Integrand size = 32, antiderivative size = 401 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\frac {2 (b C d+b c D-4 a d D) \sqrt {c+d x}}{b^5}-\frac {\left (b^3 (2 B c+A d)-a b^2 (4 c C+3 B d)-7 a^3 d D+a^2 b (5 C d+6 c D)\right ) \sqrt {c+d x}}{4 b^5 (a+b x)^2}-\frac {\left (b^3 \left (8 c^2 C+10 B c d+A d^2\right )-55 a^3 d^2 D+a^2 b d (29 C d+78 c D)-a b^2 \left (36 c C d+11 B d^2+24 c^2 D\right )\right ) \sqrt {c+d x}}{8 b^5 (b c-a d) (a+b x)}+\frac {2 D (c+d x)^{3/2}}{3 b^4}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{3/2}}{3 b^4 (a+b x)^3}+\frac {\left (105 a^3 d^3 D-35 a^2 b d^2 (C d+6 c D)+5 a b^2 d \left (12 c C d+B d^2+24 c^2 D\right )-b^3 \left (24 c^2 C d+6 B c d^2-A d^3+16 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{11/2} (b c-a d)^{3/2}} \] Output:
2*(C*b*d-4*D*a*d+D*b*c)*(d*x+c)^(1/2)/b^5-1/4*(b^3*(A*d+2*B*c)-a*b^2*(3*B* d+4*C*c)-7*a^3*d*D+a^2*b*(5*C*d+6*D*c))*(d*x+c)^(1/2)/b^5/(b*x+a)^2-1/8*(b ^3*(A*d^2+10*B*c*d+8*C*c^2)-55*a^3*d^2*D+a^2*b*d*(29*C*d+78*D*c)-a*b^2*(11 *B*d^2+36*C*c*d+24*D*c^2))*(d*x+c)^(1/2)/b^5/(-a*d+b*c)/(b*x+a)+2/3*D*(d*x +c)^(3/2)/b^4-1/3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(3/2)/b^4/(b*x+a)^ 3+1/8*(105*a^3*d^3*D-35*a^2*b*d^2*(C*d+6*D*c)+5*a*b^2*d*(B*d^2+12*C*c*d+24 *D*c^2)-b^3*(-A*d^3+6*B*c*d^2+24*C*c^2*d+16*D*c^3))*arctanh(b^(1/2)*(d*x+c )^(1/2)/(-a*d+b*c)^(1/2))/b^(11/2)/(-a*d+b*c)^(3/2)
Time = 1.55 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.15 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\frac {\sqrt {c+d x} \left (315 a^5 d^2 D-105 a^4 b d (C d+4 D (c-2 d x))+A b^3 \left (3 a^2 d^2+2 a b d (c+4 d x)-b^2 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )+2 b^5 c x \left (-3 B (2 c+5 d x)+4 x \left (-3 c C+6 C d x+8 c D x+2 d D x^2\right )\right )-a b^4 \left (B \left (4 c^2+22 c d x-33 d^2 x^2\right )+4 x \left (6 c^2 (C-11 D x)+4 d^2 x^2 (3 C+D x)+c d x (-63 C+52 D x)\right )\right )+a^3 b^2 \left (108 c^2 D+2 c d (55 C-567 D x)+d^2 (15 B+7 x (-40 C+99 D x))\right )+a^2 b^3 \left (c^2 (-8 C+300 D x)+d^2 x (40 B+3 x (-77 C+48 D x))-2 c d (4 B+x (-149 C+477 D x))\right )\right )}{24 b^5 (b c-a d) (a+b x)^3}-\frac {\left (-105 a^3 d^3 D+35 a^2 b d^2 (C d+6 c D)-5 a b^2 d \left (12 c C d+B d^2+24 c^2 D\right )+b^3 \left (24 c^2 C d+6 B c d^2-A d^3+16 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{11/2} (-b c+a d)^{3/2}} \] Input:
Integrate[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^4,x]
Output:
(Sqrt[c + d*x]*(315*a^5*d^2*D - 105*a^4*b*d*(C*d + 4*D*(c - 2*d*x)) + A*b^ 3*(3*a^2*d^2 + 2*a*b*d*(c + 4*d*x) - b^2*(8*c^2 + 14*c*d*x + 3*d^2*x^2)) + 2*b^5*c*x*(-3*B*(2*c + 5*d*x) + 4*x*(-3*c*C + 6*C*d*x + 8*c*D*x + 2*d*D*x ^2)) - a*b^4*(B*(4*c^2 + 22*c*d*x - 33*d^2*x^2) + 4*x*(6*c^2*(C - 11*D*x) + 4*d^2*x^2*(3*C + D*x) + c*d*x*(-63*C + 52*D*x))) + a^3*b^2*(108*c^2*D + 2*c*d*(55*C - 567*D*x) + d^2*(15*B + 7*x*(-40*C + 99*D*x))) + a^2*b^3*(c^2 *(-8*C + 300*D*x) + d^2*x*(40*B + 3*x*(-77*C + 48*D*x)) - 2*c*d*(4*B + x*( -149*C + 477*D*x)))))/(24*b^5*(b*c - a*d)*(a + b*x)^3) - ((-105*a^3*d^3*D + 35*a^2*b*d^2*(C*d + 6*c*D) - 5*a*b^2*d*(12*c*C*d + B*d^2 + 24*c^2*D) + b ^3*(24*c^2*C*d + 6*B*c*d^2 - A*d^3 + 16*c^3*D))*ArcTan[(Sqrt[b]*Sqrt[c + d *x])/Sqrt[-(b*c) + a*d]])/(8*b^(11/2)*(-(b*c) + a*d)^(3/2))
