Integrand size = 44, antiderivative size = 424 \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {9 (2 c d-b e)^5 (16 c e f+6 c d g-11 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^6 e}+\frac {3 (2 c d-b e)^3 (16 c e f+6 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2048 c^5 e}-\frac {(16 c e f+6 c d g-11 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{112 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}-\frac {3 (2 c d-b e) (16 c e f+6 c d g-11 b e g) (24 c d-7 b e+10 c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4480 c^4 e^2}+\frac {9 (2 c d-b e)^7 (16 c e f+6 c d g-11 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16384 c^{13/2} e^2} \] Output:
9/16384*(-b*e+2*c*d)^5*(-11*b*e*g+6*c*d*g+16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d )-b*e^2*x-c*e^2*x^2)^(1/2)/c^6/e+3/2048*(-b*e+2*c*d)^3*(-11*b*e*g+6*c*d*g+ 16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^5/e-1/112*(-1 1*b*e*g+6*c*d*g+16*c*e*f)*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2) /c^2/e^2-1/8*g*(e*x+d)^3*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c/e^2-3/44 80*(-b*e+2*c*d)*(-11*b*e*g+6*c*d*g+16*c*e*f)*(10*c*e*x-7*b*e+24*c*d)*(d*(- b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^4/e^2+9/16384*(-b*e+2*c*d)^7*(-11*b*e* g+6*c*d*g+16*c*e*f)*arctan(c^(1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 )^(1/2))/c^(13/2)/e^2
Time = 3.76 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.73 \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {(2 c d-b e)^7 ((d+e x) (-b e+c (d-e x)))^{3/2} \left (-\frac {\sqrt {c} \left (-3465 b^7 e^7 g+210 b^6 c e^6 (24 e f+218 d g+11 e g x)-84 b^5 c^2 e^5 \left (3057 d^2 g+2 e^2 x (20 f+11 g x)+d e (760 f+334 g x)\right )+128 c^7 \left (1664 d^7 g+320 d e^6 x^5 (7 f+6 g x)+80 e^7 x^6 (8 f+7 g x)-16 d^3 e^4 x^3 (175 f+136 g x)+8 d^2 e^5 x^4 (208 f+175 g x)-8 d^5 e^2 x (245 f+176 g x)+d^6 e (2944 f+945 g x)-2 d^4 e^3 x^2 (2624 f+1925 g x)\right )+24 b^4 c^3 e^4 \left (32924 d^3 g+2 e^3 x^2 (56 f+33 g x)+8 d e^2 x (203 f+107 g x)+3 d^2 e (4704 f+1963 g x)\right )+64 b c^6 e \left (-13647 d^6 g+80 e^6 x^5 (20 f+17 g x)+6 d^4 e^2 x (-116 f+123 g x)+48 d e^5 x^4 (164 f+135 g x)+8 d^3 e^3 x^2 (1574 f+1187 g x)+8 d^2 e^4 x^3 (1882 f+1483 g x)-2 d^5 e (9812 f+3263 g x)\right )-16 b^3 c^4 e^3 \left (89587 d^4 g+8 e^4 x^3 (18 f+11 g x)+8 d e^3 x^2 (222 f+125 g x)+12 d^2 e^2 x (960 f+479 g x)+4 d^3 e (15072 f+5887 g x)\right )+32 b^2 c^5 e^2 \left (47490 d^5 g+8 e^5 x^4 (8 f+5 g x)+16 d e^4 x^3 (43 f+25 g x)+12 d^2 e^3 x^2 (308 f+163 g x)+8 d^3 e^2 x (1748 f+809 g x)+d^4 e (48712 f+17401 g x)\right )\right )}{(2 c d-b e)^7 (d+e x) (-b e+c (d-e x))}-\frac {315 (16 c e f+6 c d g-11 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{3/2} (-b e+c (d-e x))^{3/2}}\right )}{573440 c^{13/2} e^2} \] Input:
Integrate[(d + e*x)^3*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2 ),x]
Output:
