Integrand size = 32, antiderivative size = 438 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=-\frac {3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )}{6 b^3 d (b c-a d)^3 (c+d x)^{3/2}}-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac {a^2 b C d^2+b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )-a^3 d^2 D+a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )}{b^2 (b c-a d)^4 \sqrt {c+d x}}-\frac {\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt {c+d x}}{4 b (b c-a d)^4 (a+b x)}-\frac {\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{9/2}} \]
1/6*(-3*a*b^2*B*d^3+3*a^2*b*C*d^3-3*a^3*d^3*D+b^3*(7*A*d^3-4*B*c*d^2+4*C*c ^2*d-4*D*c^3))/b^3/d/(-a*d+b*c)^3/(d*x+c)^(3/2)+1/2*(-A*b^3+a*(B*b^2-C*a*b +D*a^2))/b^3/(-a*d+b*c)/(b*x+a)^2/(d*x+c)^(3/2)-1/4*(b^3*(35*A*d^2-20*B*c* d+8*C*c^2)+a^3*d^2*D+3*a^2*b*d*(C*d-4*D*c)+3*a*b^2*(-5*B*d^2+8*C*c*d-8*D*c ^2))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c)^(9 /2)+(a^2*b*C*d^2+b^3*(7*A*d^2-4*B*c*d+2*C*c^2)-a^3*d^2*D+a*b^2*(-3*B*d^2+4 *C*c*d-6*D*c^2))/b^2/(-a*d+b*c)^4/(d*x+c)^(1/2)-1/4*(b^3*(-7*A*d+4*B*c)-a* b^2*(-3*B*d+8*C*c)-5*a^3*d*D+a^2*b*(C*d+12*D*c))*(d*x+c)^(1/2)/b/(-a*d+b*c )^4/(b*x+a)
Time = 1.54 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\frac {-3 a^4 d^2 D (c+d x)^2-4 b^4 c x \left (2 c x \left (-4 c C d+c^2 D-3 C d^2 x\right )+B d \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )+a^3 b d \left (-94 c^3 D+c^2 d (55 C-129 D x)+3 d^3 x \left (-8 B+5 C x+D x^2\right )-2 c d^2 \left (8 B-39 C x+12 D x^2\right )\right )-a^2 b^2 \left (8 c^4 D+3 d^4 x^2 (25 B-3 C x)+c^3 (-50 C d+164 d D x)+2 c d^3 x \left (67 B-66 C x+18 D x^2\right )+c^2 d^2 \left (83 B-149 C x+216 D x^2\right )\right )+A b d \left (-8 a^3 d^3+8 a^2 b d^2 (10 c+7 d x)+a b^2 d \left (39 c^2+238 c d x+175 d^2 x^2\right )+b^3 \left (-6 c^3+21 c^2 d x+140 c d^2 x^2+105 d^3 x^3\right )\right )-a b^3 \left (B d \left (6 c^3+145 c^2 d x+160 c d^2 x^2+45 d^3 x^3\right )+8 c x \left (2 c^3 D-9 C d^3 x^2+c d^2 x (-17 C+9 D x)+c^2 (-11 C d+8 d D x)\right )\right )}{12 b d (b c-a d)^4 (a+b x)^2 (c+d x)^{3/2}}+\frac {\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)-3 a b^2 \left (-8 c C d+5 B d^2+8 c^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{4 b^{3/2} (-b c+a d)^{9/2}} \]