Time = 1.82 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2124, 27, 1192, 1580, 25, 2345, 27, 1467, 2009}
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.
| \(\displaystyle \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {(c+d x)^{3/2} \left (6 \left (c-\frac {a d}{b}\right ) D x^2+\frac {6 (b c-a d) (b C-a D) x}{b^2}+\frac {-5 d D a^3+b (5 C d+6 c D) a^2-b^2 (6 c C+5 B d) a+b^3 (6 B c-A d)}{b^3}\right )}{2 (a+b x)^3}dx}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{3/2} \left (-\frac {5 d D a^3}{b^3}+\frac {(5 C d+6 c D) a^2}{b^2}-\frac {(6 c C+5 B d) a}{b}+6 \left (c-\frac {a d}{b}\right ) D x^2+6 B c-A d+\frac {6 (b c-a d) (b C-a D) x}{b^2}\right )}{(a+b x)^3}dx}{6 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (-6 D c^3+6 C d c^2-6 B d^2 c+A d^3-6 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+\frac {5 a d^3 \left (D a^2-b C a+b^2 B\right )}{b^3}-\frac {6 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}\right )}{(b c-a d-b (c+d x))^3}d\sqrt {c+d x}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 1580 |
| \(\displaystyle \frac {-\frac {\int -\frac {24 b^2 (b c-a d) D (c+d x)^3+24 b (b c-a d) (b C d-2 a D d-b c D) (c+d x)^2+4 d^2 \left (-17 d D a^3+b (11 C d+18 c D) a^2-b^2 (12 c C+5 B d) a+b^3 (6 B c-A d)\right ) (c+d x)+\frac {d^2 (b c-a d) \left (-17 d D a^3+b (11 C d+18 c D) a^2-b^2 (12 c C+5 B d) a+b^3 (6 B c-A d)\right )}{b}}{(b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 b^4}-\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-17 a^3 d D+a^2 b (18 c D+11 C d)-a b^2 (5 B d+12 c C)+b^3 (6 B c-A d)\right )}{4 b^5 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {24 b^2 (b c-a d) D (c+d x)^3+24 b (b c-a d) (b C d-2 a D d-b c D) (c+d x)^2+4 d^2 \left (-17 d D a^3+b (11 C d+18 c D) a^2-b^2 (12 c C+5 B d) a+b^3 (6 B c-A d)\right ) (c+d x)+\frac {d^2 (b c-a d) \left (-17 d D a^3+b (11 C d+18 c D) a^2-b^2 (12 c C+5 B d) a+b^3 (6 B c-A d)\right )}{b}}{(b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 b^4}-\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-17 a^3 d D+a^2 b (18 c D+11 C d)-a b^2 (5 B d+12 c C)+b^3 (6 B c-A d)\right )}{4 b^5 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-157 a^3 d^2 D+a^2 b d (234 c D+79 C d)-a b^2 \left (25 B d^2+72 c^2 D+108 c C d\right )+b^3 \left (-5 A d^2+30 B c d+24 c^2 C\right )\right )}{2 b (-a d-b (c+d x)+b c)}-\frac {\int \frac {3 \left (16 b D (c+d x)^2 (b c-a d)^2+16 d (b C-3 a D) (c+d x) (b c-a d)^2+\frac {d \left (-41 d^2 D a^3+b d (19 C d+66 c D) a^2-b^2 \left (24 D c^2+28 C d c+5 B d^2\right ) a+b^3 \left (8 C c^2+6 B d c-A d^2\right )\right ) (b c-a d)}{b}\right )}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{2 (b c-a d)}}{4 b^4}-\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-17 a^3 d D+a^2 b (18 c D+11 C d)-a b^2 (5 B d+12 c C)+b^3 (6 B c-A d)\right )}{4 b^5 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-157 a^3 d^2 D+a^2 b d (234 c D+79 C d)-a b^2 \left (25 B d^2+72 c^2 D+108 c C d\right )+b^3 \left (-5 A d^2+30 B c d+24 c^2 C\right )\right )}{2 b (-a d-b (c+d x)+b c)}-\frac {3 \int \frac {16 b D (c+d x)^2 (b c-a d)^2+16 d (b C-3 a D) (c+d x) (b c-a d)^2+\frac {d \left (-41 d^2 D a^3+b d (19 C d+66 c D) a^2-b^2 \left (24 D c^2+28 C d c+5 B d^2\right ) a+b^3 \left (8 C c^2+6 B d c-A d^2\right )\right ) (b c-a d)}{b}}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{2 (b c-a d)}}{4 b^4}-\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-17 a^3 d D+a^2 b (18 c D+11 C d)-a b^2 (5 B d+12 c C)+b^3 (6 B c-A d)\right )}{4 b^5 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-157 a^3 d^2 D+a^2 b d (234 c D+79 C d)-a b^2 \left (25 B d^2+72 c^2 D+108 c C d\right )+b^3 \left (-5 A d^2+30 B c d+24 c^2 C\right )\right )}{2 b (-a d-b (c+d x)+b c)}-\frac {3 \int \left (-\frac {16 (b C d-4 a D d+b c D) (b c-a d)^2}{b}-16 D (c+d x) (b c-a d)^2+\frac {-A c d^3 b^4+6 B c^2 d^2 b^4+24 c^3 C d b^4+16 c^4 D b^4+a A d^4 b^3-11 a B c d^3 b^3-84 a c^2 C d^2 b^3-136 a c^3 d D b^3+5 a^2 B d^4 b^2+95 a^2 c C d^3 b^2+330 a^2 c^2 d^2 D b^2-35 a^3 C d^4 b-315 a^3 c d^3 D b+105 a^4 d^4 D}{b (b c-a d-b (c+d x))}\right )d\sqrt {c+d x}}{2 (b c-a d)}}{4 b^4}-\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-17 a^3 d D+a^2 b (18 c D+11 C d)-a b^2 (5 B d+12 c C)+b^3 (6 B c-A d)\right )}{4 b^5 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {d \sqrt {c+d x} \left (-157 a^3 d^2 D+a^2 b d (234 c D+79 C d)-a b^2 \left (25 B d^2+72 c^2 D+108 c C d\right )+b^3 \left (-5 A d^2+30 B c d+24 c^2 C\right )\right )}{2 b (-a d-b (c+d x)+b c)}-\frac {3 \left (-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (105 a^3 d^3 D-35 a^2 b d^2 (6 c D+C d)+5 a b^2 d \left (B d^2+24 c^2 D+12 c C d\right )-\left (b^3 \left (-A d^3+6 B c d^2+16 c^3 D+24 c^2 C d\right )\right )\right )}{b^{3/2}}-\frac {16 \sqrt {c+d x} (b c-a d)^2 (-4 a d D+b c D+b C d)}{b}-\frac {16}{3} D (c+d x)^{3/2} (b c-a d)^2\right )}{2 (b c-a d)}}{4 b^4}-\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-17 a^3 d D+a^2 b (18 c D+11 C d)-a b^2 (5 B d+12 c C)+b^3 (6 B c-A d)\right )}{4 b^5 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^3 (a+b x)^3 (b c-a d)}\) |
Input:
Int[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^4,x]
Output:
-1/3*((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(5/2))/(b^3*(b*c - a*d )*(a + b*x)^3) + (-1/4*(d^2*(b*c - a*d)*(b^3*(6*B*c - A*d) - a*b^2*(12*c*C + 5*B*d) - 17*a^3*d*D + a^2*b*(11*C*d + 18*c*D))*Sqrt[c + d*x])/(b^5*(b*c - a*d - b*(c + d*x))^2) + ((d*(b^3*(24*c^2*C + 30*B*c*d - 5*A*d^2) - 157* a^3*d^2*D + a^2*b*d*(79*C*d + 234*c*D) - a*b^2*(108*c*C*d + 25*B*d^2 + 72* c^2*D))*Sqrt[c + d*x])/(2*b*(b*c - a*d - b*(c + d*x))) - (3*((-16*(b*c - a *d)^2*(b*C*d + b*c*D - 4*a*d*D)*Sqrt[c + d*x])/b - (16*(b*c - a*d)^2*D*(c + d*x)^(3/2))/3 - (Sqrt[b*c - a*d]*(105*a^3*d^3*D - 35*a^2*b*d^2*(C*d + 6* c*D) + 5*a*b^2*d*(12*c*C*d + B*d^2 + 24*c^2*D) - b^3*(24*c^2*C*d + 6*B*c*d ^2 - A*d^3 + 16*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/ b^(3/2)))/(2*(b*c - a*d)))/(4*b^4))/(3*(b*c - a*d))
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_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* (q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b *d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e }, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : > With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px , a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - a*d))), x] + Simp[1/((m + 1)*(b*c - a*d)) Int[(a + b*x)^(m + 1)*(c + d*x )^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] || ! ILtQ[n, -1])
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.88 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {-\left (\left (A \,d^{3}-6 B c \,d^{2}-24 C \,c^{2} d -16 D c^{3}\right ) b^{3}+5 a \,b^{2} d \left (B \,d^{2}+12 C c d +24 D c^{2}\right )-35 a^{2} b \,d^{2} \left (C d +6 D c \right )+105 a^{3} d^{3} D\right ) \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (\left (-A \,d^{2} x^{2}-\frac {14 x \left (-\frac {8}{7} D x^{3}-\frac {24}{7} C \,x^{2}+\frac {15}{7} B x +A \right ) c d}{3}-\frac {8 \left (-8 D x^{3}+3 C \,x^{2}+\frac {3}{2} B x +A \right ) c^{2}}{3}\right ) b^{5}+\frac {2 \left (4 x \left (-2 D x^{3}-6 C \,x^{2}+\frac {33}{8} B x +A \right ) d^{2}+c \left (-104 D x^{3}+126 C \,x^{2}-11 B x +A \right ) d -2 c^{2} \left (-66 D x^{2}+6 C x +B \right )\right ) a \,b^{4}}{3}+\left (\left (\frac {40}{3} B x +48 D x^{3}-77 C \,x^{2}+A \right ) d^{2}-\frac {8 c \left (\frac {477}{4} D x^{2}-\frac {149}{4} C x +B \right ) d}{3}-\frac {8 \left (-\frac {75 D x}{2}+C \right ) c^{2}}{3}\right ) a^{2} b^{3}+5 \left (\left (\frac {231}{5} D x^{2}-\frac {56}{3} C x +B \right ) d^{2}+\frac {22 \left (-\frac {567 D x}{55}+C \right ) c d}{3}+\frac {36 D c^{2}}{5}\right ) a^{3} b^{2}-35 \left (\left (-8 D x +C \right ) d +4 D c \right ) d \,a^{4} b +105 D a^{5} d^{2}\right ) \sqrt {x d +c}}{8 \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{3} \left (a d -b c \right ) b^{5}}\) | \(425\) |
| derivativedivides | \(\frac {\frac {2 D \left (x d +c \right )^{\frac {3}{2}} b}{3}+2 C d b \sqrt {x d +c}-8 D a d \sqrt {x d +c}+2 D b c \sqrt {x d +c}}{b^{5}}+\frac {\frac {2 \left (\frac {b^{2} d \left (b^{3} d^{2} A -11 B a \,b^{2} d^{2}+10 B \,b^{3} c d +29 C \,a^{2} b \,d^{2}-36 C a \,b^{2} c d +8 C \,b^{3} c^{2}-55 a^{3} d^{2} D+78 a^{2} b c d D-24 a \,b^{2} c^{2} D\right ) \left (x d +c \right )^{\frac {5}{2}}}{16 a d -16 b c}-\frac {\left (b^{3} d^{2} A +5 B a \,b^{2} d^{2}-6 B \,b^{3} c d -17 C \,a^{2} b \,d^{2}+24 C a \,b^{2} c d -6 C \,b^{3} c^{2}+35 