Integrand size = 32, antiderivative size = 294 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {\left (b^3 (2 B c+3 A d)-a b^2 (4 c C+5 B d)-9 a^3 d D+a^2 b (7 C d+6 c D)\right ) \sqrt {c+d x}}{b^5}+\frac {2 \left (b^2 B-2 a b C+3 a^2 D\right ) (c+d x)^{3/2}}{3 b^4}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{3/2}}{b^4 (a+b x)}+\frac {2 (b C d-b c D-2 a d D) (c+d x)^{5/2}}{5 b^3 d^2}+\frac {2 D (c+d x)^{7/2}}{7 b^2 d^2}-\frac {\sqrt {b c-a d} \left (b^3 (2 B c+3 A d)-a b^2 (4 c C+5 B d)-9 a^3 d D+a^2 b (7 C d+6 c D)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}} \] Output:
(b^3*(3*A*d+2*B*c)-a*b^2*(5*B*d+4*C*c)-9*a^3*d*D+a^2*b*(7*C*d+6*D*c))*(d*x +c)^(1/2)/b^5+2/3*(B*b^2-2*C*a*b+3*D*a^2)*(d*x+c)^(3/2)/b^4-(A*b^3-a*(B*b^ 2-C*a*b+D*a^2))*(d*x+c)^(3/2)/b^4/(b*x+a)+2/5*(C*b*d-2*D*a*d-D*b*c)*(d*x+c )^(5/2)/b^3/d^2+2/7*D*(d*x+c)^(7/2)/b^2/d^2-(-a*d+b*c)^(1/2)*(b^3*(3*A*d+2 *B*c)-a*b^2*(5*B*d+4*C*c)-9*a^3*d*D+a^2*b*(7*C*d+6*D*c))*arctanh(b^(1/2)*( d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(11/2)
Time = 0.68 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.20 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {\sqrt {c+d x} \left (-945 a^4 d^3 D+105 a^3 b d^2 (7 C d+9 c D-6 d D x)-a b^3 \left (12 c^3 D+c^2 (-42 C d+78 d D x)+c d^2 \left (-385 B+476 C x+120 D x^2\right )+d^3 \left (-315 A+350 B x+98 C x^2+54 D x^3\right )\right )-7 a^2 b^2 d \left (12 c^2 D+c d (95 C-96 D x)+d^2 (75 B-2 x (35 C+9 D x))\right )+b^4 \left (-105 A d^2 (c-2 d x)+2 x \left (-6 c^3 D+3 c^2 d (7 C+D x)+2 c d^2 (70 B+3 x (7 C+4 D x))+d^3 x (35 B+3 x (7 C+5 D x))\right )\right )\right )}{105 b^5 d^2 (a+b x)}-\frac {\sqrt {-b c+a d} \left (b^3 (2 B c+3 A d)-a b^2 (4 c C+5 B d)-9 a^3 d D+a^2 b (7 C d+6 c D)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{11/2}} \] Input:
Integrate[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]
Output:
(Sqrt[c + d*x]*(-945*a^4*d^3*D + 105*a^3*b*d^2*(7*C*d + 9*c*D - 6*d*D*x) - a*b^3*(12*c^3*D + c^2*(-42*C*d + 78*d*D*x) + c*d^2*(-385*B + 476*C*x + 12 0*D*x^2) + d^3*(-315*A + 350*B*x + 98*C*x^2 + 54*D*x^3)) - 7*a^2*b^2*d*(12 *c^2*D + c*d*(95*C - 96*D*x) + d^2*(75*B - 2*x*(35*C + 9*D*x))) + b^4*(-10 5*A*d^2*(c - 2*d*x) + 2*x*(-6*c^3*D + 3*c^2*d*(7*C + D*x) + 2*c*d^2*(70*B + 3*x*(7*C + 4*D*x)) + d^3*x*(35*B + 3*x*(7*C + 5*D*x))))))/(105*b^5*d^2*( a + b*x)) - (Sqrt[-(b*c) + a*d]*(b^3*(2*B*c + 3*A*d) - a*b^2*(4*c*C + 5*B* d) - 9*a^3*d*D + a^2*b*(7*C*d + 6*c*D))*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqr t[-(b*c) + a*d]])/b^(11/2)
