Integrand size = 32, antiderivative size = 463 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\frac {a b^2 B d^3-a^2 b C d^3+a^3 d^3 D-b^3 \left (6 c^2 C d-6 B c d^2+7 A d^3-6 c^3 D\right )}{3 b^3 (b c-a d)^4 \sqrt {c+d x}}-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}-\frac {\left (b^3 (6 B c-7 A d)-a b^2 (12 c C-B d)-11 a^3 d D+a^2 b (5 C d+18 c D)\right ) \sqrt {c+d x}}{12 b^2 (b c-a d)^3 (a+b x)^2}-\frac {\left (b^3 \left (24 c^2 C-42 B c d+49 A d^2\right )+5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (36 c C d-7 B d^2-72 c^2 D\right )\right ) \sqrt {c+d x}}{24 b^2 (b c-a d)^4 (a+b x)}-\frac {\left (a^3 d^3 D+a^2 b d^2 (C d-6 c D)-a b^2 d \left (12 c C d-5 B d^2-24 c^2 D\right )-b^3 \left (24 c^2 C d-30 B c d^2+35 A d^3-16 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{9/2}} \]
-1/8*(a^3*d^3*D+a^2*b*d^2*(C*d-6*D*c)-a*b^2*d*(-5*B*d^2+12*C*c*d-24*D*c^2) -b^3*(35*A*d^3-30*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^(5/2)/(-a*d+b*c)^(9/2)+1/3*(a*b^2*B*d^3-a^2*b*C*d^ 3+a^3*d^3*D-b^3*(7*A*d^3-6*B*c*d^2+6*C*c^2*d-6*D*c^3))/b^3/(-a*d+b*c)^4/(d *x+c)^(1/2)+1/3*(-A*b^3+a*(B*b^2-C*a*b+D*a^2))/b^3/(-a*d+b*c)/(b*x+a)^3/(d *x+c)^(1/2)-1/12*(b^3*(-7*A*d+6*B*c)-a*b^2*(-B*d+12*C*c)-11*a^3*d*D+a^2*b* (5*C*d+18*D*c))*(d*x+c)^(1/2)/b^2/(-a*d+b*c)^3/(b*x+a)^2-1/24*(b^3*(49*A*d ^2-42*B*c*d+24*C*c^2)+5*a^3*d^2*D-a^2*b*d*(11*C*d-18*D*c)+a*b^2*(-7*B*d^2+ 36*C*c*d-72*D*c^2))*(d*x+c)^(1/2)/b^2/(-a*d+b*c)^4/(b*x+a)
Time = 2.31 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\frac {-3 a^5 d^2 D (c+d x)+a^4 b d (c+d x) (-3 C d+16 c D-8 d D x)+6 b^5 c x \left (4 c x (-c C-3 C d x+2 c D x)+B \left (-2 c^2+5 c d x+15 d^2 x^2\right )\right )-A b^2 \left (48 a^3 d^3+3 a^2 b d^2 (29 c+77 d x)+2 a b^2 d \left (-19 c^2+49 c d x+140 d^2 x^2\right )+b^3 \left (8 c^3-14 c^2 d x+35 c d^2 x^2+105 d^3 x^3\right )\right )+a b^4 \left (B \left (-4 c^3+82 c^2 d x+245 c d^2 x^2+15 d^3 x^3\right )-12 c x \left (3 C d^2 x^2+2 c^2 (C-9 D x)+c d x (17 C-6 D x)\right )\right )+a^3 b^2 \left (92 c^3 D+c^2 (-94 C d+58 d D x)+d^3 x (33 B+x (8 C+3 D x))+c d^2 (81 B+x (-38 C+17 D x))\right )+a^2 b^3 \left (d^3 x^2 (40 B+3 C x)+c^3 (-8 C+252 D x)+2 c^2 d (14 B+5 x (-25 C+9 D x))+c d^2 x (212 B-x (95 C+18 D x))\right )}{24 b^2 (b c-a d)^4 (a+b x)^3 \sqrt {c+d x}}+\frac {\left (a^3 d^3 D+a^2 b d^2 (C d-6 c D)+a b^2 d \left (-12 c C d+5 B d^2+24 c^2 D\right )+b^3 \left (-24 c^2 C d+30 B c d^2-35 A d^3+16 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{5/2} (-b c+a d)^{9/2}} \]
