Integrand size = 32, antiderivative size = 495 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{4 b^3 (b c-a d) (a+b x)^4}-\frac {\left (b^3 (8 B c-7 A d)-a b^2 (16 c C+B d)-17 a^3 d D+3 a^2 b (3 C d+8 c D)\right ) \sqrt {c+d x}}{24 b^3 (b c-a d)^2 (a+b x)^3}-\frac {\left (b^3 \left (48 c^2 C-40 B c d+35 A d^2\right )-59 a^3 d^2 D+3 a^2 b d (C d+56 c D)-a b^2 \left (16 c C d-5 B d^2+144 c^2 D\right )\right ) \sqrt {c+d x}}{96 b^3 (b c-a d)^3 (a+b x)^2}+\frac {\left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \sqrt {c+d x}}{64 b^3 (b c-a d)^4 (a+b x)}-\frac {d \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (16 c C d-5 B d^2-48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{7/2} (b c-a d)^{9/2}} \]
-1/64*d*(5*a^3*d^3*D+3*a^2*b*d^2*(C*d-8*D*c)-a*b^2*d*(-5*B*d^2+16*C*c*d-48 *D*c^2)+b^3*(35*A*d^3-40*B*c*d^2+48*C*c^2*d-64*D*c^3))*arctanh(b^(1/2)*(d* x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(7/2)/(-a*d+b*c)^(9/2)-1/4*(A*b^3-a*(B*b^2- C*a*b+D*a^2))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)/(b*x+a)^4-1/24*(b^3*(-7*A*d+8*B *c)-a*b^2*(B*d+16*C*c)-17*a^3*d*D+3*a^2*b*(3*C*d+8*D*c))*(d*x+c)^(1/2)/b^3 /(-a*d+b*c)^2/(b*x+a)^3-1/96*(b^3*(35*A*d^2-40*B*c*d+48*C*c^2)-59*a^3*d^2* D+3*a^2*b*d*(C*d+56*D*c)-a*b^2*(-5*B*d^2+16*C*c*d+144*D*c^2))*(d*x+c)^(1/2 )/b^3/(-a*d+b*c)^3/(b*x+a)^2+1/64*(5*a^3*d^3*D+3*a^2*b*d^2*(C*d-8*D*c)-a*b ^2*d*(-5*B*d^2+16*C*c*d-48*D*c^2)+b^3*(35*A*d^3-40*B*c*d^2+48*C*c^2*d-64*D *c^3))*(d*x+c)^(1/2)/b^3/(-a*d+b*c)^4/(b*x+a)
Time = 3.20 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x} \left (15 a^6 d^3 D+a^5 b d^2 (9 C d-62 c D+55 d D x)+8 b^6 c x \left (6 c x (2 c C-3 C d x+4 c D x)+B \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )+a^3 b^3 \left (48 c^3 D+c^2 (-88 C d+296 d D x)+2 c d^2 \left (73 B-26 C x-119 D x^2\right )-d^3 x \left (73 B+33 C x+15 D x^2\right )\right )+a^2 b^4 \left (-d^3 x^2 (55 B+9 C x)+16 c^3 (C+12 D x)-24 c^2 d \left (3 B+15 C x-8 D x^2\right )+2 c d^2 x \left (310 B+91 C x+36 D x^2\right )\right )+a^4 b^2 d \left (104 c^2 D-6 c d (7 C+38 D x)+d^2 \left (15 B+33 C x+73 D x^2\right )\right )+A b^3 \left (-279 a^3 d^3+a^2 b d^2 (326 c-511 d x)+a b^2 d \left (-200 c^2+252 c d x-385 d^2 x^2\right )+b^3 \left (48 c^3-56 c^2 d x+70 c d^2 x^2-105 d^3 x^3\right )\right )+a b^5 \left (B \left (16 c^3-296 c^2 d x+450 c d^2 x^2-15 d^3 x^3\right )+16 c x \left (3 C d^2 x^2+2 c^2 (2 C+9 D x)-c d x (35 C+9 D x)\right )\right )\right )}{192 b^3 (b c-a d)^4 (a+b x)^4}+\frac {d \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)+a b^2 d \left (-16 c C d+5 B d^2+48 c^2 D\right )+b^3 \left (48 c^2 C d-40 B c d^2+35 A d^3-64 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{64 b^{7/2} (-b c+a d)^{9/2}} \]