((2*c*d - b*e)^7*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-((Sqrt[c]*(-34 65*b^7*e^7*g + 210*b^6*c*e^6*(24*e*f + 218*d*g + 11*e*g*x) - 84*b^5*c^2*e^ 5*(3057*d^2*g + 2*e^2*x*(20*f + 11*g*x) + d*e*(760*f + 334*g*x)) + 128*c^7 *(1664*d^7*g + 320*d*e^6*x^5*(7*f + 6*g*x) + 80*e^7*x^6*(8*f + 7*g*x) - 16 *d^3*e^4*x^3*(175*f + 136*g*x) + 8*d^2*e^5*x^4*(208*f + 175*g*x) - 8*d^5*e ^2*x*(245*f + 176*g*x) + d^6*e*(2944*f + 945*g*x) - 2*d^4*e^3*x^2*(2624*f + 1925*g*x)) + 24*b^4*c^3*e^4*(32924*d^3*g + 2*e^3*x^2*(56*f + 33*g*x) + 8 *d*e^2*x*(203*f + 107*g*x) + 3*d^2*e*(4704*f + 1963*g*x)) + 64*b*c^6*e*(-1 3647*d^6*g + 80*e^6*x^5*(20*f + 17*g*x) + 6*d^4*e^2*x*(-116*f + 123*g*x) + 48*d*e^5*x^4*(164*f + 135*g*x) + 8*d^3*e^3*x^2*(1574*f + 1187*g*x) + 8*d^ 2*e^4*x^3*(1882*f + 1483*g*x) - 2*d^5*e*(9812*f + 3263*g*x)) - 16*b^3*c^4* e^3*(89587*d^4*g + 8*e^4*x^3*(18*f + 11*g*x) + 8*d*e^3*x^2*(222*f + 125*g* x) + 12*d^2*e^2*x*(960*f + 479*g*x) + 4*d^3*e*(15072*f + 5887*g*x)) + 32*b ^2*c^5*e^2*(47490*d^5*g + 8*e^5*x^4*(8*f + 5*g*x) + 16*d*e^4*x^3*(43*f + 2 5*g*x) + 12*d^2*e^3*x^2*(308*f + 163*g*x) + 8*d^3*e^2*x*(1748*f + 809*g*x) + d^4*e*(48712*f + 17401*g*x))))/((2*c*d - b*e)^7*(d + e*x)*(-(b*e) + c*( d - e*x)))) - (315*(16*c*e*f + 6*c*d*g - 11*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/((d + e*x)^(3/2)*(-(b*e) + c*(d - e*x))^ (3/2))))/(573440*c^(13/2)*e^2)
Time = 1.28 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1221, 1134, 1134, 1160, 1087, 1087, 1092, 217}
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.
| \(\displaystyle \int (d+e x)^3 (f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \int (d+e x)^3 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \left (\frac {9 (2 c d-b e) \int (d+e x)^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right )}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 1134 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {7 (2 c d-b e) \int (d+e x) \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right )}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right )}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right )}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right )}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {(-11 b e g+6 c d g+16 c e f) \left (\frac {9 (2 c d-b e) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right )}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (\frac {9 (2 c d-b e) \left (\frac {7 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e}\right )}{12 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{6 c e}\right )}{14 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e}\right ) (-11 b e g+6 c d g+16 c e f)}{16 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2}\) |
Input:
Int[(d + e*x)^3*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
Output:
-1/8*(g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(c*e^2) + ((16*c*e*f + 6*c*d*g - 11*b*e*g)*(-1/7*((d + e*x)^2*(d*(c*d - b*e) - b*e^ 2*x - c*e^2*x^2)^(5/2))/(c*e) + (9*(2*c*d - b*e)*(-1/6*((d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(c*e) + (7*(2*c*d - b*e)*(-1/5*(d*(c* d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2)/(c*e) + ((2*c*d - b*e)*(((b + 2*c*x) *(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(8*c) + (3*(2*c*d - b*e)^2*( ((b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c) + ((2*c*d - b*e)^2*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c* e^2*x^2])])/(8*c^(3/2)*e)))/(16*c)))/(2*c)))/(12*c)))/(14*c)))/(16*c*e)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^ (m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 *p]
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]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3508\) vs. \(2(396)=792\).