(-3*a^5*d^2*D*(c + d*x) + a^4*b*d*(c + d*x)*(-3*C*d + 16*c*D - 8*d*D*x) + 6*b^5*c*x*(4*c*x*(-(c*C) - 3*C*d*x + 2*c*D*x) + B*(-2*c^2 + 5*c*d*x + 15*d ^2*x^2)) - A*b^2*(48*a^3*d^3 + 3*a^2*b*d^2*(29*c + 77*d*x) + 2*a*b^2*d*(-1 9*c^2 + 49*c*d*x + 140*d^2*x^2) + b^3*(8*c^3 - 14*c^2*d*x + 35*c*d^2*x^2 + 105*d^3*x^3)) + a*b^4*(B*(-4*c^3 + 82*c^2*d*x + 245*c*d^2*x^2 + 15*d^3*x^ 3) - 12*c*x*(3*C*d^2*x^2 + 2*c^2*(C - 9*D*x) + c*d*x*(17*C - 6*D*x))) + a^ 3*b^2*(92*c^3*D + c^2*(-94*C*d + 58*d*D*x) + d^3*x*(33*B + x*(8*C + 3*D*x) ) + c*d^2*(81*B + x*(-38*C + 17*D*x))) + a^2*b^3*(d^3*x^2*(40*B + 3*C*x) + c^3*(-8*C + 252*D*x) + 2*c^2*d*(14*B + 5*x*(-25*C + 9*D*x)) + c*d^2*x*(21 2*B - x*(95*C + 18*D*x))))/(24*b^2*(b*c - a*d)^4*(a + b*x)^3*Sqrt[c + d*x] ) + ((a^3*d^3*D + a^2*b*d^2*(C*d - 6*c*D) + a*b^2*d*(-12*c*C*d + 5*B*d^2 + 24*c^2*D) + b^3*(-24*c^2*C*d + 30*B*c*d^2 - 35*A*d^3 + 16*c^3*D))*ArcTan[ (Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(8*b^(5/2)*(-(b*c) + a*d)^(9/ 2))
Time = 1.07 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.12, 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, 1582, 27, 361, 25, 359, 221}
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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {6 \left (c-\frac {a d}{b}\right ) D x^2+\frac {6 (b c-a d) (b C-a D) x}{b^2}+\frac {d D a^3-b (C d-6 c D) a^2-b^2 (6 c C-B d) a+b^3 (6 B c-7 A d)}{b^3}}{2 (a+b x)^3 (c+d x)^{3/2}}dx}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\frac {d D a^3}{b^3}-\frac {(C d-6 c D) a^2}{b^2}-\frac {(6 c C-B d) a}{b}+6 \left (c-\frac {a d}{b}\right ) D x^2+6 B c-7 A d+\frac {6 (b c-a d) (b C-a D) x}{b^2}}{(a+b x)^3 (c+d x)^{3/2}}dx}{6 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\int \frac {-6 D c^3+6 C d c^2-6 B d^2 c-6 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+d^3 \left (7 A-\frac {a \left (D a^2-b C a+b^2 B\right )}{b^3}\right )-\frac {6 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{(c+d x) (b c-a d-b (c+d x))^3}d\sqrt {c+d x}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {-\frac {\int \frac {4 (b c-a d) \left (-\left (\left (-6 D c^3+6 C d c^2-6 B d^2 c+7 A d^3\right ) b^3\right )+a B d^3 b^2-a^2 C d^3 b+a^3 d^3 D\right )-3 b \left (-\left (\left (-8 D c^3+6 B d^2 c-7 A d^3\right ) b^3\right )+a d \left (-24 D c^2+12 C d c-B d^2\right ) b^2-a^2 d^2 (5 C d-6 c D) b+3 a^3 d^3 D\right ) (c+d x)}{b (c+d x) (b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 b^2 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x} \left (-11 a^3 d D+a^2 b (18 c D+5 C d)-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{4 b^2 (b c-a d)^2 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {4 (b c-a d) \left (-\left (\left (-6 D c^3+6 C d c^2-6 B d^2 c+7 A d^3\right ) b^3\right )+a B d^3 b^2-a^2 C d^3 b+a^3 d^3 D\right )-3 b \left (-\left (\left (-8 D c^3+6 B d^2 c-7 A d^3\right ) b^3\right )+a d \left (-24 D c^2+12 C d c-B d^2\right ) b^2-a^2 d^2 (5 C d-6 c D) b+3 a^3 d^3 D\right ) (c+d x)}{(c+d x) (b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 b^3 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x} \left (-11 a^3 d D+a^2 b (18 c D+5 C d)-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{4 b^2 (b c-a d)^2 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 361 |
| \(\displaystyle \frac {-\frac {-\frac {1}{2} \int -\frac {8 \left (-\left (\left (-6 D c^3+6 C d c^2-6 B d^2 c+7 A d^3\right ) b^3\right )+a B d^3 b^2-a^2 C d^3 b+a^3 d^3 D\right )-\frac {b d \left (5 d^2 D a^3-b d (11 C d-18 c D) a^2+b^2 \left (-72 D c^2+36 C d c-7 B d^2\right ) a+b^3 \left (24 C c^2-42 B d c+49 A d^2\right )\right ) (c+d x)}{b c-a d}}{(c+d x) (b c-a d-b (c+d x))}d\sqrt {c+d x}-\frac {b d \sqrt {c+d x} \left (5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (-7 B d^2-72 c^2 D+36 c C d\right )+b^3 \left (49 A d^2-42 B c d+24 c^2 C\right )\right )}{2 (b c-a d) (-a d-b (c+d x)+b c)}}{4 b^3 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x} \left (-11 a^3 d D+a^2 b (18 c D+5 C d)-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{4 b^2 (b c-a d)^2 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \int \frac {8 \left (-\left (\left (-6 D c^3+6 C d c^2-6 B d^2 c+7 A d^3\right ) b^3\right )+a B d^3 b^2-a^2 C d^3 b+a^3 d^3 D\right )-\frac {b d \left (5 d^2 D a^3-b d (11 C d-18 c D) a^2+b^2 \left (-72 D c^2+36 C d c-7 B d^2\right ) a+b^3 \left (24 C c^2-42 B d c+49 A d^2\right )\right ) (c+d x)}{b c-a d}}{(c+d x) (b c-a d-b (c+d x))}d\sqrt {c+d x}-\frac {b d \sqrt {c+d x} \left (5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (-7 B d^2-72 c^2 D+36 c C d\right )+b^3 \left (49 A d^2-42 B c d+24 c^2 C\right )\right )}{2 (b c-a d) (-a d-b (c+d x)+b c)}}{4 b^3 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x} \left (-11 a^3 d D+a^2 b (18 c D+5 C d)-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{4 b^2 (b c-a d)^2 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \left (\frac {3 b \left (a^3 d^3 D+a^2 b d^2 (C d-6 c D)-a b^2 d \left (-5 B d^2-24 c^2 D+12 c C d\right )-\left (b^3 \left (35 A d^3-30 B c d^2-16 c^3 D+24 c^2 C d\right )\right )\right ) \int \frac {1}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{b c-a d}-\frac {8 \left (a^3 d^3 D-a^2 b C d^3+a b^2 B d^3-\left (b^3 \left (7 A d^3-6 B c d^2-6 c^3 D+6 c^2 C d\right )\right )\right )}{\sqrt {c+d x} (b c-a d)}\right )-\frac {b d \sqrt {c+d x} \left (5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (-7 B d^2-72 c^2 D+36 c C d\right )+b^3 \left (49 A d^2-42 B c d+24 c^2 C\right )\right )}{2 (b c-a d) (-a d-b (c+d x)+b c)}}{4 b^3 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x} \left (-11 a^3 d D+a^2 b (18 c D+5 C d)-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{4 b^2 (b c-a d)^2 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \left (\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 d^3 D+a^2 b d^2 (C d-6 c D)-a b^2 d \left (-5 B d^2-24 c^2 D+12 c C d\right )-\left (b^3 \left (35 A d^3-30 B c d^2-16 c^3 D+24 c^2 C d\right )\right )\right )}{(b c-a d)^{3/2}}-\frac {8 \left (a^3 d^3 D-a^2 b C d^3+a b^2 B d^3-\left (b^3 \left (7 A d^3-6 B c d^2-6 c^3 D+6 c^2 C d\right )\right )\right )}{\sqrt {c+d x} (b c-a d)}\right )-\frac {b d \sqrt {c+d x} \left (5 a^3 d^2 D-a^2 b d (11 C d-18 c D)+a b^2 \left (-7 B d^2-72 c^2 D+36 c C d\right )+b^3 \left (49 A d^2-42 B c d+24 c^2 C\right )\right )}{2 (b c-a d) (-a d-b (c+d x)+b c)}}{4 b^3 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x} \left (-11 a^3 d D+a^2 b (18 c D+5 C d)-a b^2 (12 c C-B d)+b^3 (6 B c-7 A d)\right )}{4 b^2 (b c-a d)^2 (-a d-b (c+d x)+b c)^2}}{3 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^3 (a+b x)^3 \sqrt {c+d x} (b c-a d)}\) |
-1/3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(b^3*(b*c - a*d)*(a + b*x)^3*Sqrt [c + d*x]) + (-1/4*(d^2*(b^3*(6*B*c - 7*A*d) - a*b^2*(12*c*C - B*d) - 11*a ^3*d*D + a^2*b*(5*C*d + 18*c*D))*Sqrt[c + d*x])/(b^2*(b*c - a*d)^2*(b*c - a*d - b*(c + d*x))^2) - (-1/2*(b*d*(b^3*(24*c^2*C - 42*B*c*d + 49*A*d^2) + 5*a^3*d^2*D - a^2*b*d*(11*C*d - 18*c*D) + a*b^2*(36*c*C*d - 7*B*d^2 - 72* c^2*D))*Sqrt[c + d*x])/((b*c - a*d)*(b*c - a*d - b*(c + d*x))) + ((-8*(a*b ^2*B*d^3 - a^2*b*C*d^3 + a^3*d^3*D - b^3*(6*c^2*C*d - 6*B*c*d^2 + 7*A*d^3 - 6*c^3*D)))/((b*c - a*d)*Sqrt[c + d*x]) + (3*Sqrt[b]*(a^3*d^3*D + a^2*b*d ^2*(C*d - 6*c*D) - a*b^2*d*(12*c*C*d - 5*B*d^2 - 24*c^2*D) - b^3*(24*c^2*C *d - 30*B*c*d^2 + 35*A*d^3 - 16*c^3*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sq rt[b*c - a*d]])/(b*c - a*d)^(3/2))/2)/(4*b^3*(b*c - a*d)^2))/(3*(b*c - a*d ))