-1/192*(Sqrt[c + d*x]*(15*a^6*d^3*D + a^5*b*d^2*(9*C*d - 62*c*D + 55*d*D*x ) + 8*b^6*c*x*(6*c*x*(2*c*C - 3*C*d*x + 4*c*D*x) + B*(8*c^2 - 10*c*d*x + 1 5*d^2*x^2)) + a^3*b^3*(48*c^3*D + c^2*(-88*C*d + 296*d*D*x) + 2*c*d^2*(73* B - 26*C*x - 119*D*x^2) - d^3*x*(73*B + 33*C*x + 15*D*x^2)) + a^2*b^4*(-(d ^3*x^2*(55*B + 9*C*x)) + 16*c^3*(C + 12*D*x) - 24*c^2*d*(3*B + 15*C*x - 8* D*x^2) + 2*c*d^2*x*(310*B + 91*C*x + 36*D*x^2)) + a^4*b^2*d*(104*c^2*D - 6 *c*d*(7*C + 38*D*x) + d^2*(15*B + 33*C*x + 73*D*x^2)) + A*b^3*(-279*a^3*d^ 3 + a^2*b*d^2*(326*c - 511*d*x) + a*b^2*d*(-200*c^2 + 252*c*d*x - 385*d^2* x^2) + b^3*(48*c^3 - 56*c^2*d*x + 70*c*d^2*x^2 - 105*d^3*x^3)) + a*b^5*(B* (16*c^3 - 296*c^2*d*x + 450*c*d^2*x^2 - 15*d^3*x^3) + 16*c*x*(3*C*d^2*x^2 + 2*c^2*(2*C + 9*D*x) - c*d*x*(35*C + 9*D*x)))))/(b^3*(b*c - a*d)^4*(a + b *x)^4) + (d*(5*a^3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) + a*b^2*d*(-16*c*C*d + 5*B*d^2 + 48*c^2*D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D ))*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(64*b^(7/2)*(-(b*c) + a*d)^(9/2))
Time = 0.96 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2124, 27, 1192, 25, 1471, 25, 27, 298, 215, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {8 \left (c-\frac {a d}{b}\right ) D x^2+\frac {8 (b c-a d) (b C-a D) x}{b^2}+\frac {-d D a^3+b (C d+8 c D) a^2-b^2 (8 c C+B d) a+b^3 (8 B c-7 A d)}{b^3}}{2 (a+b x)^4 \sqrt {c+d x}}dx}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\frac {d D a^3}{b^3}+\frac {(C d+8 c D) a^2}{b^2}-\frac {(8 c C+B d) a}{b}+8 \left (c-\frac {a d}{b}\right ) D x^2+8 B c-7 A d+\frac {8 (b c-a d) (b C-a D) x}{b^2}}{(a+b x)^4 \sqrt {c+d x}}dx}{8 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {d \int -\frac {-8 D c^3+8 C d c^2-8 B d^2 c+7 A d^3-8 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+\frac {a d^3 \left (D a^2-b C a+b^2 B\right )}{b^3}-\frac {8 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{(b c-a d-b (c+d x))^4}d\sqrt {c+d x}}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {d \int \frac {-8 D c^3+8 C d c^2-8 B d^2 c-8 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+d^3 \left (7 A+\frac {a \left (D a^2-b C a+b^2 B\right )}{b^3}\right )-\frac {8 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{(b c-a d-b (c+d x))^4}d\sqrt {c+d x}}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {d \left (\frac {\int -\frac {\left (-48 D c^3+48 C d c^2-\frac {8 d^2 \left (-3 D a^2+2 b C a+5 b^2 B\right ) c}{b^2}+35 A d^3+\frac {a d^3 \left (-11 D a^2+3 b C a+5 b^2 B\right )}{b^3}\right ) b^2+48 (b c-a d)^2 D (c+d x)}{b^2 (b c-a d-b (c+d x))^3}d\sqrt {c+d x}}{6 (b c-a d)}+\frac {d^2 \sqrt {c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{6 b^3 (b c-a d) (-a d-b (c+d x)+b c)^3}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{6 b^3 (b c-a d) (-a d-b (c+d x)+b c)^3}-\frac {\int \frac {\left (-48 D c^3+48 C d c^2-\frac {8 d^2 \left (-3 D a^2+2 b C a+5 b^2 B\right ) c}{b^2}+d^3 \left (35 A+\frac {a \left (-11 D a^2+3 b C a+5 b^2 B\right )}{b^3}\right )\right ) b^2+48 (b c-a d)^2 D (c+d x)}{b^2 (b c-a d-b (c+d x))^3}d\sqrt {c+d x}}{6 (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{6 b^3 (b c-a d) (-a d-b (c+d x)+b c)^3}-\frac {\int \frac {\left (-48 D c^3+48 C d c^2-\frac {8 d^2 \left (-3 D a^2+2 b C a+5 b^2 B\right ) c}{b^2}+d^3 \left (35 A+\frac {a \left (-11 D a^2+3 b C a+5 b^2 B\right )}{b^3}\right )\right ) b^2+48 (b c-a d)^2 D (c+d x)}{(b c-a d-b (c+d x))^3}d\sqrt {c+d x}}{6 b^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{6 b^3 (b c-a d) (-a d-b (c+d x)+b c)^3}-\frac {\frac {3 \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right ) \int \frac {1}{(b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{4 b (b c-a d)}+\frac {d \sqrt {c+d x} \left (-59 a^3 d^2 D+3 a^2 b d (56 c D+C d)-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{4 b (b c-a d) (-a d-b (c+d x)+b c)^2}}{6 b^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{6 b^3 (b c-a d) (-a d-b (c+d x)+b c)^3}-\frac {\frac {3 \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right ) \left (\frac {\int \frac {1}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{2 (b c-a d)}+\frac {\sqrt {c+d x}}{2 (b c-a d) (-a d-b (c+d x)+b c)}\right )}{4 b (b c-a d)}+\frac {d \sqrt {c+d x} \left (-59 a^3 d^2 D+3 a^2 b d (56 c D+C d)-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{4 b (b c-a d) (-a d-b (c+d x)+b c)^2}}{6 b^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (\frac {d^2 \sqrt {c+d x} \left (-17 a^3 d D+3 a^2 b (8 c D+3 C d)-a b^2 (B d+16 c C)+b^3 (8 B c-7 A d)\right )}{6 b^3 (b c-a d) (-a d-b (c+d x)+b c)^3}-\frac {\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x}}{2 (b c-a d) (-a d-b (c+d x)+b c)}\right ) \left (5 a^3 d^3 D+3 a^2 b d^2 (C d-8 c D)-a b^2 d \left (-5 B d^2-48 c^2 D+16 c C d\right )+b^3 \left (35 A d^3-40 B c d^2-64 c^3 D+48 c^2 C d\right )\right )}{4 b (b c-a d)}+\frac {d \sqrt {c+d x} \left (-59 a^3 d^2 D+3 a^2 b d (56 c D+C d)-a b^2 \left (-5 B d^2+144 c^2 D+16 c C d\right )+b^3 \left (35 A d^2-40 B c d+48 c^2 C\right )\right )}{4 b (b c-a d) (-a d-b (c+d x)+b c)^2}}{6 b^2 (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^3 (a+b x)^4 (b c-a d)}\) |
-1/4*((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/(b^3*(b*c - a*d)* (a + b*x)^4) + (d*((d^2*(b^3*(8*B*c - 7*A*d) - a*b^2*(16*c*C + B*d) - 17*a ^3*d*D + 3*a^2*b*(3*C*d + 8*c*D))*Sqrt[c + d*x])/(6*b^3*(b*c - a*d)*(b*c - a*d - b*(c + d*x))^3) - ((d*(b^3*(48*c^2*C - 40*B*c*d + 35*A*d^2) - 59*a^ 3*d^2*D + 3*a^2*b*d*(C*d + 56*c*D) - a*b^2*(16*c*C*d - 5*B*d^2 + 144*c^2*D ))*Sqrt[c + d*x])/(4*b*(b*c - a*d)*(b*c - a*d - b*(c + d*x))^2) + (3*(5*a^ 3*d^3*D + 3*a^2*b*d^2*(C*d - 8*c*D) - a*b^2*d*(16*c*C*d - 5*B*d^2 - 48*c^2 *D) + b^3*(48*c^2*C*d - 40*B*c*d^2 + 35*A*d^3 - 64*c^3*D))*(Sqrt[c + d*x]/ (2*(b*c - a*d)*(b*c - a*d - b*(c + d*x))) + ArcTanh[(Sqrt[b]*Sqrt[c + d*x] )/Sqrt[b*c - a*d]]/(2*Sqrt[b]*(b*c - a*d)^(3/2))))/(4*b*(b*c - a*d)))/(6*b ^2*(b*c - a*d))))/(4*(b*c - a*d))