Time = 3.53 (sec) , antiderivative size = 3509, normalized size of antiderivative = 8.28
Input:
int((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RET URNVERBOSE)
Output:
d^3*f*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2 )-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/ e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^ 2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b* e^2*x-b*d*e+c*d^2)^(1/2))))+e^2*(3*d*g+e*f)*(-1/7*x^2*(-c*e^2*x^2-b*e^2*x- b*d*e+c*d^2)^(5/2)/c/e^2-9/14*b/c*(-1/6*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2) ^(5/2)/c/e^2-7/12*b/c*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c/e^2-1 /2*b/c*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/ 2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c /e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b ^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b *e^2*x-b*d*e+c*d^2)^(1/2)))))+1/6*(-b*d*e+c*d^2)/c/e^2*(-1/8*(-2*c*e^2*x-b *e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+ c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b *d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2 )*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)) )))+2/7*(-b*d*e+c*d^2)/c/e^2*(-1/5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/ c/e^2-1/2*b/c*(-1/8*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d ^2)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/4*(-2*c*e^2*x-b *e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*...
Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (396) = 792\).
Time = 3.43 (sec) , antiderivative size = 2337, normalized size of antiderivative = 5.51 \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo rithm="fricas")
Output:
[-1/2293760*(315*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5* e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14* b^6*c^2*d*e^7 - b^7*c*e^8)*f + (768*c^8*d^8 - 4096*b*c^7*d^7*e + 8960*b^2* c^6*d^6*e^2 - 10752*b^3*c^5*d^5*e^3 + 7840*b^4*c^4*d^4*e^4 - 3584*b^5*c^3* d^3*e^5 + 1008*b^6*c^2*d^2*e^6 - 160*b^7*c*d*e^7 + 11*b^8*e^8)*g)*sqrt(-c) *log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 - 4*sqr t(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*(716 80*c^8*e^7*g*x^7 + 5120*(16*c^8*e^7*f + (48*c^8*d*e^6 + 17*b*c^7*e^7)*g)*x ^6 + 1280*(16*(14*c^8*d*e^6 + 5*b*c^7*e^7)*f + (140*c^8*d^2*e^5 + 324*b*c^ 7*d*e^6 + b^2*c^6*e^7)*g)*x^5 + 128*(16*(104*c^8*d^2*e^5 + 246*b*c^7*d*e^6 + b^2*c^6*e^7)*f - (2176*c^8*d^3*e^4 - 5932*b*c^7*d^2*e^5 - 100*b^2*c^6*d *e^6 + 11*b^3*c^5*e^7)*g)*x^4 - 16*(16*(1400*c^8*d^3*e^4 - 3764*b*c^7*d^2* e^5 - 86*b^2*c^6*d*e^6 + 9*b^3*c^5*e^7)*f + (30800*c^8*d^4*e^3 - 37984*b*c ^7*d^3*e^4 - 3912*b^2*c^6*d^2*e^5 + 1000*b^3*c^5*d*e^6 - 99*b^4*c^4*e^7)*g )*x^3 - 8*(16*(5248*c^8*d^4*e^3 - 6296*b*c^7*d^3*e^4 - 924*b^2*c^6*d^2*e^5 + 222*b^3*c^5*d*e^6 - 21*b^4*c^4*e^7)*f + (22528*c^8*d^5*e^2 - 5904*b*c^7 *d^4*e^3 - 25888*b^2*c^6*d^3*e^4 + 11496*b^3*c^5*d^2*e^5 - 2568*b^4*c^4*d* e^6 + 231*b^5*c^3*e^7)*g)*x^2 + 16*(23552*c^8*d^6*e - 78496*b*c^7*d^5*e^2 + 97424*b^2*c^6*d^4*e^3 - 60288*b^3*c^5*d^3*e^4 + 21168*b^4*c^4*d^2*e^5 - 3990*b^5*c^3*d*e^6 + 315*b^6*c^2*e^7)*f + (212992*c^8*d^7 - 873408*b*c^...