Time = 1.01 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2124, 27, 1192, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {(c+d x)^{3/2} \left (2 \left (c-\frac {a d}{b}\right ) D x^2+\frac {2 (b c-a d) (b C-a D) x}{b^2}+\frac {-5 d D a^3+b (5 C d+2 c D) a^2-b^2 (2 c C+5 B d) a+b^3 (2 B c+3 A d)}{b^3}\right )}{2 (a+b x)}dx}{b c-a d}-\frac {(c+d x)^{5/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{3/2} \left (-\frac {5 d D a^3}{b^3}+\frac {(5 C d+2 c D) a^2}{b^2}-\frac {(2 c C+5 B d) a}{b}+2 \left (c-\frac {a d}{b}\right ) D x^2+2 B c+3 A d+\frac {2 (b c-a d) (b C-a D) x}{b^2}\right )}{a+b x}dx}{2 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (-2 D c^3+2 C d c^2-2 B d^2 c-2 \left (c-\frac {a d}{b}\right ) D (c+d x)^2-d^3 \left (3 A-\frac {5 a \left (D a^2-b C a+b^2 B\right )}{b^3}\right )-\frac {2 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}\right )}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\int \left (\frac {2 (b c-a d) D (c+d x)^3}{b^2}+\frac {2 (b c-a d) (b C d-2 a D d-b c D) (c+d x)^2}{b^3}+\frac {d^2 \left (-9 d D a^3+b (7 C d+6 c D) a^2-b^2 (4 c C+5 B d) a+b^3 (2 B c+3 A d)\right ) (c+d x)}{b^4}+\frac {d^2 (b c-a d) \left (-9 d D a^3+b (7 C d+6 c D) a^2-b^2 (4 c C+5 B d) a+b^3 (2 B c+3 A d)\right )}{b^5}+\frac {-3 A c^2 d^3 b^5-2 B c^3 d^2 b^5+6 a A c d^4 b^4+9 a B c^2 d^3 b^4+4 a c^3 C d^2 b^4-3 a^2 A d^5 b^3-12 a^2 B c d^4 b^3-15 a^2 c^2 C d^3 b^3-6 a^2 c^3 d^2 D b^3+5 a^3 B d^5 b^2+18 a^3 c C d^4 b^2+21 a^3 c^2 d^3 D b^2-7 a^4 C d^5 b-24 a^4 c d^4 D b+9 a^5 d^5 D}{b^5 (b c-a d-b (c+d x))}\right )d\sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {d^2 (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (-9 a^3 d D+a^2 b (6 c D+7 C d)-a b^2 (5 B d+4 c C)+b^3 (3 A d+2 B c)\right )}{b^{11/2}}+\frac {d^2 \sqrt {c+d x} (b c-a d) \left (-9 a^3 d D+a^2 b (6 c D+7 C d)-a b^2 (5 B d+4 c C)+b^3 (3 A d+2 B c)\right )}{b^5}+\frac {d^2 (c+d x)^{3/2} \left (-9 a^3 d D+a^2 b (6 c D+7 C d)-a b^2 (5 B d+4 c C)+b^3 (3 A d+2 B c)\right )}{3 b^4}+\frac {2 (c+d x)^{5/2} (b c-a d) (-2 a d D-b c D+b C d)}{5 b^3}+\frac {2 D (c+d x)^{7/2} (b c-a d)}{7 b^2}}{d^2 (b c-a d)}-\frac {(c+d x)^{5/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
Input:
Int[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]
Output:
-(((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*(c + d*x)^(5/2))/((b*c - a*d)*(a + b*x))) + ((d^2*(b*c - a*d)*(b^3*(2*B*c + 3*A*d) - a*b^2*(4*c*C + 5*B*d) - 9*a^3*d*D + a^2*b*(7*C*d + 6*c*D))*Sqrt[c + d*x])/b^5 + (d^2*(b^3*(2*B*c + 3*A*d) - a*b^2*(4*c*C + 5*B*d) - 9*a^3*d*D + a^2*b*(7*C*d + 6*c*D))*(c + d*x)^(3/2))/(3*b^4) + (2*(b*c - a*d)*(b*C*d - b*c*D - 2*a*d*D)*(c + d*x) ^(5/2))/(5*b^3) + (2*(b*c - a*d)*D*(c + d*x)^(7/2))/(7*b^2) - (d^2*(b*c - a*d)^(3/2)*(b^3*(2*B*c + 3*A*d) - a*b^2*(4*c*C + 5*B*d) - 9*a^3*d*D + a^2* b*(7*C*d + 6*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(11 /2))/(d^2*(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[((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 = 0.81 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(\frac {-3 \left (\left (A d +\frac {2 B c}{3}\right ) b^{3}-\frac {5 \left (B d +\frac {4 C c}{5}\right ) a \,b^{2}}{3}+a^{2} \left (\frac {7 C d}{3}+2 D c \right ) b -3 a^{3} d D\right ) \left (a d -b c \right ) d^{2} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+3 \sqrt {\left (a d -b c \right ) b}\, \left (\frac {\left (2 x \left (\frac {1}{5} C \,x^{2}+\frac {1}{3} B x +\frac {1}{7} D x^{3}+A \right ) d^{3}-\left (-\frac {16}{35} D x^{3}-\frac {4}{5} C \,x^{2}-\frac {8}{3} B x +A \right ) c \,d^{2}+\frac {2 x \left (\frac {D x}{7}+C \right ) c^{2} d}{5}-\frac {4 D c^{3} x}{35}\right ) b^{4}}{3}+a \left (\left (-\frac {6}{35} D x^{3}-\frac {14}{45} C \,x^{2}-\frac {10}{9} B x +A \right ) d^{3}+\frac {11 \left (-\frac {24}{77} D x^{2}-\frac {68}{55} C x +B \right ) c \,d^{2}}{9}+\frac {2 \left (-\frac {13 D x}{7}+C \right ) c^{2} d}{15}-\frac {4 D c^{3}}{105}\right ) b^{3}-\frac {5 \left (\left (-\frac {6}{25} D x^{2}-\frac {14}{15} C x +B \right ) d^{2}+\frac {19 \left (-\frac {96 D x}{95}+C \right ) c d}{15}+\frac {4 D c^{2}}{25}\right ) d \,a^{2} b^{2}}{3}+\frac {7 \left (\left (-\frac {6 D x}{7}+C \right ) d +\frac {9 D c}{7}\right ) d^{2} a^{3} b}{3}-3 D a^{4} d^{3}\right ) \sqrt {x d +c}}{d^{2} b^{5} \left (b x +a \right ) \sqrt {\left (a d -b c \right ) b}}\) | \(349\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {D \left (x d +c \right )^{\frac {7}{2}} b^{3}}{7}+\frac {C \,b^{3} d \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {2 D a \,b^{2} d \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {D b^{3} c \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {B \,b^{3} d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {2 C a \,b^{2} d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}+D a^{2} b \,d^{2} \left (x d +c \right )^{\frac {3}{2}}+A \,b^{3} d^{3} \sqrt {x d +c}-2 B a \,b^{2} d^{3} \sqrt {x d +c}+B \,b^{3} c \,d^{2} \sqrt {x d +c}+3 C \,a^{2} b \,d^{3} \sqrt {x d +c}-2 C a \,b^{2} c \,d^{2} \sqrt {x d +c}-4 D a^{3} d^{3} \sqrt {x d +c}+3 D a^{2} b c \,d^{2} \sqrt {x d +c}\right )}{b^{5}}-\frac {2 d^{2} \left (\frac {\left (-\frac {1}{2} A a \,b^{3} d^{2}+\frac {1}{2} A \,b^{4} c d +\frac {1}{2} B \,a^{2} b^{2} d^{2}-\frac {1}{2} B a \,b^{3} c d -\frac {1}{2} C \,a^{3} b \,d^{2}+\frac {1}{2} C \,a^{2} b^{2} c d +\frac {1}{2} D a^{4} d^{2}-\frac {1}{2} D a^{3} b c d \right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {\left (3 A a \,b^{3} d^{2}-3 A \,b^{4} c d -5 B \,a^{2} b^{2} d^{2}+7 B a \,b^{3} c d -2 B \,b^{4} c^{2}+7 C \,a^{3} b \,d^{2}-11 C \,a^{2} b^{2} c d +4 C a \,b^{3} c^{2}-9 D a^{4} d^{2}+15 D a^{3} b c d -6 D a^{2} b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{5}}}{d^{2}}\) | \(493\) |