(-3*a^4*d^2*D*(c + d*x)^2 - 4*b^4*c*x*(2*c*x*(-4*c*C*d + c^2*D - 3*C*d^2*x ) + B*d*(3*c^2 + 20*c*d*x + 15*d^2*x^2)) + a^3*b*d*(-94*c^3*D + c^2*d*(55* C - 129*D*x) + 3*d^3*x*(-8*B + 5*C*x + D*x^2) - 2*c*d^2*(8*B - 39*C*x + 12 *D*x^2)) - a^2*b^2*(8*c^4*D + 3*d^4*x^2*(25*B - 3*C*x) + c^3*(-50*C*d + 16 4*d*D*x) + 2*c*d^3*x*(67*B - 66*C*x + 18*D*x^2) + c^2*d^2*(83*B - 149*C*x + 216*D*x^2)) + A*b*d*(-8*a^3*d^3 + 8*a^2*b*d^2*(10*c + 7*d*x) + a*b^2*d*( 39*c^2 + 238*c*d*x + 175*d^2*x^2) + b^3*(-6*c^3 + 21*c^2*d*x + 140*c*d^2*x ^2 + 105*d^3*x^3)) - a*b^3*(B*d*(6*c^3 + 145*c^2*d*x + 160*c*d^2*x^2 + 45* d^3*x^3) + 8*c*x*(2*c^3*D - 9*C*d^3*x^2 + c*d^2*x*(-17*C + 9*D*x) + c^2*(- 11*C*d + 8*d*D*x))))/(12*b*d*(b*c - a*d)^4*(a + b*x)^2*(c + d*x)^(3/2)) + ((b^3*(8*c^2*C - 20*B*c*d + 35*A*d^2) + a^3*d^2*D + 3*a^2*b*d*(C*d - 4*c*D ) - 3*a*b^2*(-8*c*C*d + 5*B*d^2 + 8*c^2*D))*ArcTan[(Sqrt[b]*Sqrt[c + d*x]) /Sqrt[-(b*c) + a*d]])/(4*b^(3/2)*(-(b*c) + a*d)^(9/2))
Time = 1.26 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2124, 27, 1192, 25, 1582, 25, 1584, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {4 \left (c-\frac {a d}{b}\right ) D x^2+\frac {4 (b c-a d) (b C-a D) x}{b^2}+\frac {3 d D a^3-b (3 C d-4 c D) a^2-b^2 (4 c C-3 B d) a+b^3 (4 B c-7 A d)}{b^3}}{2 (a+b x)^2 (c+d x)^{5/2}}dx}{2 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\frac {3 d D a^3}{b^3}-\frac {(3 C d-4 c D) a^2}{b^2}-\frac {(4 c C-3 B d) a}{b}+4 \left (c-\frac {a d}{b}\right ) D x^2+4 B c-7 A d+\frac {4 (b c-a d) (b C-a D) x}{b^2}}{(a+b x)^2 (c+d x)^{5/2}}dx}{4 (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\int -\frac {-4 D c^3+4 C d c^2-4 B d^2 c-4 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+d^3 \left (7 A-\frac {3 a \left (D a^2-b C a+b^2 B\right )}{b^3}\right )-\frac {4 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{(c+d x)^2 (b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{2 d (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {-4 D c^3+4 C d c^2-4 B d^2 c-4 \left (c-\frac {a d}{b}\right ) D (c+d x)^2+d^3 \left (7 A-\frac {3 a \left (D a^2-b C a+b^2 B\right )}{b^3}\right )-\frac {4 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}}{(c+d x)^2 (b c-a d-b (c+d x))^2}d\sqrt {c+d x}}{2 d (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{2 b (b c-a d)^3 (-a d-b (c+d x)+b c)}-\frac {\int -\frac {\frac {2 \left (-\left (\left (-4 D c^3+4 C d c^2-4 B d^2 c+7 A d^3\right ) b^3\right )+3 a B d^3 b^2-3 a^2 C d^3 b+3 a^3 d^3 D\right ) (b c-a d)^2}{b}+2 \left (\left (-4 D c^3+4 B d^2 c-7 A d^3\right ) b^3-a d \left (-12 D c^2+8 C d c-3 B d^2\right ) b^2+a^2 C d^3 b-a^3 d^3 D\right ) (c+d x) (b c-a d)+b d^2 \left (-5 d D a^3+b (C d+12 c D) a^2-b^2 (8 c C-3 B d) a+b^3 (4 B c-7 A d)\right ) (c+d x)^2}{(c+d x)^2 (b c-a d-b (c+d x))}d\sqrt {c+d x}}{2 b^2 (b c-a d)^3}}{2 d (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {2 \left (-\left (\left (-4 D c^3+4 C d c^2-4 B d^2 c+7 A d^3\right ) b^3\right )+3 a B d^3 b^2-3 a^2 C d^3 b+3 a^3 d^3 D\right ) (b c-a d)^2}{b}+2 \left (\left (-4 D c^3+4 B d^2 c-7 A d^3\right ) b^3-a d \left (-12 D c^2+8 C d c-3 B d^2\right ) b^2+a^2 C d^3 b-a^3 d^3 D\right ) (c+d x) (b c-a d)+b d^2 \left (-5 d D a^3+b (C d+12 c D) a^2-b^2 (8 c C-3 B d) a+b^3 (4 B c-7 A d)\right ) (c+d x)^2}{(c+d x)^2 (b c-a d-b (c+d x))}d\sqrt {c+d x}}{2 b^2 (b c-a d)^3}+\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{2 b (b c-a d)^3 (-a d-b (c+d x)+b c)}}{2 d (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {\int \left (\frac {b d \left (-d^2 D a^3-3 b d (C d-4 c D) a^2-3 b^2 \left (-8 D c^2+8 C d c-5 B d^2\right ) a-b^3 \left (8 C c^2-20 B d c+35 A d^2\right )\right )}{b c-a d-b (c+d x)}+\frac {4 d \left (d^2 D a^3-b C d^2 a^2-b^2 \left (-6 D c^2+4 C d c-3 B d^2\right ) a-b^3 \left (2 C c^2-4 B d c+7 A d^2\right )\right )}{c+d x}+\frac {2 (b c-a d) \left (-\left (\left (-4 D c^3+4 C d c^2-4 B d^2 c+7 A d^3\right ) b^3\right )+3 a B d^3 b^2-3 a^2 C d^3 b+3 a^3 d^3 D\right )}{b (c+d x)^2}\right )d\sqrt {c+d x}}{2 b^2 (b c-a d)^3}+\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{2 b (b c-a d)^3 (-a d-b (c+d x)+b c)}}{2 d (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {\sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (-5 B d^2-8 c^2 D+8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{\sqrt {b c-a d}}+\frac {4 d \left (a^3 \left (-d^2\right ) D+a^2 b C d^2+a b^2 \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (7 A d^2-4 B c d+2 c^2 C\right )\right )}{\sqrt {c+d x}}-\frac {2 (b c-a d) \left (3 a^3 d^3 D-3 a^2 b C d^3+3 a b^2 B d^3-\left (b^3 \left (7 A d^3-4 B c d^2-4 c^3 D+4 c^2 C d\right )\right )\right )}{3 b (c+d x)^{3/2}}}{2 b^2 (b c-a d)^3}+\frac {d^2 \sqrt {c+d x} \left (-5 a^3 d D+a^2 b (12 c D+C d)-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{2 b (b c-a d)^3 (-a d-b (c+d x)+b c)}}{2 d (b c-a d)}-\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}\) |
-1/2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(b^3*(b*c - a*d)*(a + b*x)^2*(c + d*x)^(3/2)) + ((d^2*(b^3*(4*B*c - 7*A*d) - a*b^2*(8*c*C - 3*B*d) - 5*a^3* d*D + a^2*b*(C*d + 12*c*D))*Sqrt[c + d*x])/(2*b*(b*c - a*d)^3*(b*c - a*d - b*(c + d*x))) + ((-2*(b*c - a*d)*(3*a*b^2*B*d^3 - 3*a^2*b*C*d^3 + 3*a^3*d ^3*D - b^3*(4*c^2*C*d - 4*B*c*d^2 + 7*A*d^3 - 4*c^3*D)))/(3*b*(c + d*x)^(3 /2)) + (4*d*(a^2*b*C*d^2 + b^3*(2*c^2*C - 4*B*c*d + 7*A*d^2) - a^3*d^2*D + a*b^2*(4*c*C*d - 3*B*d^2 - 6*c^2*D)))/Sqrt[c + d*x] - (Sqrt[b]*d*(b^3*(8* c^2*C - 20*B*c*d + 35*A*d^2) + a^3*d^2*D + 3*a^2*b*d*(C*d - 4*c*D) + 3*a*b ^2*(8*c*C*d - 5*B*d^2 - 8*c^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/Sqrt[b*c - a*d])/(2*b^2*(b*c - a*d)^3))/(2*d*(b*c - a*d))