Leaf count of result is larger than twice the leaf count of optimal. 9527 vs. \(2 (418) = 836\).
Time = 1.77 (sec) , antiderivative size = 9527, normalized size of antiderivative = 22.47 \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**3*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x )
Output:
Piecewise((sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(-c*e**5*g*x**7/ 8 - x**6*(17*b*c*e**7*g/16 + 3*c**2*d*e**6*g + c**2*e**7*f)/(7*c*e**2) - x **5*(b**2*e**7*g + 8*b*c*d*e**6*g + 2*b*c*e**7*f - 13*b*(17*b*c*e**7*g/16 + 3*c**2*d*e**6*g + c**2*e**7*f)/(14*c) + c**2*d**2*e**5*g + 3*c**2*d*e**6 *f + c*e**5*g*(-7*b*d*e + 7*c*d**2)/8)/(6*c*e**2) - x**4*(5*b**2*d*e**6*g + b**2*e**7*f + 10*b*c*d**2*e**5*g + 8*b*c*d*e**6*f - 11*b*(b**2*e**7*g + 8*b*c*d*e**6*g + 2*b*c*e**7*f - 13*b*(17*b*c*e**7*g/16 + 3*c**2*d*e**6*g + c**2*e**7*f)/(14*c) + c**2*d**2*e**5*g + 3*c**2*d*e**6*f + c*e**5*g*(-7*b *d*e + 7*c*d**2)/8)/(12*c) - 5*c**2*d**3*e**4*g + c**2*d**2*e**5*f + (-6*b *d*e + 6*c*d**2)*(17*b*c*e**7*g/16 + 3*c**2*d*e**6*g + c**2*e**7*f)/(7*c*e **2))/(5*c*e**2) - x**3*(10*b**2*d**2*e**5*g + 5*b**2*d*e**6*f + 10*b*c*d* *2*e**5*f - 9*b*(5*b**2*d*e**6*g + b**2*e**7*f + 10*b*c*d**2*e**5*g + 8*b* c*d*e**6*f - 11*b*(b**2*e**7*g + 8*b*c*d*e**6*g + 2*b*c*e**7*f - 13*b*(17* b*c*e**7*g/16 + 3*c**2*d*e**6*g + c**2*e**7*f)/(14*c) + c**2*d**2*e**5*g + 3*c**2*d*e**6*f + c*e**5*g*(-7*b*d*e + 7*c*d**2)/8)/(12*c) - 5*c**2*d**3* e**4*g + c**2*d**2*e**5*f + (-6*b*d*e + 6*c*d**2)*(17*b*c*e**7*g/16 + 3*c* *2*d*e**6*g + c**2*e**7*f)/(7*c*e**2))/(10*c) - 5*c**2*d**4*e**3*g - 5*c** 2*d**3*e**4*f + (-5*b*d*e + 5*c*d**2)*(b**2*e**7*g + 8*b*c*d*e**6*g + 2*b* c*e**7*f - 13*b*(17*b*c*e**7*g/16 + 3*c**2*d*e**6*g + c**2*e**7*f)/(14*c) + c**2*d**2*e**5*g + 3*c**2*d*e**6*f + c*e**5*g*(-7*b*d*e + 7*c*d**2)/8...
Exception generated. \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo rithm="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(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 1229 vs. \(2 (396) = 792\).