3.1.17.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
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[(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[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[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])
Time = 1.95 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.10
| method | result | size |
| pseudoelliptic | \(-\frac {35 \left (\left (\left (A \,d^{3}-\frac {6}{7} B c \,d^{2}+\frac {24}{35} C \,c^{2} d -\frac {16}{35} D c^{3}\right ) b^{3}-\frac {a \left (B \,d^{2}-\frac {12}{5} C c d +\frac {24}{5} D c^{2}\right ) d \,b^{2}}{7}-\frac {a^{2} b \,d^{2} \left (C d -6 D c \right )}{35}-\frac {a^{3} d^{3} D}{35}\right ) \sqrt {d x +c}\, \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\frac {16 \sqrt {\left (a d -b c \right ) b}\, \left (\left (\frac {35 A \,d^{3} x^{3}}{16}+\frac {35 x^{2} c \left (-\frac {18 B x}{7}+A \right ) d^{2}}{48}-\frac {7 x \,c^{2} \left (-\frac {36}{7} C \,x^{2}+\frac {15}{7} B x +A \right ) d}{24}+\frac {c^{3} \left (-6 D x^{3}+3 C \,x^{2}+\frac {3}{2} B x +A \right )}{6}\right ) b^{5}-\frac {19 a \left (-\frac {140 x^{2} \left (-\frac {3 B x}{56}+A \right ) d^{3}}{19}-\frac {49 x \left (\frac {18}{49} C \,x^{2}-\frac {5}{2} B x +A \right ) c \,d^{2}}{19}+c^{2} \left (\frac {36}{19} D x^{3}-\frac {102}{19} C \,x^{2}+\frac {41}{19} B x +A \right ) d -\frac {2 c^{3} \left (-54 D x^{2}+6 C x +B \right )}{19}\right ) b^{4}}{24}+\frac {29 a^{2} \left (\frac {77 x \left (-\frac {1}{77} C \,x^{2}-\frac {40}{231} B x +A \right ) d^{3}}{29}+c \left (\frac {6}{29} D x^{3}+\frac {95}{87} C \,x^{2}-\frac {212}{87} B x +A \right ) d^{2}-\frac {28 \left (\frac {45}{14} D x^{2}-\frac {125}{14} C x +B \right ) c^{2} d}{87}+\frac {8 \left (-\frac {63 D x}{2}+C \right ) c^{3}}{87}\right ) b^{3}}{16}+\left (\left (A -\frac {1}{16} D x^{3}-\frac {1}{6} C \,x^{2}-\frac {11}{16} B x \right ) d^{3}-\frac {27 \left (\frac {17}{81} D x^{2}-\frac {38}{81} C x +B \right ) c \,d^{2}}{16}+\frac {47 \left (-\frac {29 D x}{47}+C \right ) c^{2} d}{24}-\frac {23 D c^{3}}{12}\right ) a^{3} b^{2}+\frac {a^{4} \left (d x +c \right ) \left (\left (\frac {8 D x}{3}+C \right ) d -\frac {16 D c}{3}\right ) d b}{16}+\frac {D a^{5} d^{2} \left (d x +c \right )}{16}\right )}{35}\right )}{8 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}\, \left (b x +a \right )^{3} \left (a d -b c \right )^{4} b^{2}}\) | \(511\) |