3.1.9.3.1 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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]
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 = 1.92 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {\frac {35 \left (b x +a \right )^{4} \left (\left (A \,d^{3}-\frac {8}{7} B c \,d^{2}+\frac {48}{35} C \,c^{2} d -\frac {64}{35} D c^{3}\right ) b^{3}+\frac {a \left (B \,d^{2}-\frac {16}{5} C c d +\frac {48}{5} D c^{2}\right ) d \,b^{2}}{7}+\frac {3 a^{2} b \,d^{2} \left (C d -8 D c \right )}{35}+\frac {a^{3} d^{3} D}{7}\right ) d \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}+\frac {93 \sqrt {\left (a d -b c \right ) b}\, \left (\frac {\left (35 A \,d^{3} x^{3}-\frac {70 \left (\frac {12 B x}{7}+A \right ) x^{2} c \,d^{2}}{3}+\frac {56 x \,c^{2} \left (\frac {18}{7} C \,x^{2}+\frac {10}{7} B x +A \right ) d}{3}-16 c^{3} \left (4 D x^{3}+2 C \,x^{2}+\frac {4}{3} B x +A \right )\right ) b^{6}}{93}+\frac {200 a \left (\frac {77 x^{2} \left (\frac {3 B x}{77}+A \right ) d^{3}}{40}-\frac {63 \left (\frac {4}{21} C \,x^{2}+\frac {25}{14} B x +A \right ) x c \,d^{2}}{50}+c^{2} \left (\frac {18}{25} D x^{3}+\frac {14}{5} C \,x^{2}+\frac {37}{25} B x +A \right ) d -\frac {2 c^{3} \left (18 D x^{2}+4 C x +B \right )}{25}\right ) b^{5}}{279}-\frac {326 a^{2} \left (-\frac {511 x \left (\frac {9}{511} C \,x^{2}+\frac {55}{511} B x +A \right ) d^{3}}{326}+c \left (\frac {36}{163} D x^{3}+\frac {91}{163} C \,x^{2}+\frac {310}{163} B x +A \right ) d^{2}-\frac {36 \left (-\frac {8}{3} D x^{2}+5 C x +B \right ) c^{2} d}{163}+\frac {8 c^{3} \left (12 D x +C \right )}{163}\right ) b^{4}}{279}+a^{3} \left (\left (\frac {5}{93} D x^{3}+\frac {11}{93} C \,x^{2}+\frac {73}{279} B x +A \right ) d^{3}-\frac {146 c \left (-\frac {119}{73} D x^{2}-\frac {26}{73} C x +B \right ) d^{2}}{279}+\frac {88 c^{2} \left (-\frac {37 D x}{11}+C \right ) d}{279}-\frac {16 D c^{3}}{93}\right ) b^{3}-\frac {5 a^{4} \left (\left (\frac {73}{15} D x^{2}+\frac {11}{5} C x +B \right ) d^{2}-\frac {14 c \left (\frac {38 D x}{7}+C \right ) d}{5}+\frac {104 D c^{2}}{15}\right ) d \,b^{2}}{93}-\frac {a^{5} \left (\left (\frac {55 D x}{9}+C \right ) d -\frac {62 D c}{9}\right ) d^{2} b}{31}-\frac {5 D a^{6} d^{3}}{93}\right ) \sqrt {d x +c}}{64}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{4} \left (a d -b c \right )^{4} b^{3}}\) | \(540\) |