Time = 0.42 (sec) , antiderivative size = 1229, normalized size of antiderivative = 2.90 \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algo rithm="giac")
Output:
-1/573440*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(2*(8*(10*(4*(1 4*c*e^5*g*x + (16*c^8*e^17*f + 48*c^8*d*e^16*g + 17*b*c^7*e^17*g)/(c^7*e^1 2))*x + (224*c^8*d*e^16*f + 80*b*c^7*e^17*f + 140*c^8*d^2*e^15*g + 324*b*c ^7*d*e^16*g + b^2*c^6*e^17*g)/(c^7*e^12))*x + (1664*c^8*d^2*e^15*f + 3936* b*c^7*d*e^16*f + 16*b^2*c^6*e^17*f - 2176*c^8*d^3*e^14*g + 5932*b*c^7*d^2* e^15*g + 100*b^2*c^6*d*e^16*g - 11*b^3*c^5*e^17*g)/(c^7*e^12))*x - (22400* c^8*d^3*e^14*f - 60224*b*c^7*d^2*e^15*f - 1376*b^2*c^6*d*e^16*f + 144*b^3* c^5*e^17*f + 30800*c^8*d^4*e^13*g - 37984*b*c^7*d^3*e^14*g - 3912*b^2*c^6* d^2*e^15*g + 1000*b^3*c^5*d*e^16*g - 99*b^4*c^4*e^17*g)/(c^7*e^12))*x - (8 3968*c^8*d^4*e^13*f - 100736*b*c^7*d^3*e^14*f - 14784*b^2*c^6*d^2*e^15*f + 3552*b^3*c^5*d*e^16*f - 336*b^4*c^4*e^17*f + 22528*c^8*d^5*e^12*g - 5904* b*c^7*d^4*e^13*g - 25888*b^2*c^6*d^3*e^14*g + 11496*b^3*c^5*d^2*e^15*g - 2 568*b^4*c^4*d*e^16*g + 231*b^5*c^3*e^17*g)/(c^7*e^12))*x - (125440*c^8*d^5 *e^12*f + 22272*b*c^7*d^4*e^13*f - 223744*b^2*c^6*d^3*e^14*f + 92160*b^3*c ^5*d^2*e^15*f - 19488*b^4*c^4*d*e^16*f + 1680*b^5*c^3*e^17*f - 60480*c^8*d ^6*e^11*g + 208832*b*c^7*d^5*e^12*g - 278416*b^2*c^6*d^4*e^13*g + 188384*b ^3*c^5*d^3*e^14*g - 70668*b^4*c^4*d^2*e^15*g + 14028*b^5*c^3*d*e^16*g - 11 55*b^6*c^2*e^17*g)/(c^7*e^12))*x + (376832*c^8*d^6*e^11*f - 1255936*b*c^7* d^5*e^12*f + 1558784*b^2*c^6*d^4*e^13*f - 964608*b^3*c^5*d^3*e^14*f + 3386 88*b^4*c^4*d^2*e^15*f - 63840*b^5*c^3*d*e^16*f + 5040*b^6*c^2*e^17*f + ...
Timed out. \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\int \left (f+g\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2} \,d x \] Input:
int((f + g*x)*(d + e*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)
Output:
int((f + g*x)*(d + e*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2), x)
Time = 11.20 (sec) , antiderivative size = 4284, normalized size of antiderivative = 10.10 \[ \int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(i*(3465*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d) )*b**9*e**9*g - 57330*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**8*c*d*e**8*g - 5040*sqrt(c)*asinh((sqrt( - b*e + c*d - c *e*x)*i)/sqrt( - b*e + 2*c*d))*b**8*c*e**9*f + 418320*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**7*c**2*d**2*e**7*g + 806 40*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**7 *c**2*d*e**8*f - 1764000*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt ( - b*e + 2*c*d))*b**6*c**3*d**3*e**6*g - 564480*sqrt(c)*asinh((sqrt( - b* e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**6*c**3*d**2*e**7*f + 4727520* sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c* *4*d**4*e**5*g + 2257920*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt ( - b*e + 2*c*d))*b**5*c**4*d**3*e**6*f - 8326080*sqrt(c)*asinh((sqrt( - b *e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**5*d**5*e**4*g - 5644800 *sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c **5*d**4*e**5*f + 9596160*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqr t( - b*e + 2*c*d))*b**3*c**6*d**6*e**3*g + 9031680*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**6*d**5*e**4*f - 693504 0*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2* c**7*d**7*e**2*g - 9031680*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sq rt( - b*e + 2*c*d))*b**2*c**7*d**6*e**3*f + 2822400*sqrt(c)*asinh((sqrt...