a^{3} d^{2} D-54 a^{2} b c d D+18 a \,b^{2} c^{2} D\right ) b d \left (x d +c \right )^{\frac {3}{2}}}{6}-\frac {\left (A a \,b^{3} d^{3}-A \,b^{4} c \,d^{2}+5 B \,a^{2} b^{2} d^{3}-11 B a \,b^{3} c \,d^{2}+6 B \,b^{4} c^{2} d -19 C \,a^{3} b \,d^{3}+47 C \,a^{2} b^{2} c \,d^{2}-36 C a \,b^{3} c^{2} d +8 c^{3} C \,b^{4}+41 D a^{4} d^{3}-107 D a^{3} b c \,d^{2}+90 D a^{2} b^{2} c^{2} d -24 D b^{3} c^{3} a \right ) d \sqrt {x d +c}}{16}\right )}{\left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\left (A \,b^{3} d^{3}+5 B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}-35 a^{2} b C \,d^{3}+60 C a \,b^{2} c \,d^{2}-24 C \,b^{3} c^{2} d +105 a^{3} d^{3} D-210 D a^{2} b c \,d^{2}+120 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}}{b^{5}}\) | \(573\) |
| default | \(\frac {\frac {2 D \left (x d +c \right )^{\frac {3}{2}} b}{3}+2 C d b \sqrt {x d +c}-8 D a d \sqrt {x d +c}+2 D b c \sqrt {x d +c}}{b^{5}}+\frac {\frac {2 \left (\frac {b^{2} d \left (b^{3} d^{2} A -11 B a \,b^{2} d^{2}+10 B \,b^{3} c d +29 C \,a^{2} b \,d^{2}-36 C a \,b^{2} c d +8 C \,b^{3} c^{2}-55 a^{3} d^{2} D+78 a^{2} b c d D-24 a \,b^{2} c^{2} D\right ) \left (x d +c \right )^{\frac {5}{2}}}{16 a d -16 b c}-\frac {\left (b^{3} d^{2} A +5 B a \,b^{2} d^{2}-6 B \,b^{3} c d -17 C \,a^{2} b \,d^{2}+24 C a \,b^{2} c d -6 C \,b^{3} c^{2}+35 a^{3} d^{2} D-54 a^{2} b c d D+18 a \,b^{2} c^{2} D\right ) b d \left (x d +c \right )^{\frac {3}{2}}}{6}-\frac {\left (A a \,b^{3} d^{3}-A \,b^{4} c \,d^{2}+5 B \,a^{2} b^{2} d^{3}-11 B a \,b^{3} c \,d^{2}+6 B \,b^{4} c^{2} d -19 C \,a^{3} b \,d^{3}+47 C \,a^{2} b^{2} c \,d^{2}-36 C a \,b^{3} c^{2} d +8 c^{3} C \,b^{4}+41 D a^{4} d^{3}-107 D a^{3} b c \,d^{2}+90 D a^{2} b^{2} c^{2} d -24 D b^{3} c^{3} a \right ) d \sqrt {x d +c}}{16}\right )}{\left (\left (x d +c \right ) b +a d -b c \right )^{3}}+\frac {\left (A \,b^{3} d^{3}+5 B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}-35 a^{2} b C \,d^{3}+60 C a \,b^{2} c \,d^{2}-24 C \,b^{3} c^{2} d +105 a^{3} d^{3} D-210 D a^{2} b c \,d^{2}+120 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}}{b^{5}}\) | \(573\) |
Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
-1/8*(-((A*d^3-6*B*c*d^2-24*C*c^2*d-16*D*c^3)*b^3+5*a*b^2*d*(B*d^2+12*C*c* d+24*D*c^2)-35*a^2*b*d^2*(C*d+6*D*c)+105*a^3*d^3*D)*(b*x+a)^3*arctan(b*(d* x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*((-A*d^2*x^2-14/3*x*(- 8/7*D*x^3-24/7*C*x^2+15/7*B*x+A)*c*d-8/3*(-8*D*x^3+3*C*x^2+3/2*B*x+A)*c^2) *b^5+2/3*(4*x*(-2*D*x^3-6*C*x^2+33/8*B*x+A)*d^2+c*(-104*D*x^3+126*C*x^2-11 *B*x+A)*d-2*c^2*(-66*D*x^2+6*C*x+B))*a*b^4+((40/3*B*x+48*D*x^3-77*C*x^2+A) *d^2-8/3*c*(477/4*D*x^2-149/4*C*x+B)*d-8/3*(-75/2*D*x+C)*c^2)*a^2*b^3+5*(( 231/5*D*x^2-56/3*C*x+B)*d^2+22/3*(-567/55*D*x+C)*c*d+36/5*D*c^2)*a^3*b^2-3 5*((-8*D*x+C)*d+4*D*c)*d*a^4*b+105*D*a^5*d^2)*(d*x+c)^(1/2))/((a*d-b*c)*b) ^(1/2)/(b*x+a)^3/(a*d-b*c)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 1174 vs. \(2 (373) = 746\).