3.1.24.3.1 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[((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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
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 = 1.96 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {\frac {2 d \left (\frac {\left (\frac {11}{8} A \,b^{3} d^{2}-\frac {7}{8} B a \,b^{2} d^{2}-\frac {1}{2} B \,b^{3} c d +\frac {3}{8} a^{2} b C \,d^{2}+C a \,b^{2} c d +\frac {1}{8} a^{3} d^{2} D-\frac {3}{2} D a^{2} b c d \right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {d \left (13 A a \,b^{3} d^{2}-13 A \,b^{4} c d -9 B \,a^{2} b^{2} d^{2}+5 B a \,b^{3} c d +4 B \,b^{4} c^{2}+5 C \,a^{3} b \,d^{2}+3 C \,a^{2} b^{2} c d -8 C a \,b^{3} c^{2}-D a^{4} d^{2}-11 D a^{3} b c d +12 D a^{2} b^{2} c^{2}\right ) \sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\left (35 A \,b^{3} d^{2}-15 B a \,b^{2} d^{2}-20 B \,b^{3} c d +3 a^{2} b C \,d^{2}+24 C a \,b^{2} c d +8 C \,b^{3} c^{2}+a^{3} d^{2} D-12 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 d \left (3 A b \,d^{2}-B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-3 D a \,c^{2}\right )}{\left (a d -b c \right )^{4} \sqrt {d x +c}}}{d}\) | \(459\) |
| default | \(\frac {\frac {2 d \left (\frac {\left (\frac {11}{8} A \,b^{3} d^{2}-\frac {7}{8} B a \,b^{2} d^{2}-\frac {1}{2} B \,b^{3} c d +\frac {3}{8} a^{2} b C \,d^{2}+C a \,b^{2} c d +\frac {1}{8} a^{3} d^{2} D-\frac {3}{2} D a^{2} b c d \right ) \left (d x +c \right )^{\frac {3}{2}}+\frac {d \left (13 A a \,b^{3} d^{2}-13 A \,b^{4} c d -9 B \,a^{2} b^{2} d^{2}+5 B a \,b^{3} c d +4 B \,b^{4} c^{2}+5 C \,a^{3} b \,d^{2}+3 C \,a^{2} b^{2} c d -8 C a \,b^{3} c^{2}-D a^{4} d^{2}-11 D a^{3} b c d +12 D a^{2} b^{2} c^{2}\right ) \sqrt {d x +c}}{8 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{2}}+\frac {\left (35 A \,b^{3} d^{2}-15 B a \,b^{2} d^{2}-20 B \,b^{3} c d +3 a^{2} b C \,d^{2}+24 C a \,b^{2} c d +8 C \,b^{3} c^{2}+a^{3} d^{2} D-12 D a^{2} b c d -24 D a \,b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 b \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}-\frac {2 \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 d \left (3 A b \,d^{2}-B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-3 D a \,c^{2}\right )}{\left (a d -b c \right )^{4} \sqrt {d x +c}}}{d}\) | \(459\) |