| derivativedivides | \(2 d \left (\frac {\frac {\left (35 A \,b^{3} d^{3}+5 B a \,b^{2} d^{3}-40 B \,b^{3} c \,d^{2}+3 a^{2} b C \,d^{3}-16 C a \,b^{2} c \,d^{2}+48 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-24 D a^{2} b c \,d^{2}+48 D a \,b^{2} c^{2} d -64 D b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {7}{2}}}{128 a^{4} d^{4}-512 a^{3} b c \,d^{3}+768 a^{2} b^{2} c^{2} d^{2}-512 a \,b^{3} c^{3} d +128 b^{4} c^{4}}+\frac {\left (385 A \,b^{3} d^{3}+55 B a \,b^{2} d^{3}-440 B \,b^{3} c \,d^{2}+33 a^{2} b C \,d^{3}-176 C a \,b^{2} c \,d^{2}+528 C \,b^{3} c^{2} d -73 a^{3} d^{3} D+120 D a^{2} b c \,d^{2}+144 D a \,b^{2} c^{2} d -576 D b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {5}{2}}}{384 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (511 A \,b^{3} d^{3}+73 B a \,b^{2} d^{3}-584 B \,b^{3} c \,d^{2}-33 a^{2} b C \,d^{3}-80 C a \,b^{2} c \,d^{2}+624 C \,b^{3} c^{2} d -55 a^{3} d^{3} D+264 D a^{2} b c \,d^{2}-144 D a \,b^{2} c^{2} d -576 D b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {3}{2}}}{384 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (93 A \,b^{3} d^{3}-5 B a \,b^{2} d^{3}-88 B \,b^{3} c \,d^{2}-3 a^{2} b C \,d^{3}+16 C a \,b^{2} c \,d^{2}+80 C \,b^{3} c^{2} d -5 a^{3} d^{3} D+24 D a^{2} b c \,d^{2}-48 D a \,b^{2} c^{2} d -64 D b^{3} c^{3}\right ) \sqrt {d x +c}}{128 \left (a d -b c \right ) b^{3}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {\left (35 A \,b^{3} d^{3}+5 B a \,b^{2} d^{3}-40 B \,b^{3} c \,d^{2}+3 a^{2} b C \,d^{3}-16 C a \,b^{2} c \,d^{2}+48 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-24 D a^{2} b c \,d^{2}+48 D a \,b^{2} c^{2} d -64 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(786\) |
| default | \(2 d \left (\frac {\frac {\left (35 A \,b^{3} d^{3}+5 B a \,b^{2} d^{3}-40 B \,b^{3} c \,d^{2}+3 a^{2} b C \,d^{3}-16 C a \,b^{2} c \,d^{2}+48 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-24 D a^{2} b c \,d^{2}+48 D a \,b^{2} c^{2} d -64 D b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {7}{2}}}{128 a^{4} d^{4}-512 a^{3} b c \,d^{3}+768 a^{2} b^{2} c^{2} d^{2}-512 a \,b^{3} c^{3} d +128 b^{4} c^{4}}+\frac {\left (385 A \,b^{3} d^{3}+55 B a \,b^{2} d^{3}-440 B \,b^{3} c \,d^{2}+33 a^{2} b C \,d^{3}-176 C a \,b^{2} c \,d^{2}+528 C \,b^{3} c^{2} d -73 a^{3} d^{3} D+120 D a^{2} b c \,d^{2}+144 D a \,b^{2} c^{2} d -576 D b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {5}{2}}}{384 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (511 A \,b^{3} d^{3}+73 B a \,b^{2} d^{3}-584 B \,b^{3} c \,d^{2}-33 a^{2} b C \,d^{3}-80 C a \,b^{2} c \,d^{2}+624 C \,b^{3} c^{2} d -55 a^{3} d^{3} D+264 D a^{2} b c \,d^{2}-144 D a \,b^{2} c^{2} d -576 D b^{3} c^{3}\right ) \left (d x +c \right )^{\frac {3}{2}}}{384 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (93 A \,b^{3} d^{3}-5 B a \,b^{2} d^{3}-88 B \,b^{3} c \,d^{2}-3 a^{2} b C \,d^{3}+16 C a \,b^{2} c \,d^{2}+80 C \,b^{3} c^{2} d -5 a^{3} d^{3} D+24 D a^{2} b c \,d^{2}-48 D a \,b^{2} c^{2} d -64 D b^{3} c^{3}\right ) \sqrt {d x +c}}{128 \left (a d -b c \right ) b^{3}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {\left (35 A \,b^{3} d^{3}+5 B a \,b^{2} d^{3}-40 B \,b^{3} c \,d^{2}+3 a^{2} b C \,d^{3}-16 C a \,b^{2} c \,d^{2}+48 C \,b^{3} c^{2} d +5 a^{3} d^{3} D-24 D a^{2} b c \,d^{2}+48 D a \,b^{2} c^{2} d -64 D b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(786\) |