| derivativedivides | \(-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{\left (a d -b c \right )^{4} \sqrt {d x +c}}-\frac {2 \left (\frac {\left (\frac {19}{16} A \,b^{3} d^{3}-\frac {7}{8} B \,b^{3} c \,d^{2}+\frac {1}{2} C \,b^{3} c^{2} d -\frac {5}{16} B a \,b^{2} d^{3}-\frac {1}{16} a^{2} b C \,d^{3}+\frac {3}{4} C a \,b^{2} c \,d^{2}-\frac {1}{16} a^{3} d^{3} D+\frac {3}{8} D a^{2} b c \,d^{2}-\frac {3}{2} D a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\frac {d \left (17 A a \,b^{3} d^{3}-17 A \,b^{4} c \,d^{2}-5 B \,a^{2} b^{2} d^{3}-7 B a \,b^{3} c \,d^{2}+12 B \,b^{4} c^{2} d -C \,a^{3} b \,d^{3}+13 C \,a^{2} b^{2} c \,d^{2}-6 C a \,b^{3} c^{2} d -6 C \,b^{4} c^{3}+D a^{4} d^{3}-D a^{3} b c \,d^{2}-18 D a^{2} c^{2} d \,b^{2}+18 D a \,b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {3}{2}}}{6 b}+\frac {d \left (29 A \,a^{2} b^{3} d^{4}-58 A a \,b^{4} c \,d^{3}+29 A \,b^{5} c^{2} d^{2}-11 B \,a^{3} b^{2} d^{4}+4 B \,a^{2} b^{3} c \,d^{3}+25 B a \,b^{4} c^{2} d^{2}-18 B \,b^{5} c^{3} d +d^{4} C \,a^{4} b +18 C \,a^{3} b^{2} c \,d^{3}-31 C \,a^{2} b^{3} c^{2} d^{2}+4 C a \,b^{4} c^{3} d +8 C \,b^{5} c^{4}+D a^{5} d^{4}-8 D a^{4} c \,d^{3} b -11 D a^{3} b^{2} c^{2} d^{2}+42 D a^{2} b^{3} c^{3} d -24 D a \,b^{4} c^{4}\right ) \sqrt {d x +c}}{16 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\left (35 A \,b^{3} d^{3}-5 B a \,b^{2} d^{3}-30 B \,b^{3} c \,d^{2}-a^{2} b C \,d^{3}+12 C a \,b^{2} c \,d^{2}+24 C \,b^{3} c^{2} d -a^{3} d^{3} D+6 D a^{2} b c \,d^{2}-24 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b^{2} \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}\) | \(681\) |
| default | \(-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{\left (a d -b c \right )^{4} \sqrt {d x +c}}-\frac {2 \left (\frac {\left (\frac {19}{16} A \,b^{3} d^{3}-\frac {7}{8} B \,b^{3} c \,d^{2}+\frac {1}{2} C \,b^{3} c^{2} d -\frac {5}{16} B a \,b^{2} d^{3}-\frac {1}{16} a^{2} b C \,d^{3}+\frac {3}{4} C a \,b^{2} c \,d^{2}-\frac {1}{16} a^{3} d^{3} D+\frac {3}{8} D a^{2} b c \,d^{2}-\frac {3}{2} D a \,b^{2} c^{2} d \right ) \left (d x +c \right )^{\frac {5}{2}}+\frac {d \left (17 A a \,b^{3} d^{3}-17 A \,b^{4} c \,d^{2}-5 B \,a^{2} b^{2} d^{3}-7 B a \,b^{3} c \,d^{2}+12 B \,b^{4} c^{2} d -C \,a^{3} b \,d^{3}+13 C \,a^{2} b^{2} c \,d^{2}-6 C a \,b^{3} c^{2} d -6 C \,b^{4} c^{3}+D a^{4} d^{3}-D a^{3} b c \,d^{2}-18 D a^{2} c^{2} d \,b^{2}+18 D a \,b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {3}{2}}}{6 b}+\frac {d \left (29 A \,a^{2} b^{3} d^{4}-58 A a \,b^{4} c \,d^{3}+29 A \,b^{5} c^{2} d^{2}-11 B \,a^{3} b^{2} d^{4}+4 B \,a^{2} b^{3} c \,d^{3}+25 B a \,b^{4} c^{2} d^{2}-18 B \,b^{5} c^{3} d +d^{4} C \,a^{4} b +18 C \,a^{3} b^{2} c \,d^{3}-31 C \,a^{2} b^{3} c^{2} d^{2}+4 C a \,b^{4} c^{3} d +8 C \,b^{5} c^{4}+D a^{5} d^{4}-8 D a^{4} c \,d^{3} b -11 D a^{3} b^{2} c^{2} d^{2}+42 D a^{2} b^{3} c^{3} d -24 D a \,b^{4} c^{4}\right ) \sqrt {d x +c}}{16 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\left (35 A \,b^{3} d^{3}-5 B a \,b^{2} d^{3}-30 B \,b^{3} c \,d^{2}-a^{2} b C \,d^{3}+12 C a \,b^{2} c \,d^{2}+24 C \,b^{3} c^{2} d -a^{3} d^{3} D+6 D a^{2} b c \,d^{2}-24 D a \,b^{2} c^{2} d -16 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b^{2} \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}\) | \(681\) |
-35/8/((a*d-b*c)*b)^(1/2)/(d*x+c)^(1/2)*(((A*d^3-6/7*B*c*d^2+24/35*C*c^2*d -16/35*D*c^3)*b^3-1/7*a*(B*d^2-12/5*C*c*d+24/5*D*c^2)*d*b^2-1/35*a^2*b*d^2 *(C*d-6*D*c)-1/35*a^3*d^3*D)*(d*x+c)^(1/2)*(b*x+a)^3*arctan(b*(d*x+c)^(1/2 )/((a*d-b*c)*b)^(1/2))+16/35*((a*d-b*c)*b)^(1/2)*((35/16*A*d^3*x^3+35/48*x ^2*c*(-18/7*B*x+A)*d^2-7/24*x*c^2*(-36/7*C*x^2+15/7*B*x+A)*d+1/6*c^3*(-6*D *x^3+3*C*x^2+3/2*B*x+A))*b^5-19/24*a*(-140/19*x^2*(-3/56*B*x+A)*d^3-49/19* x*(18/49*C*x^2-5/2*B*x+A)*c*d^2+c^2*(36/19*D*x^3-102/19*C*x^2+41/19*B*x+A) *d-2/19*c^3*(-54*D*x^2+6*C*x+B))*b^4+29/16*a^2*(77/29*x*(-1/77*C*x^2-40/23 1*B*x+A)*d^3+c*(6/29*D*x^3+95/87*C*x^2-212/87*B*x+A)*d^2-28/87*(45/14*D*x^ 2-125/14*C*x+B)*c^2*d+8/87*(-63/2*D*x+C)*c^3)*b^3+((A-1/16*D*x^3-1/6*C*x^2 -11/16*B*x)*d^3-27/16*(17/81*D*x^2-38/81*C*x+B)*c*d^2+47/24*(-29/47*D*x+C) *c^2*d-23/12*D*c^3)*a^3*b^2+1/16*a^4*(d*x+c)*((8/3*D*x+C)*d-16/3*D*c)*d*b+ 1/16*D*a^5*d^2*(d*x+c)))/(b*x+a)^3/(a*d-b*c)^4/b^2
Leaf count of result is larger than twice the leaf count of optimal. 1910 vs. \(2 (435) = 870\).
Time = 0.56 (sec) , antiderivative size = 3834, normalized size of antiderivative = 8.28 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
[1/48*(3*(16*D*a^3*b^3*c^4 + (16*D*b^6*c^3*d + (D*a^3*b^3 + C*a^2*b^4 + 5* B*a*b^5 - 35*A*b^6)*d^4 - 6*(D*a^2*b^4*c + (2*C*a*b^5 - 5*B*b^6)*c)*d^3 + 24*(D*a*b^5*c^2 - C*b^6*c^2)*d^2)*x^4 + (D*a^6*c + (C*a^5*b + 5*B*a^4*b^2 - 35*A*a^3*b^3)*c)*d^3 + (16*D*b^6*c^4 + 3*(D*a^4*b^2 + C*a^3*b^3 + 5*B*a^ 2*b^4 - 35*A*a*b^5)*d^4 - (17*D*a^3*b^3*c + 5*(7*C*a^2*b^4 - 19*B*a*b^5 + 7*A*b^6)*c)*d^3 + 6*(11*D*a^2*b^4*c^2 - (14*C*a*b^5 - 5*B*b^6)*c^2)*d^2 + 24*(3*D*a*b^5*c^3 - C*b^6*c^3)*d)*x^3 - 6*(D*a^5*b*c^2 + (2*C*a^4*b^2 - 5* B*a^3*b^3)*c^2)*d^2 + 3*(16*D*a*b^5*c^4 + (D*a^5*b + C*a^4*b^2 + 