| pseudoelliptic | \(\frac {\frac {35 \left (d x +c \right )^{\frac {3}{2}} \left (\left (b^{3} A -\frac {3}{7} a \,b^{2} B +\frac {3}{35} C \,a^{2} b +\frac {1}{35} D a^{3}\right ) d^{2}-\frac {4 \left (B \,b^{2}-\frac {6}{5} C a b +\frac {3}{5} D a^{2}\right ) b c d}{7}+\frac {8 b^{2} c^{2} \left (C b -3 D a \right )}{35}\right ) \left (b x +a \right )^{2} d \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{4}-\frac {2 \left (\left (-\frac {105 A \,x^{3} b^{4}}{8}-\frac {175 a \,x^{2} \left (-\frac {9 B x}{35}+A \right ) b^{3}}{8}-7 a^{2} x \left (\frac {9}{56} C \,x^{2}-\frac {75}{56} B x +A \right ) b^{2}+a^{3} \left (A -\frac {3}{8} D x^{3}-\frac {15}{8} C \,x^{2}+3 B x \right ) b +\frac {3 D a^{4} x^{2}}{8}\right ) d^{4}-10 \left (\frac {7 x^{2} \left (-\frac {3 B x}{7}+A \right ) b^{4}}{4}+\frac {119 a x \left (\frac {36}{119} C \,x^{2}-\frac {80}{119} B x +A \right ) b^{3}}{40}+a^{2} \left (-\frac {9}{20} D x^{3}+\frac {33}{20} C \,x^{2}-\frac {67}{40} B x +A \right ) b^{2}-\frac {a^{3} \left (\frac {3}{2} D x^{2}-\frac {39}{8} C x +B \right ) b}{5}-\frac {3 D a^{4} x}{40}\right ) c \,d^{3}-\frac {39 c^{2} \left (\frac {7 x \left (\frac {8}{7} C \,x^{2}-\frac {80}{21} B x +A \right ) b^{4}}{13}+a \left (-\frac {24}{13} D x^{3}+\frac {136}{39} C \,x^{2}-\frac {145}{39} B x +A \right ) b^{3}-\frac {83 a^{2} \left (\frac {216}{83} D x^{2}-\frac {149}{83} C x +B \right ) b^{2}}{39}+\frac {55 a^{3} \left (-\frac {129 D x}{55}+C \right ) b}{39}-\frac {D a^{4}}{13}\right ) d^{2}}{8}+\frac {3 b \,c^{3} \left (\left (-\frac {16}{3} C \,x^{2}+2 B x +A \right ) b^{3}+a \left (\frac {32}{3} D x^{2}-\frac {44}{3} C x +B \right ) b^{2}-\frac {25 a^{2} \left (-\frac {82 D x}{25}+C \right ) b}{3}+\frac {47 D a^{3}}{3}\right ) d}{4}+D b^{2} c^{4} \left (b x +a \right )^{2}\right ) \sqrt {\left (a d -b c \right ) b}}{3}}{\left (a d -b c \right )^{4} \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{2} \left (d x +c \right )^{\frac {3}{2}} b d}\) | \(502\) |
2/d*(-1/3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)/(a*d-b*c)^3/(d*x+c)^(3/2)+d*(3*A*b *d^2-B*a*d^2-2*B*b*c*d+2*C*a*c*d+C*b*c^2-3*D*a*c^2)/(a*d-b*c)^4/(d*x+c)^(1 /2)+d/(a*d-b*c)^4*(((11/8*A*b^3*d^2-7/8*B*a*b^2*d^2-1/2*B*b^3*c*d+3/8*a^2* b*C*d^2+C*a*b^2*c*d+1/8*a^3*d^2*D-3/2*D*a^2*b*c*d)*(d*x+c)^(3/2)+1/8*d*(13 *A*a*b^3*d^2-13*A*b^4*c*d-9*B*a^2*b^2*d^2+5*B*a*b^3*c*d+4*B*b^4*c^2+5*C*a^ 3*b*d^2+3*C*a^2*b^2*c*d-8*C*a*b^3*c^2-D*a^4*d^2-11*D*a^3*b*c*d+12*D*a^2*b^ 2*c^2)/b*(d*x+c)^(1/2))/((d*x+c)*b+a*d-b*c)^2+1/8*(35*A*b^3*d^2-15*B*a*b^2 *d^2-20*B*b^3*c*d+3*C*a^2*b*d^2+24*C*a*b^2*c*d+8*C*b^3*c^2+D*a^3*d^2-12*D* a^2*b*c*d-24*D*a*b^2*c^2)/b/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a *d-b*c)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1938 vs. \(2 (415) = 830\).
