Integrand size = 20, antiderivative size = 207 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}} \] Output:
5/1024*(-4*a*c+b^2)^2*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4-5/384 *(-4*a*c+b^2)*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+1/24*(-b*e+2* c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^2+1/7*e*(c*x^2+b*x+a)^(7/2)/c-5/2048* (-4*a*c+b^2)^3*(-b*e+2*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1 /2))/c^(9/2)
Time = 3.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.50 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^6 e+70 b^5 c (3 d+e x)+28 b^4 c (40 a e-c x (5 d+2 e x))+16 b^3 c^2 \left (c x^2 (7 d+3 e x)-14 a (10 d+3 e x)\right )+64 c^3 \left (48 a^3 e+8 c^3 x^5 (7 d+6 e x)+3 a^2 c x (77 d+48 e x)+2 a c^2 x^3 (91 d+72 e x)\right )+16 b^2 c^2 \left (-231 a^2 e+6 a c x (14 d+5 e x)+2 c^2 x^3 (189 d+148 e x)\right )+32 b c^3 \left (3 a^2 (77 d+19 e x)+8 c^2 x^4 (35 d+29 e x)+2 a c x^2 (273 d+197 e x)\right )\right )+105 \left (b^2-4 a c\right )^3 (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{21504 c^{9/2}} \] Input:
Integrate[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]
Output:
(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^6*e + 70*b^5*c*(3*d + e*x) + 28*b^4 *c*(40*a*e - c*x*(5*d + 2*e*x)) + 16*b^3*c^2*(c*x^2*(7*d + 3*e*x) - 14*a*( 10*d + 3*e*x)) + 64*c^3*(48*a^3*e + 8*c^3*x^5*(7*d + 6*e*x) + 3*a^2*c*x*(7 7*d + 48*e*x) + 2*a*c^2*x^3*(91*d + 72*e*x)) + 16*b^2*c^2*(-231*a^2*e + 6* a*c*x*(14*d + 5*e*x) + 2*c^2*x^3*(189*d + 148*e*x)) + 32*b*c^3*(3*a^2*(77* d + 19*e*x) + 8*c^2*x^4*(35*d + 29*e*x) + 2*a*c*x^2*(273*d + 197*e*x))) + 105*(b^2 - 4*a*c)^3*(-2*c*d + b*e)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(21504*c^(9/2))
Time = 0.31 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1160, 1087, 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.
| \(\displaystyle \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {(2 c d-b e) \int \left (c x^2+b x+a\right )^{5/2}dx}{2 c}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {e \left (a+b x+c x^2\right )^{7/2}}{7 c}\) |
Input:
Int[(d + e*x)*(a + b*x + c*x^2)^(5/2),x]
Output:
(e*(a + b*x + c*x^2)^(7/2))/(7*c) + ((2*c*d - b*e)*(((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(12*c) - (5*(b^2 - 4*a*c)*(((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)))/(24*c)))/(2*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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Time = 0.94 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(d \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\) | \(314\) |
| risch | \(\frac {\left (3072 c^{6} e \,x^{6}+7424 b \,c^{5} e \,x^{5}+3584 c^{6} d \,x^{5}+9216 a \,c^{5} e \,x^{4}+4736 b^{2} c^{4} e \,x^{4}+8960 b \,c^{5} d \,x^{4}+12608 a b \,c^{4} e \,x^{3}+11648 a \,c^{5} d \,x^{3}+48 b^{3} c^{3} e \,x^{3}+6048 b^{2} c^{4} d \,x^{3}+9216 a^{2} c^{4} e \,x^{2}+480 a \,b^{2} c^{3} e \,x^{2}+17472 a b \,c^{4} d \,x^{2}-56 b^{4} c^{2} e \,x^{2}+112 b^{3} c^{3} d \,x^{2}+1824 a^{2} b \,c^{3} e x +14784 a^{2} c^{4} d x -672 a \,b^{3} c^{2} e x +1344 a \,b^{2} c^{3} d x +70 b^{5} c e x -140 b^{4} c^{2} d x +3072 a^{3} c^{3} e -3696 a^{2} b^{2} c^{2} e +7392 a^{2} b \,c^{3} d +1120 a \,b^{4} c e -2240 a \,b^{3} c^{2} d -105 b^{6} e +210 b^{5} c d \right ) \sqrt {c \,x^{2}+b x +a}}{21504 c^{4}}-\frac {5 \left (64 a^{3} b \,c^{3} e -128 a^{3} c^{4} d -48 a^{2} b^{3} c^{2} e +96 a^{2} b^{2} c^{3} d +12 a \,b^{5} c e -24 c^{2} d a \,b^{4}-b^{7} e +2 b^{6} c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {9}{2}}}\) | \(413\) |
Input:
int((e*x+d)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
d*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)* (c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2 )/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))) +e*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/ c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2) /c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*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. 427 vs. \(2 (181) = 362\).
