Integrand size = 32, antiderivative size = 329 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {2 (b c-a d)^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {c+d x}}{b^6}+\frac {2 (b c-a d) \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{3/2}}{3 b^5}+\frac {2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{5/2}}{5 b^4}+\frac {2 \left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) (c+d x)^{7/2}}{7 b^3 d^3}+\frac {2 (b C d-2 b c D-a d D) (c+d x)^{9/2}}{9 b^2 d^3}+\frac {2 D (c+d x)^{11/2}}{11 b d^3}-\frac {2 (b c-a d)^{5/2} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \] Output:
2*(-a*d+b*c)^2*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(1/2)/b^6+2/3*(-a*d+b *c)*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(3/2)/b^5+2/5*(A*b^3-a*(B*b^2-C* a*b+D*a^2))*(d*x+c)^(5/2)/b^4+2/7*(a^2*d^2*D-a*b*d*(C*d-D*c)-b^2*(-B*d^2+C *c*d-D*c^2))*(d*x+c)^(7/2)/b^3/d^3+2/9*(C*b*d-D*a*d-2*D*b*c)*(d*x+c)^(9/2) /b^2/d^3+2/11*D*(d*x+c)^(11/2)/b/d^3-2*(-a*d+b*c)^(5/2)*(A*b^3-a*(B*b^2-C* a*b+D*a^2))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(13/2)
Time = 0.63 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {2 \sqrt {c+d x} \left (-3465 a^5 d^5 D+1155 a^4 b d^4 (3 C d+7 c D+d D x)-231 a^3 b^2 d^3 \left (23 c^2 D+c d (35 C+11 D x)+d^2 (15 B+x (5 C+3 D x))\right )+b^5 \left (40 c^5 D-10 c^4 d (11 C+2 D x)+5 c^3 d^2 (99 B+x (11 C+3 D x))+d^5 x^2 \left (693 A+5 x \left (99 B+77 C x+63 D x^2\right )\right )+c^2 d^3 \left (5313 A+5 x \left (297 B+165 C x+113 D x^2\right )\right )+c d^4 x \left (2541 A+5 x \left (297 B+209 C x+161 D x^2\right )\right )\right )+33 a^2 b^3 d^2 \left (15 c^3 D+c^2 d (161 C+45 D x)+c d^2 (245 B+x (77 C+45 D x))+d^3 (105 A+x (35 B+3 x (7 C+5 D x)))\right )-11 a b^4 d \left (-10 c^4 D+5 c^3 d (9 C+D x)+3 c^2 d^2 (161 B+5 x (9 C+5 D x))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))+c d^3 (735 A+x (231 B+5 x (27 C+19 D x)))\right )\right )}{3465 b^6 d^3}-\frac {2 (-b c+a d)^{5/2} \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \] Input:
Integrate[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]
Output:
(2*Sqrt[c + d*x]*(-3465*a^5*d^5*D + 1155*a^4*b*d^4*(3*C*d + 7*c*D + d*D*x) - 231*a^3*b^2*d^3*(23*c^2*D + c*d*(35*C + 11*D*x) + d^2*(15*B + x*(5*C + 3*D*x))) + b^5*(40*c^5*D - 10*c^4*d*(11*C + 2*D*x) + 5*c^3*d^2*(99*B + x*( 11*C + 3*D*x)) + d^5*x^2*(693*A + 5*x*(99*B + 77*C*x + 63*D*x^2)) + c^2*d^ 3*(5313*A + 5*x*(297*B + 165*C*x + 113*D*x^2)) + c*d^4*x*(2541*A + 5*x*(29 7*B + 209*C*x + 161*D*x^2))) + 33*a^2*b^3*d^2*(15*c^3*D + c^2*d*(161*C + 4 5*D*x) + c*d^2*(245*B + x*(77*C + 45*D*x)) + d^3*(105*A + x*(35*B + 3*x*(7 *C + 5*D*x)))) - 11*a*b^4*d*(-10*c^4*D + 5*c^3*d*(9*C + D*x) + 3*c^2*d^2*( 161*B + 5*x*(9*C + 5*D*x)) + d^4*x*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) + c*d^3*(735*A + x*(231*B + 5*x*(27*C + 19*D*x))))))/(3465*b^6*d^3) - (2*( -(b*c) + a*d)^(5/2)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*ArcTan[(Sqrt[b]*Sq rt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(13/2)
Time = 0.63 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^3 (a+b x)}+\frac {(c+d x)^{5/2} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^2}+\frac {(c+d x)^{7/2} (-a d D-2 b c D+b C d)}{b^2 d^2}+\frac {D (c+d x)^{9/2}}{b d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (b c-a d)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}+\frac {2 \sqrt {c+d x} (b c-a d)^2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^6}+\frac {2 (c+d x)^{3/2} (b c-a d) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{3 b^5}+\frac {2 (c+d x)^{5/2} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{5 b^4}+\frac {2 (c+d x)^{7/2} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{7 b^3 d^3}+\frac {2 (c+d x)^{9/2} (-a d D-2 b c D+b C d)}{9 b^2 d^3}+\frac {2 D (c+d x)^{11/2}}{11 b d^3}\) |
Input:
Int[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x),x]