| default | \(\frac {\frac {2 \left (\frac {D \left (x d +c \right )^{\frac {7}{2}} b^{3}}{7}+\frac {C \,b^{3} d \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {2 D a \,b^{2} d \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {D b^{3} c \left (x d +c \right )^{\frac {5}{2}}}{5}+\frac {B \,b^{3} d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {2 C a \,b^{2} d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}+D a^{2} b \,d^{2} \left (x d +c \right )^{\frac {3}{2}}+A \,b^{3} d^{3} \sqrt {x d +c}-2 B a \,b^{2} d^{3} \sqrt {x d +c}+B \,b^{3} c \,d^{2} \sqrt {x d +c}+3 C \,a^{2} b \,d^{3} \sqrt {x d +c}-2 C a \,b^{2} c \,d^{2} \sqrt {x d +c}-4 D a^{3} d^{3} \sqrt {x d +c}+3 D a^{2} b c \,d^{2} \sqrt {x d +c}\right )}{b^{5}}-\frac {2 d^{2} \left (\frac {\left (-\frac {1}{2} A a \,b^{3} d^{2}+\frac {1}{2} A \,b^{4} c d +\frac {1}{2} B \,a^{2} b^{2} d^{2}-\frac {1}{2} B a \,b^{3} c d -\frac {1}{2} C \,a^{3} b \,d^{2}+\frac {1}{2} C \,a^{2} b^{2} c d +\frac {1}{2} D a^{4} d^{2}-\frac {1}{2} D a^{3} b c d \right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {\left (3 A a \,b^{3} d^{2}-3 A \,b^{4} c d -5 B \,a^{2} b^{2} d^{2}+7 B a \,b^{3} c d -2 B \,b^{4} c^{2}+7 C \,a^{3} b \,d^{2}-11 C \,a^{2} b^{2} c d +4 C a \,b^{3} c^{2}-9 D a^{4} d^{2}+15 D a^{3} b c d -6 D a^{2} b^{2} c^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{5}}}{d^{2}}\) | \(493\) |
Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
3/((a*d-b*c)*b)^(1/2)*(-((A*d+2/3*B*c)*b^3-5/3*(B*d+4/5*C*c)*a*b^2+a^2*(7/ 3*C*d+2*D*c)*b-3*a^3*d*D)*(a*d-b*c)*d^2*(b*x+a)*arctan(b*(d*x+c)^(1/2)/((a *d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*(1/3*(2*x*(1/5*C*x^2+1/3*B*x+1/7*D*x ^3+A)*d^3-(-16/35*D*x^3-4/5*C*x^2-8/3*B*x+A)*c*d^2+2/5*x*(1/7*D*x+C)*c^2*d -4/35*D*c^3*x)*b^4+a*((-6/35*D*x^3-14/45*C*x^2-10/9*B*x+A)*d^3+11/9*(-24/7 7*D*x^2-68/55*C*x+B)*c*d^2+2/15*(-13/7*D*x+C)*c^2*d-4/105*D*c^3)*b^3-5/3*( (-6/25*D*x^2-14/15*C*x+B)*d^2+19/15*(-96/95*D*x+C)*c*d+4/25*D*c^2)*d*a^2*b ^2+7/3*((-6/7*D*x+C)*d+9/7*D*c)*d^2*a^3*b-3*D*a^4*d^3)*(d*x+c)^(1/2))/d^2/ b^5/(b*x+a)
Time = 0.10 (sec) , antiderivative size = 1066, normalized size of antiderivative = 3.63 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="fricas ")
Output:
[1/210*(105*(2*(3*D*a^3*b - 2*C*a^2*b^2 + B*a*b^3)*c*d^2 - (9*D*a^4 - 7*C* a^3*b + 5*B*a^2*b^2 - 3*A*a*b^3)*d^3 + (2*(3*D*a^2*b^2 - 2*C*a*b^3 + B*b^4 )*c*d^2 - (9*D*a^3*b - 7*C*a^2*b^2 + 5*B*a*b^3 - 3*A*b^4)*d^3)*x)*sqrt((b* c - a*d)/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/ b))/(b*x + a)) + 2*(30*D*b^4*d^3*x^4 - 12*D*a*b^3*c^3 - 42*(2*D*a^2*b^2 - C*a*b^3)*c^2*d + 35*(27*D*a^3*b - 19*C*a^2*b^2 + 11*B*a*b^3 - 3*A*b^4)*c*d ^2 - 105*(9*D*a^4 - 7*C*a^3*b + 5*B*a^2*b^2 - 3*A*a*b^3)*d^3 + 6*(8*D*b^4* c*d^2 - (9*D*a*b^3 - 7*C*b^4)*d^3)*x^3 + 2*(3*D*b^4*c^2*d - 6*(10*D*a*b^3 - 7*C*b^4)*c*d^2 + 7*(9*D*a^2*b^2 - 7*C*a*b^3 + 5*B*b^4)*d^3)*x^2 - 2*(6*D *b^4*c^3 + 3*(13*D*a*b^3 - 7*C*b^4)*c^2*d - 14*(24*D*a^2*b^2 - 17*C*a*b^3 + 10*B*b^4)*c*d^2 + 35*(9*D*a^3*b - 7*C*a^2*b^2 + 5*B*a*b^3 - 3*A*b^4)*d^3 )*x)*sqrt(d*x + c))/(b^6*d^2*x + a*b^5*d^2), -1/105*(105*(2*(3*D*a^3*b - 2 *C*a^2*b^2 + B*a*b^3)*c*d^2 - (9*D*a^4 - 7*C*a^3*b + 5*B*a^2*b^2 - 3*A*a*b ^3)*d^3 + (2*(3*D*a^2*b^2 - 2*C*a*b^3 + B*b^4)*c*d^2 - (9*D*a^3*b - 7*C*a^ 2*b^2 + 5*B*a*b^3 - 3*A*b^4)*d^3)*x)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) - (30*D*b^4*d^3*x^4 - 12*D*a*b^3 *c^3 - 42*(2*D*a^2*b^2 - C*a*b^3)*c^2*d + 35*(27*D*a^3*b - 19*C*a^2*b^2 + 11*B*a*b^3 - 3*A*b^4)*c*d^2 - 105*(9*D*a^4 - 7*C*a^3*b + 5*B*a^2*b^2 - 3*A *a*b^3)*d^3 + 6*(8*D*b^4*c*d^2 - (9*D*a*b^3 - 7*C*b^4)*d^3)*x^3 + 2*(3*D*b ^4*c^2*d - 6*(10*D*a*b^3 - 7*C*b^4)*c*d^2 + 7*(9*D*a^2*b^2 - 7*C*a*b^3 ...
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(3/2)*(D*x**3+C*x**2+B*x+A)/(b*x+a)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,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. 556 vs. \(2 (270) = 540\).
Time = 0.14 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.89 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {{\left (6 \, D a^{2} b^{2} c^{2} - 4 \, C a b^{3} c^{2} + 2 \, B b^{4} c^{2} - 15 \, D a^{3} b c d + 11 \, C a^{2} b^{2} c d - 7 \, B a b^{3} c d + 3 \, A b^{4} c d + 9 \, D a^{4} d^{2} - 7 \, C a^{3} b d^{2} + 5 \, B a^{2} b^{2} d^{2} - 3 \, A a b^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{5}} + \frac {\sqrt {d x + c} D a^{3} b c d - \sqrt {d x + c} C a^{2} b^{2} c d + \sqrt {d x + c} B a b^{3} c d - \sqrt {d x + c} A b^{4} c d - \sqrt {d x + c} D a^{4} d^{2} + \sqrt {d x + c} C a^{3} b d^{2} - \sqrt {d x + c} B a^{2} b^{2} d^{2} + \sqrt {d x + c} A a b^{3} d^{2}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{12} d^{12} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} D b^{12} c d^{12} - 42 \, {\left (d x + c\right )}^{\frac {5}{2}} D a b^{11} d^{13} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} C b^{12} d^{13} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} b^{10} d^{14} - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b^{11} d^{14} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{12} d^{14} + 315 \, \sqrt {d x + c} D a^{2} b^{10} c d^{14} - 210 \, \sqrt {d x + c} C a b^{11} c d^{14} + 105 \, \sqrt {d x + c} B b^{12} c d^{14} - 420 \, \sqrt {d x + c} D a^{3} b^{9} d^{15} + 315 \, \sqrt {d x + c} C a^{2} b^{10} d^{15} - 210 \, \sqrt {d x + c} B a b^{11} d^{15} + 105 \, \sqrt {d x + c} A b^{12} d^{15}\right )}}{105 \, b^{14} d^{14}} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="giac")
Output:
(6*D*a^2*b^2*c^2 - 4*C*a*b^3*c^2 + 2*B*b^4*c^2 - 15*D*a^3*b*c*d + 11*C*a^2 *b^2*c*d - 7*B*a*b^3*c*d + 3*A*b^4*c*d + 9*D*a^4*d^2 - 7*C*a^3*b*d^2 + 5*B *a^2*b^2*d^2 - 3*A*a*b^3*d^2)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d)) /(sqrt(-b^2*c + a*b*d)*b^5) + (sqrt(d*x + c)*D*a^3*b*c*d - sqrt(d*x + c)*C *a^2*b^2*c*d + sqrt(d*x + c)*B*a*b^3*c*d - sqrt(d*x + c)*A*b^4*c*d - sqrt( d*x + c)*D*a^4*d^2 + sqrt(d*x + c)*C*a^3*b*d^2 - sqrt(d*x + c)*B*a^2*b^2*d ^2 + sqrt(d*x + c)*A*a*b^3*d^2)/(((d*x + c)*b - b*c + a*d)*b^5) + 2/105*(1 5*(d*x + c)^(7/2)*D*b^12*d^12 - 21*(d*x + c)^(5/2)*D*b^12*c*d^12 - 42*(d*x + c)^(5/2)*D*a*b^11*d^13 + 21*(d*x + c)^(5/2)*C*b^12*d^13 + 105*(d*x + c) ^(3/2)*D*a^2*b^10*d^14 - 70*(d*x + c)^(3/2)*C*a*b^11*d^14 + 35*(d*x + c)^( 3/2)*B*b^12*d^14 + 315*sqrt(d*x + c)*D*a^2*b^10*c*d^14 - 210*sqrt(d*x + c) *C*a*b^11*c*d^14 + 105*sqrt(d*x + c)*B*b^12*c*d^14 - 420*sqrt(d*x + c)*D*a ^3*b^9*d^15 + 315*sqrt(d*x + c)*C*a^2*b^10*d^15 - 210*sqrt(d*x + c)*B*a*b^ 11*d^15 + 105*sqrt(d*x + c)*A*b^12*d^15)/(b^14*d^14)
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^2} \,d x \] Input:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^2,x)
Output:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^2, x)
Time = 0.14 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.69 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x)
Output:
(945*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b* c)))*a**4*d**3 - 1365*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt (b)*sqrt(a*d - b*c)))*a**3*b*c*d**2 + 945*sqrt(b)*sqrt(a*d - b*c)*atan((sq rt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b*d**3*x + 210*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* *2 + 420*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**2*c**2*d - 1365*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d* x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**2*c*d**2*x - 210*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 + 210 *sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c))) *a*b**4*d**2*x + 420*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt( b)*sqrt(a*d - b*c)))*a*b**3*c**2*d*x - 210*sqrt(b)*sqrt(a*d - b*c)*atan((s qrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**5*c*d*x - 945*sqrt(c + d*x)* a**4*b*d**3 + 1680*sqrt(c + d*x)*a**3*b**2*c*d**2 - 630*sqrt(c + d*x)*a**3 *b**2*d**3*x - 210*sqrt(c + d*x)*a**2*b**4*d**2 - 749*sqrt(c + d*x)*a**2*b **3*c**2*d + 1162*sqrt(c + d*x)*a**2*b**3*c*d**2*x + 126*sqrt(c + d*x)*a** 2*b**3*d**3*x**2 + 280*sqrt(c + d*x)*a*b**5*c*d - 140*sqrt(c + d*x)*a*b**5 *d**2*x + 30*sqrt(c + d*x)*a*b**4*c**3 - 554*sqrt(c + d*x)*a*b**4*c**2*d*x - 218*sqrt(c + d*x)*a*b**4*c*d**2*x**2 - 54*sqrt(c + d*x)*a*b**4*d**3*x** 3 + 280*sqrt(c + d*x)*b**6*c*d*x + 70*sqrt(c + d*x)*b**6*d**2*x**2 + 30...