Time = 0.13 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.14 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
[1/86016*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e)*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*(3072*c^7*e*x^6 + 256*(14*c^7*d + 29*b*c^6*e)*x^5 + 128*(70*b*c^6*d + (37*b^2*c^5 + 72*a*c^6)*e)*x^4 + 16*(14*(27*b^2*c^5 + 52*a*c^6)*d + (3*b^3 *c^4 + 788*a*b*c^5)*e)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*d - (7*b^4*c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*e)*x^2 + 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 5 28*a^2*b*c^4)*d - (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^ 3*c^4)*e - 2*(14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d - (35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/43008 *(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d - (b^7 - 1 2*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e)*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*(3072*c^7* e*x^6 + 256*(14*c^7*d + 29*b*c^6*e)*x^5 + 128*(70*b*c^6*d + (37*b^2*c^5 + 72*a*c^6)*e)*x^4 + 16*(14*(27*b^2*c^5 + 52*a*c^6)*d + (3*b^3*c^4 + 788*a*b *c^5)*e)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*d - (7*b^4*c^3 - 60*a*b^2*c^4 - 1152*a^2*c^5)*e)*x^2 + 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)* d - (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e - 2*( 14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*d - (35*b^5*c^2 - 336*a*b^3*c^ 3 + 912*a^2*b*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5]
Leaf count of result is larger than twice the leaf count of optimal. 2428 vs. \(2 (197) = 394\).
Time = 0.86 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.73 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)*(c*x**2+b*x+a)**(5/2),x)
Output:
Piecewise((sqrt(a + b*x + c*x**2)*(c**2*e*x**6/7 + x**5*(29*b*c**2*e/14 + c**3*d)/(6*c) + x**4*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b *c**2*e/14 + c**3*d)/(12*c))/(5*c) + x**3*(6*a*b*c*e + 3*a*c**2*d - 5*a*(2 9*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/ (4*c) + x**2*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3 *b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b* *3*d - 7*b*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b *(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(8*c))/(3*c) + x*(3*a**2*b*e + 3*a**2*c*d + 3*a*b**2*d - 3*a*(6*a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/1 4 + c**3*d)/(6*c) + b**3*e + 3*b**2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(10*c))/(4*c) - 5*b* (3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*b**2*c*e + 3 *b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d - 7*b*(6 *a*b*c*e + 3*a*c**2*d - 5*a*(29*b*c**2*e/14 + c**3*d)/(6*c) + b**3*e + 3*b **2*c*d - 9*b*(15*a*c**2*e/7 + 3*b**2*c*e + 3*b*c**2*d - 11*b*(29*b*c**2*e /14 + c**3*d)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) + (a**3*e + 3*a**2*b*d - 2*a*(3*a**2*c*e + 3*a*b**2*e + 6*a*b*c*d - 4*a*(15*a*c**2*e/7 + 3*b**2*c* e + 3*b*c**2*d - 11*b*(29*b*c**2*e/14 + c**3*d)/(12*c))/(5*c) + b**3*d ...
Exception generated. \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^(5/2),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. 423 vs. \(2 (181) = 362\).