93/64*(35/93*(b*x+a)^4*((A*d^3-8/7*B*c*d^2+48/35*C*c^2*d-64/35*D*c^3)*b^3+ 1/7*a*(B*d^2-16/5*C*c*d+48/5*D*c^2)*d*b^2+3/35*a^2*b*d^2*(C*d-8*D*c)+1/7*a ^3*d^3*D)*d*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2 )*(1/93*(35*A*d^3*x^3-70/3*(12/7*B*x+A)*x^2*c*d^2+56/3*x*c^2*(18/7*C*x^2+1 0/7*B*x+A)*d-16*c^3*(4*D*x^3+2*C*x^2+4/3*B*x+A))*b^6+200/279*a*(77/40*x^2* (3/77*B*x+A)*d^3-63/50*(4/21*C*x^2+25/14*B*x+A)*x*c*d^2+c^2*(18/25*D*x^3+1 4/5*C*x^2+37/25*B*x+A)*d-2/25*c^3*(18*D*x^2+4*C*x+B))*b^5-326/279*a^2*(-51 1/326*x*(9/511*C*x^2+55/511*B*x+A)*d^3+c*(36/163*D*x^3+91/163*C*x^2+310/16 3*B*x+A)*d^2-36/163*(-8/3*D*x^2+5*C*x+B)*c^2*d+8/163*c^3*(12*D*x+C))*b^4+a ^3*((5/93*D*x^3+11/93*C*x^2+73/279*B*x+A)*d^3-146/279*c*(-119/73*D*x^2-26/ 73*C*x+B)*d^2+88/279*c^2*(-37/11*D*x+C)*d-16/93*D*c^3)*b^3-5/93*a^4*((73/1 5*D*x^2+11/5*C*x+B)*d^2-14/5*c*(38/7*D*x+C)*d+104/15*D*c^2)*d*b^2-1/31*a^5 *((55/9*D*x+C)*d-62/9*D*c)*d^2*b-5/93*D*a^6*d^3)*(d*x+c)^(1/2))/((a*d-b*c) *b)^(1/2)/(b*x+a)^4/(a*d-b*c)^4/b^3
Leaf count of result is larger than twice the leaf count of optimal. 1805 vs. \(2 (469) = 938\).
Time = 0.49 (sec) , antiderivative size = 3624, normalized size of antiderivative = 7.32 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
[-1/384*(3*(64*D*a^4*b^3*c^3*d - (5*D*a^7 + 3*C*a^6*b + 5*B*a^5*b^2 + 35*A *a^4*b^3)*d^4 + (64*D*b^7*c^3*d - (5*D*a^3*b^4 + 3*C*a^2*b^5 + 5*B*a*b^6 + 35*A*b^7)*d^4 + 8*(3*D*a^2*b^5*c + (2*C*a*b^6 + 5*B*b^7)*c)*d^3 - 48*(D*a *b^6*c^2 + C*b^7*c^2)*d^2)*x^4 + 8*(3*D*a^6*b*c + (2*C*a^5*b^2 + 5*B*a^4*b ^3)*c)*d^3 + 4*(64*D*a*b^6*c^3*d - (5*D*a^4*b^3 + 3*C*a^3*b^4 + 5*B*a^2*b^ 5 + 35*A*a*b^6)*d^4 + 8*(3*D*a^3*b^4*c + (2*C*a^2*b^5 + 5*B*a*b^6)*c)*d^3 - 48*(D*a^2*b^5*c^2 + C*a*b^6*c^2)*d^2)*x^3 - 48*(D*a^5*b^2*c^2 + C*a^4*b^ 3*c^2)*d^2 + 6*(64*D*a^2*b^5*c^3*d - (5*D*a^5*b^2 + 3*C*a^4*b^3 + 5*B*a^3* b^4 + 35*A*a^2*b^5)*d^4 + 8*(3*D*a^4*b^3*c + (2*C*a^3*b^4 + 5*B*a^2*b^5)*c )*d^3 - 48*(D*a^3*b^4*c^2 + C*a^2*b^5*c^2)*d^2)*x^2 + 4*(64*D*a^3*b^4*c^3* d - (5*D*a^6*b + 3*C*a^5*b^2 + 5*B*a^4*b^3 + 35*A*a^3*b^4)*d^4 + 8*(3*D*a^ 5*b^2*c + (2*C*a^4*b^3 + 5*B*a^3*b^4)*c)*d^3 - 48*(D*a^4*b^3*c^2 + C*a^3*b ^4*c^2)*d^2)*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*(48*D*a^3*b^5*c^4 + 16*(C*a^2*b^6 + B*a*b^7 + 3*A*b^8)*c^4 - 3*(5*D*a^7*b + 3*C*a^6*b^2 + 5*B*a^5*b^3 - 93* A*a^4*b^4)*d^4 + (77*D*a^6*b^2*c + (51*C*a^5*b^3 - 131*B*a^4*b^4 - 605*A*a ^3*b^5)*c)*d^3 + 3*(64*D*b^8*c^4 + (5*D*a^4*b^4 + 3*C*a^3*b^5 + 5*B*a^2*b^ 6 + 35*A*a*b^7)*d^4 - (29*D*a^3*b^5*c + (19*C*a^2*b^6 + 45*B*a*b^7 + 35*A* b^8)*c)*d^3 + 8*(9*D*a^2*b^6*c^2 + (8*C*a*b^7 + 5*B*b^8)*c^2)*d^2 - 16*(7* D*a*b^7*c^3 + 3*C*b^8*c^3)*d)*x^3 - 2*(83*D*a^5*b^3*c^2 - (23*C*a^4*b^4...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 1512 vs. \(2 (469) = 938\).