5*B*a^3*b ^3 - 35*A*a^2*b^4)*d^4 - (5*D*a^4*b^2*c + (11*C*a^3*b^3 - 35*B*a^2*b^4 + 3 5*A*a*b^5)*c)*d^3 + 6*(3*D*a^3*b^3*c^2 - (6*C*a^2*b^4 - 5*B*a*b^5)*c^2)*d^ 2 + 8*(5*D*a^2*b^4*c^3 - 3*C*a*b^5*c^3)*d)*x^2 + 24*(D*a^4*b^2*c^3 - C*a^3 *b^3*c^3)*d + (48*D*a^2*b^4*c^4 + (D*a^6 + C*a^5*b + 5*B*a^4*b^2 - 35*A*a^ 3*b^3)*d^4 - 3*(D*a^5*b*c + (3*C*a^4*b^2 - 15*B*a^3*b^3 + 35*A*a^2*b^4)*c) *d^3 + 6*(D*a^4*b^2*c^2 - 5*(2*C*a^3*b^3 - 3*B*a^2*b^4)*c^2)*d^2 + 8*(11*D *a^3*b^3*c^3 - 9*C*a^2*b^4*c^3)*d)*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*(92*D*a^3*b ^4*c^4 + 48*A*a^4*b^3*d^4 - 4*(2*C*a^2*b^5 + B*a*b^6 + 2*A*b^7)*c^4 + 3*(D *a^6*b*c + (C*a^5*b^2 - 27*B*a^4*b^3 + 13*A*a^3*b^4)*c)*d^3 + 3*(16*D*b^7* c^4 - (D*a^4*b^3 + C*a^3*b^4 + 5*B*a^2*b^5 - 35*A*a*b^6)*d^4 + (7*D*a^3*b^ 4*c + (13*C*a^2*b^5 - 25*B*a*b^6 - 35*A*b^7)*c)*d^3 - 6*(5*D*a^2*b^5*c^...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (435) = 870\).
Time = 0.32 (sec) , antiderivative size = 1085, normalized size of antiderivative = 2.34 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
1/8*(16*D*b^3*c^3 + 24*D*a*b^2*c^2*d - 24*C*b^3*c^2*d - 6*D*a^2*b*c*d^2 - 12*C*a*b^2*c*d^2 + 30*B*b^3*c*d^2 + D*a^3*d^3 + C*a^2*b*d^3 + 5*B*a*b^2*d^ 3 - 35*A*b^3*d^3)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*sqrt(- b^2*c + a*b*d)) + 2*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)/((b^4*c^4 - 4*a*b^ 3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(d*x + c)) + 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 - 14 4*(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 - 18*(d*x + c)^(5/2 )*D*a^2*b^3*c*d^2 - 36*(d*x + c)^(5/2)*C*a*b^4*c*d^2 + 42*(d*x + c)^(5/2)* B*b^5*c*d^2 + 144*(d*x + c)^(3/2)*D*a^2*b^3*c^2*d^2 + 48*(d*x + c)^(3/2)*C *a*b^4*c^2*d^2 - 96*(d*x + c)^(3/2)*B*b^5*c^2*d^2 - 126*sqrt(d*x + c)*D*a^ 2*b^3*c^3*d^2 - 12*sqrt(d*x + c)*C*a*b^4*c^3*d^2 + 54*sqrt(d*x + c)*B*b^5* c^3*d^2 + 3*(d*x + c)^(5/2)*D*a^3*b^2*d^3 + 3*(d*x + c)^(5/2)*C*a^2*b^3*d^ 3 + 15*(d*x + c)^(5/2)*B*a*b^4*d^3 - 57*(d*x + c)^(5/2)*A*b^5*d^3 + 8*(d*x + c)^(3/2)*D*a^3*b^2*c*d^3 - 104*(d*x + c)^(3/2)*C*a^2*b^3*c*d^3 + 56*(d* x + c)^(3/2)*B*a*b^4*c*d^3 + 136*(d*x + c)^(3/2)*A*b^5*c*d^3 + 33*sqrt(d*x + c)*D*a^3*b^2*c^2*d^3 + 93*sqrt(d*x + c)*C*a^2*b^3*c^2*d^3 - 75*sqrt(d*x + c)*B*a*b^4*c^2*d^3 - 87*sqrt(d*x + c)*A*b^5*c^2*d^3 - 8*(d*x + c)^(3/2) *D*a^4*b*d^4 + 8*(d*x + c)^(3/2)*C*a^3*b^2*d^4 + 40*(d*x + c)^(3/2)*B*a...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^4\,{\left (c+d\,x\right )}^{3/2}} \,d x \]