^5 *e)*x^4 + 16*(192*b*c^5*d + (69*b^2*c^4 + 140*a*c^5)*e)*x^3 + 8*(192*(b^2* c^4 + 2*a*c^5)*d - (b^3*c^3 - 228*a*b*c^4)*e)*x^2 - (15*b^5*c - 160*a*b^3* c^2 + 528*a^2*b*c^3)*e + 2*(1536*a*b*c^4*d + (5*b^4*c^2 - 48*a*b^2*c^3 + 2 40*a^2*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^4, -1/7680*(15*(b^6 - 12*a*b^4* c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-c)*e*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*(1280*c^6*e*x^5 + 153 6*a^2*c^4*d + 128*(12*c^6*d + 19*b*c^5*e)*x^4 + 16*(192*b*c^5*d + (69*b^2* c^4 + 140*a*c^5)*e)*x^3 + 8*(192*(b^2*c^4 + 2*a*c^5)*d - (b^3*c^3 - 228*a* b*c^4)*e)*x^2 - (15*b^5*c - 160*a*b^3*c^2 + 528*a^2*b*c^3)*e + 2*(1536*a*b *c^4*d + (5*b^4*c^2 - 48*a*b^2*c^3 + 240*a^2*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^4]
Leaf count of result is larger than twice the leaf count of optimal. 1564 vs. \(2 (151) = 302\).
Time = 1.33 (sec) , antiderivative size = 1564, normalized size of antiderivative = 9.78 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)*(e*x+d)*(c*x**2+b*x+a)**(3/2),x)
Output:
Piecewise((sqrt(a + b*x + c*x**2)*(c**2*e*x**5/3 + x**4*(19*b*c**2*e/6 + 2 *c**3*d)/(5*c) + x**3*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2*d - 9*b*(19*b* c**2*e/6 + 2*c**3*d)/(10*c))/(4*c) + x**2*(6*a*b*c*e + 4*a*c**2*d - 4*a*(1 9*b*c**2*e/6 + 2*c**3*d)/(5*c) + b**3*e + 4*b**2*c*d - 7*b*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2*d - 9*b*(19*b*c**2*e/6 + 2*c**3*d)/(10*c))/(8*c))/( 3*c) + x*(2*a**2*c*e + 2*a*b**2*e + 6*a*b*c*d - 3*a*(7*a*c**2*e/3 + 4*b**2 *c*e + 5*b*c**2*d - 9*b*(19*b*c**2*e/6 + 2*c**3*d)/(10*c))/(4*c) + b**3*d - 5*b*(6*a*b*c*e + 4*a*c**2*d - 4*a*(19*b*c**2*e/6 + 2*c**3*d)/(5*c) + b** 3*e + 4*b**2*c*d - 7*b*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2*d - 9*b*(19*b *c**2*e/6 + 2*c**3*d)/(10*c))/(8*c))/(6*c))/(2*c) + (a**2*b*e + 2*a**2*c*d + 2*a*b**2*d - 2*a*(6*a*b*c*e + 4*a*c**2*d - 4*a*(19*b*c**2*e/6 + 2*c**3* d)/(5*c) + b**3*e + 4*b**2*c*d - 7*b*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2 *d - 9*b*(19*b*c**2*e/6 + 2*c**3*d)/(10*c))/(8*c))/(3*c) - 3*b*(2*a**2*c*e + 2*a*b**2*e + 6*a*b*c*d - 3*a*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2*d - 9*b*(19*b*c**2*e/6 + 2*c**3*d)/(10*c))/(4*c) + b**3*d - 5*b*(6*a*b*c*e + 4 *a*c**2*d - 4*a*(19*b*c**2*e/6 + 2*c**3*d)/(5*c) + b**3*e + 4*b**2*c*d - 7 *b*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2*d - 9*b*(19*b*c**2*e/6 + 2*c**3*d )/(10*c))/(8*c))/(6*c))/(4*c))/c) + (a**2*b*d - a*(2*a**2*c*e + 2*a*b**2*e + 6*a*b*c*d - 3*a*(7*a*c**2*e/3 + 4*b**2*c*e + 5*b*c**2*d - 9*b*(19*b*c** 2*e/6 + 2*c**3*d)/(10*c))/(4*c) + b**3*d - 5*b*(6*a*b*c*e + 4*a*c**2*d ...