Time = 0.16 (sec) , antiderivative size = 2362, normalized size of antiderivative = 5.89 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x, algorithm="fricas ")
Output:
[1/48*(3*(16*D*a^3*b^3*c^3 - 24*(5*D*a^4*b^2 - C*a^3*b^3)*c^2*d + 6*(35*D* a^5*b - 10*C*a^4*b^2 + B*a^3*b^3)*c*d^2 - (105*D*a^6 - 35*C*a^5*b + 5*B*a^ 4*b^2 + A*a^3*b^3)*d^3 + (16*D*b^6*c^3 - 24*(5*D*a*b^5 - C*b^6)*c^2*d + 6* (35*D*a^2*b^4 - 10*C*a*b^5 + B*b^6)*c*d^2 - (105*D*a^3*b^3 - 35*C*a^2*b^4 + 5*B*a*b^5 + A*b^6)*d^3)*x^3 + 3*(16*D*a*b^5*c^3 - 24*(5*D*a^2*b^4 - C*a* b^5)*c^2*d + 6*(35*D*a^3*b^3 - 10*C*a^2*b^4 + B*a*b^5)*c*d^2 - (105*D*a^4* b^2 - 35*C*a^3*b^3 + 5*B*a^2*b^4 + A*a*b^5)*d^3)*x^2 + 3*(16*D*a^2*b^4*c^3 - 24*(5*D*a^3*b^3 - C*a^2*b^4)*c^2*d + 6*(35*D*a^4*b^2 - 10*C*a^3*b^3 + B *a^2*b^4)*c*d^2 - (105*D*a^5*b - 35*C*a^4*b^2 + 5*B*a^3*b^3 + A*a^2*b^4)*d ^3)*x)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d )*sqrt(d*x + c))/(b*x + a)) + 2*(16*(D*b^7*c^2*d - 2*D*a*b^6*c*d^2 + D*a^2 *b^5*d^3)*x^4 + 4*(27*D*a^3*b^4 - 2*C*a^2*b^5 - B*a*b^6 - 2*A*b^7)*c^3 - 2 *(264*D*a^4*b^3 - 59*C*a^3*b^4 + 2*B*a^2*b^5 - 5*A*a*b^6)*c^2*d + (735*D*a ^5*b^2 - 215*C*a^4*b^3 + 23*B*a^3*b^4 + A*a^2*b^5)*c*d^2 - 3*(105*D*a^6*b - 35*C*a^5*b^2 + 5*B*a^4*b^3 + A*a^3*b^4)*d^3 + 16*(4*D*b^7*c^3 - (17*D*a* b^6 - 3*C*b^7)*c^2*d + 2*(11*D*a^2*b^5 - 3*C*a*b^6)*c*d^2 - 3*(3*D*a^3*b^4 - C*a^2*b^5)*d^3)*x^3 + 3*(8*(11*D*a*b^6 - C*b^7)*c^3 - 2*(203*D*a^2*b^5 - 46*C*a*b^6 + 5*B*b^7)*c^2*d + (549*D*a^3*b^4 - 161*C*a^2*b^5 + 21*B*a*b^ 6 - A*b^7)*c*d^2 - (231*D*a^4*b^3 - 77*C*a^3*b^4 + 11*B*a^2*b^5 - A*a*b^6) *d^3)*x^2 + 2*(6*(25*D*a^2*b^5 - 2*C*a*b^6 - B*b^7)*c^3 - (717*D*a^3*b^...
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(3/2)*(D*x**3+C*x**2+B*x+A)/(b*x+a)**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^4,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(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (373) = 746\).