Time = 0.32 (sec) , antiderivative size = 1512, normalized size of antiderivative = 3.05 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=\text {Too large to display} \]
-1/64*(64*D*b^3*c^3*d - 48*D*a*b^2*c^2*d^2 - 48*C*b^3*c^2*d^2 + 24*D*a^2*b *c*d^3 + 16*C*a*b^2*c*d^3 + 40*B*b^3*c*d^3 - 5*D*a^3*d^4 - 3*C*a^2*b*d^4 - 5*B*a*b^2*d^4 - 35*A*b^3*d^4)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d) )/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^ 3*d^4)*sqrt(-b^2*c + a*b*d)) - 1/192*(192*(d*x + c)^(7/2)*D*b^6*c^3*d - 57 6*(d*x + c)^(5/2)*D*b^6*c^4*d + 576*(d*x + c)^(3/2)*D*b^6*c^5*d - 192*sqrt (d*x + c)*D*b^6*c^6*d - 144*(d*x + c)^(7/2)*D*a*b^5*c^2*d^2 - 144*(d*x + c )^(7/2)*C*b^6*c^2*d^2 + 720*(d*x + c)^(5/2)*D*a*b^5*c^3*d^2 + 528*(d*x + c )^(5/2)*C*b^6*c^3*d^2 - 1008*(d*x + c)^(3/2)*D*a*b^5*c^4*d^2 - 624*(d*x + c)^(3/2)*C*b^6*c^4*d^2 + 432*sqrt(d*x + c)*D*a*b^5*c^5*d^2 + 240*sqrt(d*x + c)*C*b^6*c^5*d^2 + 72*(d*x + c)^(7/2)*D*a^2*b^4*c*d^3 + 48*(d*x + c)^(7/ 2)*C*a*b^5*c*d^3 + 120*(d*x + c)^(7/2)*B*b^6*c*d^3 - 24*(d*x + c)^(5/2)*D* a^2*b^4*c^2*d^3 - 704*(d*x + c)^(5/2)*C*a*b^5*c^2*d^3 - 440*(d*x + c)^(5/2 )*B*b^6*c^2*d^3 + 24*(d*x + c)^(3/2)*D*a^2*b^4*c^3*d^3 + 1328*(d*x + c)^(3 /2)*C*a*b^5*c^3*d^3 + 584*(d*x + c)^(3/2)*B*b^6*c^3*d^3 - 72*sqrt(d*x + c) *D*a^2*b^4*c^4*d^3 - 672*sqrt(d*x + c)*C*a*b^5*c^4*d^3 - 264*sqrt(d*x + c) *B*b^6*c^4*d^3 - 15*(d*x + c)^(7/2)*D*a^3*b^3*d^4 - 9*(d*x + c)^(7/2)*C*a^ 2*b^4*d^4 - 15*(d*x + c)^(7/2)*B*a*b^5*d^4 - 105*(d*x + c)^(7/2)*A*b^6*d^4 - 193*(d*x + c)^(5/2)*D*a^3*b^3*c*d^4 + 209*(d*x + c)^(5/2)*C*a^2*b^4*c*d ^4 + 495*(d*x + c)^(5/2)*B*a*b^5*c*d^4 + 385*(d*x + c)^(5/2)*A*b^6*c*d^...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^5 \sqrt {c+d x}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^5\,\sqrt {c+d\,x}} \,d x \]