Exception generated. \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*x+b)*(e*x+d)*(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
Time = 0.20 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.72 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{3840} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c^{2} e x + \frac {12 \, c^{7} d + 19 \, b c^{6} e}{c^{5}}\right )} x + \frac {192 \, b c^{6} d + 69 \, b^{2} c^{5} e + 140 \, a c^{6} e}{c^{5}}\right )} x + \frac {192 \, b^{2} c^{5} d + 384 \, a c^{6} d - b^{3} c^{4} e + 228 \, a b c^{5} e}{c^{5}}\right )} x + \frac {1536 \, a b c^{5} d + 5 \, b^{4} c^{3} e - 48 \, a b^{2} c^{4} e + 240 \, a^{2} c^{5} e}{c^{5}}\right )} x + \frac {1536 \, a^{2} c^{5} d - 15 \, b^{5} c^{2} e + 160 \, a b^{3} c^{3} e - 528 \, a^{2} b c^{4} e}{c^{5}}\right )} - \frac {{\left (b^{6} e - 12 \, a b^{4} c e + 48 \, a^{2} b^{2} c^{2} e - 64 \, a^{3} c^{3} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{512 \, c^{\frac {7}{2}}} \] Input:
integrate((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
1/3840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c^2*e*x + (12*c^7*d + 19*b*c^ 6*e)/c^5)*x + (192*b*c^6*d + 69*b^2*c^5*e + 140*a*c^6*e)/c^5)*x + (192*b^2 *c^5*d + 384*a*c^6*d - b^3*c^4*e + 228*a*b*c^5*e)/c^5)*x + (1536*a*b*c^5*d + 5*b^4*c^3*e - 48*a*b^2*c^4*e + 240*a^2*c^5*e)/c^5)*x + (1536*a^2*c^5*d - 15*b^5*c^2*e + 160*a*b^3*c^3*e - 528*a^2*b*c^4*e)/c^5) - 1/512*(b^6*e - 12*a*b^4*c*e + 48*a^2*b^2*c^2*e - 64*a^3*c^3*e)*log(abs(2*(sqrt(c)*x - sqr t(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2)
Timed out. \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\int \left (b+2\,c\,x\right )\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \] Input:
int((b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^(3/2),x)
Output:
int((b + 2*c*x)*(d + e*x)*(a + b*x + c*x^2)^(3/2), x)
Time = 1.40 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.58 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {-1056 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{3} e +3072 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{4} d +960 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{4} e x +320 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{2} e -192 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{3} e x +6144 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{4} d x +3648 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{4} e \,x^{2}+6144 \sqrt {c \,x^{2}+b x +a}\, a \,c^{5} d \,x^{2}+4480 \sqrt {c \,x^{2}+b x +a}\, a \,c^{5} e \,x^{3}-30 \sqrt {c \,x^{2}+b x +a}\, b^{5} c e +20 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{2} e x -16 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{3} e \,x^{2}+3072 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{4} d \,x^{2}+2208 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{4} e \,x^{3}+6144 \sqrt {c \,x^{2}+b x +a}\, b \,c^{5} d \,x^{3}+4864 \sqrt {c \,x^{2}+b x +a}\, b \,c^{5} e \,x^{4}+3072 \sqrt {c \,x^{2}+b x +a}\, c^{6} d \,x^{4}+2560 \sqrt {c \,x^{2}+b x +a}\, c^{6} e \,x^{5}-960 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{3} c^{3} e +720 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{2} c^{2} e -180 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{4} c e +15 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{6} e}{7680 c^{4}} \] Input:
int((2*c*x+b)*(e*x+d)*(c*x^2+b*x+a)^(3/2),x)
Output:
( - 1056*sqrt(a + b*x + c*x**2)*a**2*b*c**3*e + 3072*sqrt(a + b*x + c*x**2 )*a**2*c**4*d + 960*sqrt(a + b*x + c*x**2)*a**2*c**4*e*x + 320*sqrt(a + b* x + c*x**2)*a*b**3*c**2*e - 192*sqrt(a + b*x + c*x**2)*a*b**2*c**3*e*x + 6 144*sqrt(a + b*x + c*x**2)*a*b*c**4*d*x + 3648*sqrt(a + b*x + c*x**2)*a*b* c**4*e*x**2 + 6144*sqrt(a + b*x + c*x**2)*a*c**5*d*x**2 + 4480*sqrt(a + b* x + c*x**2)*a*c**5*e*x**3 - 30*sqrt(a + b*x + c*x**2)*b**5*c*e + 20*sqrt(a + b*x + c*x**2)*b**4*c**2*e*x - 16*sqrt(a + b*x + c*x**2)*b**3*c**3*e*x** 2 + 3072*sqrt(a + b*x + c*x**2)*b**2*c**4*d*x**2 + 2208*sqrt(a + b*x + c*x **2)*b**2*c**4*e*x**3 + 6144*sqrt(a + b*x + c*x**2)*b*c**5*d*x**3 + 4864*s qrt(a + b*x + c*x**2)*b*c**5*e*x**4 + 3072*sqrt(a + b*x + c*x**2)*c**6*d*x **4 + 2560*sqrt(a + b*x + c*x**2)*c**6*e*x**5 - 960*sqrt(c)*log((2*sqrt(c) *sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*c**3*e + 720 *sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b **2))*a**2*b**2*c**2*e - 180*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*e + 15*sqrt(c)*log((2*sqrt(c)*s qrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**6*e)/(7680*c**4)