Time = 0.15 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.45 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x, algorithm="giac")
Output:
1/8*(16*D*b^3*c^3 - 120*D*a*b^2*c^2*d + 24*C*b^3*c^2*d + 210*D*a^2*b*c*d^2 - 60*C*a*b^2*c*d^2 + 6*B*b^3*c*d^2 - 105*D*a^3*d^3 + 35*C*a^2*b*d^3 - 5*B *a*b^2*d^3 - A*b^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^6 *c - a*b^5*d)*sqrt(-b^2*c + a*b*d)) + 1/24*(72*(d*x + c)^(5/2)*D*a*b^4*c^2 *d - 24*(d*x + c)^(5/2)*C*b^5*c^2*d - 144*(d*x + c)^(3/2)*D*a*b^4*c^3*d + 48*(d*x + c)^(3/2)*C*b^5*c^3*d + 72*sqrt(d*x + c)*D*a*b^4*c^4*d - 24*sqrt( d*x + c)*C*b^5*c^4*d - 234*(d*x + c)^(5/2)*D*a^2*b^3*c*d^2 + 108*(d*x + c) ^(5/2)*C*a*b^4*c*d^2 - 30*(d*x + c)^(5/2)*B*b^5*c*d^2 + 576*(d*x + c)^(3/2 )*D*a^2*b^3*c^2*d^2 - 240*(d*x + c)^(3/2)*C*a*b^4*c^2*d^2 + 48*(d*x + c)^( 3/2)*B*b^5*c^2*d^2 - 342*sqrt(d*x + c)*D*a^2*b^3*c^3*d^2 + 132*sqrt(d*x + c)*C*a*b^4*c^3*d^2 - 18*sqrt(d*x + c)*B*b^5*c^3*d^2 + 165*(d*x + c)^(5/2)* D*a^3*b^2*d^3 - 87*(d*x + c)^(5/2)*C*a^2*b^3*d^3 + 33*(d*x + c)^(5/2)*B*a* b^4*d^3 - 3*(d*x + c)^(5/2)*A*b^5*d^3 - 712*(d*x + c)^(3/2)*D*a^3*b^2*c*d^ 3 + 328*(d*x + c)^(3/2)*C*a^2*b^3*c*d^3 - 88*(d*x + c)^(3/2)*B*a*b^4*c*d^3 - 8*(d*x + c)^(3/2)*A*b^5*c*d^3 + 591*sqrt(d*x + c)*D*a^3*b^2*c^2*d^3 - 2 49*sqrt(d*x + c)*C*a^2*b^3*c^2*d^3 + 51*sqrt(d*x + c)*B*a*b^4*c^2*d^3 + 3* sqrt(d*x + c)*A*b^5*c^2*d^3 + 280*(d*x + c)^(3/2)*D*a^4*b*d^4 - 136*(d*x + c)^(3/2)*C*a^3*b^2*d^4 + 40*(d*x + c)^(3/2)*B*a^2*b^3*d^4 + 8*(d*x + c)^( 3/2)*A*a*b^4*d^4 - 444*sqrt(d*x + c)*D*a^4*b*c*d^4 + 198*sqrt(d*x + c)*C*a ^3*b^2*c*d^4 - 48*sqrt(d*x + c)*B*a^2*b^3*c*d^4 - 6*sqrt(d*x + c)*A*a*b...
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^4} \,d x \] Input:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^4,x)
Output:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^4, x)
Time = 0.15 (sec) , antiderivative size = 1272, normalized size of antiderivative = 3.17 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x)
Output:
(315*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b* c)))*a**5*d**3 - 420*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( b)*sqrt(a*d - b*c)))*a**4*b*c*d**2 + 945*sqrt(b)*sqrt(a*d - b*c)*atan((sqr t(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**4*b*d**3*x + 18*sqrt(b)*sqrt(a *d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**3*d**2 + 120*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*c**2*d - 1260*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x) *b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*c*d**2*x + 945*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*d**3*x** 2 + 54*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**4*d**2*x + 360*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)* b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*c**2*d*x - 1260*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*c*d**2*x **2 + 315*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*d**3*x**3 + 54*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**5*d**2*x**2 + 360*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**4*c**2*d*x* *2 - 420*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**4*c*d**2*x**3 + 18*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d* x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**6*d**2*x**3 + 120*sqrt(b)*sqrt(a*d ...