Time = 0.34 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.04 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} e x + \frac {14 \, c^{8} d + 29 \, b c^{7} e}{c^{6}}\right )} x + \frac {70 \, b c^{7} d + 37 \, b^{2} c^{6} e + 72 \, a c^{7} e}{c^{6}}\right )} x + \frac {378 \, b^{2} c^{6} d + 728 \, a c^{7} d + 3 \, b^{3} c^{5} e + 788 \, a b c^{6} e}{c^{6}}\right )} x + \frac {14 \, b^{3} c^{5} d + 2184 \, a b c^{6} d - 7 \, b^{4} c^{4} e + 60 \, a b^{2} c^{5} e + 1152 \, a^{2} c^{6} e}{c^{6}}\right )} x - \frac {70 \, b^{4} c^{4} d - 672 \, a b^{2} c^{5} d - 7392 \, a^{2} c^{6} d - 35 \, b^{5} c^{3} e + 336 \, a b^{3} c^{4} e - 912 \, a^{2} b c^{5} e}{c^{6}}\right )} x + \frac {210 \, b^{5} c^{3} d - 2240 \, a b^{3} c^{4} d + 7392 \, a^{2} b c^{5} d - 105 \, b^{6} c^{2} e + 1120 \, a b^{4} c^{3} e - 3696 \, a^{2} b^{2} c^{4} e + 3072 \, a^{3} c^{5} e}{c^{6}}\right )} + \frac {5 \, {\left (2 \, b^{6} c d - 24 \, a b^{4} c^{2} d + 96 \, a^{2} b^{2} c^{3} d - 128 \, a^{3} c^{4} d - b^{7} e + 12 \, a b^{5} c e - 48 \, a^{2} b^{3} c^{2} e + 64 \, a^{3} b c^{3} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*c^2*e*x + (14*c^8*d + 29* b*c^7*e)/c^6)*x + (70*b*c^7*d + 37*b^2*c^6*e + 72*a*c^7*e)/c^6)*x + (378*b ^2*c^6*d + 728*a*c^7*d + 3*b^3*c^5*e + 788*a*b*c^6*e)/c^6)*x + (14*b^3*c^5 *d + 2184*a*b*c^6*d - 7*b^4*c^4*e + 60*a*b^2*c^5*e + 1152*a^2*c^6*e)/c^6)* x - (70*b^4*c^4*d - 672*a*b^2*c^5*d - 7392*a^2*c^6*d - 35*b^5*c^3*e + 336* a*b^3*c^4*e - 912*a^2*b*c^5*e)/c^6)*x + (210*b^5*c^3*d - 2240*a*b^3*c^4*d + 7392*a^2*b*c^5*d - 105*b^6*c^2*e + 1120*a*b^4*c^3*e - 3696*a^2*b^2*c^4*e + 3072*a^3*c^5*e)/c^6) + 5/2048*(2*b^6*c*d - 24*a*b^4*c^2*d + 96*a^2*b^2* c^3*d - 128*a^3*c^4*d - b^7*e + 12*a*b^5*c*e - 48*a^2*b^3*c^2*e + 64*a^3*b *c^3*e)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2 )
Timed out. \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:
int((d + e*x)*(a + b*x + c*x^2)^(5/2),x)
Output:
int((d + e*x)*(a + b*x + c*x^2)^(5/2), x)
Time = 2.43 (sec) , antiderivative size = 987, normalized size of antiderivative = 4.77 \[ \int (d+e x) \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)*(c*x^2+b*x+a)^(5/2),x)
Output:
(6144*sqrt(a + b*x + c*x**2)*a**3*c**4*e - 7392*sqrt(a + b*x + c*x**2)*a** 2*b**2*c**3*e + 14784*sqrt(a + b*x + c*x**2)*a**2*b*c**4*d + 3648*sqrt(a + b*x + c*x**2)*a**2*b*c**4*e*x + 29568*sqrt(a + b*x + c*x**2)*a**2*c**5*d* x + 18432*sqrt(a + b*x + c*x**2)*a**2*c**5*e*x**2 + 2240*sqrt(a + b*x + c* x**2)*a*b**4*c**2*e - 4480*sqrt(a + b*x + c*x**2)*a*b**3*c**3*d - 1344*sqr t(a + b*x + c*x**2)*a*b**3*c**3*e*x + 2688*sqrt(a + b*x + c*x**2)*a*b**2*c **4*d*x + 960*sqrt(a + b*x + c*x**2)*a*b**2*c**4*e*x**2 + 34944*sqrt(a + b *x + c*x**2)*a*b*c**5*d*x**2 + 25216*sqrt(a + b*x + c*x**2)*a*b*c**5*e*x** 3 + 23296*sqrt(a + b*x + c*x**2)*a*c**6*d*x**3 + 18432*sqrt(a + b*x + c*x* *2)*a*c**6*e*x**4 - 210*sqrt(a + b*x + c*x**2)*b**6*c*e + 420*sqrt(a + b*x + c*x**2)*b**5*c**2*d + 140*sqrt(a + b*x + c*x**2)*b**5*c**2*e*x - 280*sq rt(a + b*x + c*x**2)*b**4*c**3*d*x - 112*sqrt(a + b*x + c*x**2)*b**4*c**3* e*x**2 + 224*sqrt(a + b*x + c*x**2)*b**3*c**4*d*x**2 + 96*sqrt(a + b*x + c *x**2)*b**3*c**4*e*x**3 + 12096*sqrt(a + b*x + c*x**2)*b**2*c**5*d*x**3 + 9472*sqrt(a + b*x + c*x**2)*b**2*c**5*e*x**4 + 17920*sqrt(a + b*x + c*x**2 )*b*c**6*d*x**4 + 14848*sqrt(a + b*x + c*x**2)*b*c**6*e*x**5 + 7168*sqrt(a + b*x + c*x**2)*c**7*d*x**5 + 6144*sqrt(a + b*x + c*x**2)*c**7*e*x**6 - 6 720*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**3*e + 13440*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**4*d + 5040*sqrt(c)*log((2*s...