Time = 0.51 (sec) , antiderivative size = 3889, normalized size of antiderivative = 8.88 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
[1/24*(3*(((D*a^3*b^2 + 3*C*a^2*b^3 - 15*B*a*b^4 + 35*A*b^5)*d^5 - 4*(3*D* a^2*b^3*c - (6*C*a*b^4 - 5*B*b^5)*c)*d^4 - 8*(3*D*a*b^4*c^2 - C*b^5*c^2)*d ^3)*x^4 + (D*a^5*c^2 + (3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^3)*c^2)*d^3 + 2*((D*a^4*b + 3*C*a^3*b^2 - 15*B*a^2*b^3 + 35*A*a*b^4)*d^5 - (11*D*a^3*b ^2*c - (27*C*a^2*b^3 - 35*B*a*b^4 + 35*A*b^5)*c)*d^4 - 4*(9*D*a^2*b^3*c^2 - (8*C*a*b^4 - 5*B*b^5)*c^2)*d^3 - 8*(3*D*a*b^4*c^3 - C*b^5*c^3)*d^2)*x^3 - 4*(3*D*a^4*b*c^3 - (6*C*a^3*b^2 - 5*B*a^2*b^3)*c^3)*d^2 + ((D*a^5 + 3*C* a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^3)*d^5 - 4*(2*D*a^4*b*c - (9*C*a^3*b^2 - 20*B*a^2*b^3 + 35*A*a*b^4)*c)*d^4 - (71*D*a^3*b^2*c^2 - (107*C*a^2*b^3 - 95*B*a*b^4 + 35*A*b^5)*c^2)*d^3 - 4*(27*D*a^2*b^3*c^3 - (14*C*a*b^4 - 5*B* b^5)*c^3)*d^2 - 8*(3*D*a*b^4*c^4 - C*b^5*c^4)*d)*x^2 - 8*(3*D*a^3*b^2*c^4 - C*a^2*b^3*c^4)*d + 2*((D*a^5*c + (3*C*a^4*b - 15*B*a^3*b^2 + 35*A*a^2*b^ 3)*c)*d^4 - (11*D*a^4*b*c^2 - (27*C*a^3*b^2 - 35*B*a^2*b^3 + 35*A*a*b^4)*c ^2)*d^3 - 4*(9*D*a^3*b^2*c^3 - (8*C*a^2*b^3 - 5*B*a*b^4)*c^3)*d^2 - 8*(3*D *a^2*b^3*c^4 - C*a*b^4*c^4)*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*(8*D*a^2*b^4*c^ 5 - 8*A*a^4*b^2*d^5 - 8*(2*B*a^4*b^2 - 11*A*a^3*b^3)*c*d^4 - (3*D*a^5*b*c^ 2 - (55*C*a^4*b^2 - 67*B*a^3*b^3 - 41*A*a^2*b^4)*c^2)*d^3 + 3*((D*a^4*b^2 + 3*C*a^3*b^3 - 15*B*a^2*b^4 + 35*A*a*b^5)*d^5 - (13*D*a^3*b^3*c - (21*C*a ^2*b^4 - 5*B*a*b^5 - 35*A*b^6)*c)*d^4 - 4*(3*D*a^2*b^4*c^2 + (4*C*a*b^5...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.33 (sec) , antiderivative size = 767, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=-\frac {{\left (24 \, D a b^{2} c^{2} - 8 \, C b^{3} c^{2} + 12 \, D a^{2} b c d - 24 \, C a b^{2} c d + 20 \, B b^{3} c d - D a^{3} d^{2} - 3 \, C a^{2} b d^{2} + 15 \, B a b^{2} d^{2} - 35 \, A b^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (D b c^{4} + 9 \, {\left (d x + c\right )} D a c^{2} d - 3 \, {\left (d x + c\right )} C b c^{2} d - D a c^{3} d - C b c^{3} d - 6 \, {\left (d x + c\right )} C a c d^{2} + 6 \, {\left (d x + c\right )} B b c d^{2} + C a c^{2} d^{2} + B b c^{2} d^{2} + 3 \, {\left (d x + c\right )} B a d^{3} - 9 \, {\left (d x + c\right )} A b d^{3} - B a c d^{3} - A b c d^{3} + A a d^{4}\right )}}{3 \, {\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} - \frac {12 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} b^{2} c d - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b^{3} c d + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{4} c d - 12 \, \sqrt {d x + c} D a^{2} b^{2} c^{2} d + 8 \, \sqrt {d x + c} C a b^{3} c^{2} d - 4 \, \sqrt {d x + c} B b^{4} c^{2} d - {\left (d x + c\right )}^{\frac {3}{2}} D a^{3} b d^{2} - 3 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{2} b^{2} d^{2} + 7 \, {\left (d x + c\right )}^{\frac {3}{2}} B a b^{3} d^{2} - 11 \, {\left (d x + c\right )}^{\frac {3}{2}} A b^{4} d^{2} + 11 \, \sqrt {d x + c} D a^{3} b c d^{2} - 3 \, \sqrt {d x + c} C a^{2} b^{2} c d^{2} - 5 \, \sqrt {d x + c} B a b^{3} c d^{2} + 13 \, \sqrt {d x + c} A b^{4} c d^{2} + \sqrt {d x + c} D a^{4} d^{3} - 5 \, \sqrt {d x + c} C a^{3} b d^{3} + 9 \, \sqrt {d x + c} B a^{2} b^{2} d^{3} - 13 \, \sqrt {d x + c} A a b^{3} d^{3}}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
-1/4*(24*D*a*b^2*c^2 - 8*C*b^3*c^2 + 12*D*a^2*b*c*d - 24*C*a*b^2*c*d + 20* B*b^3*c*d - D*a^3*d^2 - 3*C*a^2*b*d^2 + 15*B*a*b^2*d^2 - 35*A*b^3*d^2)*arc tan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^ 2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*sqrt(-b^2*c + a*b*d)) - 2/3*( D*b*c^4 + 9*(d*x + c)*D*a*c^2*d - 3*(d*x + c)*C*b*c^2*d - D*a*c^3*d - C*b* c^3*d - 6*(d*x + c)*C*a*c*d^2 + 6*(d*x + c)*B*b*c*d^2 + C*a*c^2*d^2 + B*b* c^2*d^2 + 3*(d*x + c)*B*a*d^3 - 9*(d*x + c)*A*b*d^3 - B*a*c*d^3 - A*b*c*d^ 3 + A*a*d^4)/((b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c *d^4 + a^4*d^5)*(d*x + c)^(3/2)) - 1/4*(12*(d*x + c)^(3/2)*D*a^2*b^2*c*d - 8*(d*x + c)^(3/2)*C*a*b^3*c*d + 4*(d*x + c)^(3/2)*B*b^4*c*d - 12*sqrt(d*x + c)*D*a^2*b^2*c^2*d + 8*sqrt(d*x + c)*C*a*b^3*c^2*d - 4*sqrt(d*x + c)*B* b^4*c^2*d - (d*x + c)^(3/2)*D*a^3*b*d^2 - 3*(d*x + c)^(3/2)*C*a^2*b^2*d^2 + 7*(d*x + c)^(3/2)*B*a*b^3*d^2 - 11*(d*x + c)^(3/2)*A*b^4*d^2 + 11*sqrt(d *x + c)*D*a^3*b*c*d^2 - 3*sqrt(d*x + c)*C*a^2*b^2*c*d^2 - 5*sqrt(d*x + c)* B*a*b^3*c*d^2 + 13*sqrt(d*x + c)*A*b^4*c*d^2 + sqrt(d*x + c)*D*a^4*d^3 - 5 *sqrt(d*x + c)*C*a^3*b*d^3 + 9*sqrt(d*x + c)*B*a^2*b^2*d^3 - 13*sqrt(d*x + c)*A*a*b^3*d^3)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2 *c*d^3 + a^4*b*d^4)*((d*x + c)*b - b*c + a*d)^2)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2}} \,d x \]