Output:
(2*(b*c - a*d)^2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Sqrt[c + d*x])/b^6 + (2*(b*c - a*d)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(3/2))/(3*b^5 ) + (2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(c + d*x)^(5/2))/(5*b^4) + (2*( a^2*d^2*D - a*b*d*(C*d - c*D) - b^2*(c*C*d - B*d^2 - c^2*D))*(c + d*x)^(7/ 2))/(7*b^3*d^3) + (2*(b*C*d - 2*b*c*D - a*d*D)*(c + d*x)^(9/2))/(9*b^2*d^3 ) + (2*D*(c + d*x)^(11/2))/(11*b*d^3) - (2*(b*c - a*d)^(5/2)*(A*b^3 - a*(b ^2*B - a*b*C + a^2*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b ^(13/2)
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 0.85 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.39
| method | result | size |
| pseudoelliptic | \(\frac {-2 d^{3} \left (a d -b c \right )^{3} \left (b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+2 \sqrt {\left (a d -b c \right ) b}\, \left (\left (\frac {x^{2} \left (\frac {5}{11} D x^{3}+\frac {5}{9} C \,x^{2}+\frac {5}{7} B x +A \right ) b^{5}}{5}-\frac {x \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) a \,b^{4}}{3}+a^{2} \left (\frac {1}{5} C \,x^{2}+\frac {1}{3} B x +\frac {1}{7} D x^{3}+A \right ) b^{3}-\left (\frac {1}{5} D x^{2}+\frac {1}{3} C x +B \right ) a^{3} b^{2}+a^{4} \left (\frac {D x}{3}+C \right ) b -D a^{5}\right ) d^{5}-\frac {7 \left (-\frac {11 x \left (\frac {115}{363} D x^{3}+\frac {95}{231} C \,x^{2}+\frac {45}{77} B x +A \right ) b^{4}}{35}+a \left (\frac {19}{147} D x^{3}+\frac {9}{49} C \,x^{2}+\frac {11}{35} B x +A \right ) b^{3}-\left (\frac {9}{49} D x^{2}+\frac {11}{35} C x +B \right ) a^{2} b^{2}+a^{3} \left (\frac {11 D x}{35}+C \right ) b -D a^{4}\right ) c b \,d^{4}}{3}+\frac {23 \left (\left (\frac {565}{5313} D x^{3}+\frac {25}{161} C \,x^{2}+\frac {45}{161} B x +A \right ) b^{3}-\left (\frac {25}{161} D x^{2}+\frac {45}{161} C x +B \right ) a \,b^{2}+a^{2} \left (\frac {45 D x}{161}+C \right ) b -a^{3} D\right ) c^{2} b^{2} d^{3}}{15}+\frac {\left (\left (\frac {1}{33} D x^{2}+\frac {1}{9} C x +B \right ) b^{2}-\left (\frac {D x}{9}+C \right ) a b +D a^{2}\right ) c^{3} b^{3} d^{2}}{7}-\frac {2 \left (\left (\frac {2 D x}{11}+C \right ) b -D a \right ) c^{4} b^{4} d}{63}+\frac {8 D b^{5} c^{5}}{693}\right ) \sqrt {x d +c}}{d^{3} b^{6} \sqrt {\left (a d -b c \right ) b}}\) | \(458\) |
| derivativedivides | \(\frac {\frac {2 \left (-B \,a^{3} b^{2} d^{5} \sqrt {x d +c}+\frac {C \,a^{2} b^{3} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {D a^{3} b^{2} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {B a \,b^{4} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}+A \,a^{2} b^{3} d^{5} \sqrt {x d +c}+A \,b^{5} c^{2} d^{3} \sqrt {x d +c}+\frac {D a^{2} b^{3} d^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}-B a \,b^{4} c^{2} d^{3} \sqrt {x d +c}+2 D a^{4} b c \,d^{4} \sqrt {x d +c}+\frac {B \,b^{5} d^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {D b^{5} c^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{5} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}-D a^{5} d^{5} \sqrt {x d +c}+\frac {C \,b^{5} d \left (x d +c \right )^{\frac {9}{2}}}{9}-\frac {2 D b^{5} c \left (x d +c \right )^{\frac {9}{2}}}{9}-D a^{3} b^{2} c^{2} d^{3} \sqrt {x d +c}-2 C \,a^{3} b^{2} c \,d^{4} \sqrt {x d +c}+C \,a^{2} b^{3} c^{2} d^{3} \sqrt {x d +c}-\frac {B a \,b^{4} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-2 A a \,b^{4} c \,d^{4} \sqrt {x d +c}+2 B \,a^{2} b^{3} c \,d^{4} \sqrt {x d +c}-\frac {A a \,b^{4} d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{2} b^{3} d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {C \,a^{3} b^{2} d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {D a^{4} b \,d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {C \,a^{2} b^{3} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {D a \,b^{4} d \left (x d +c \right )^{\frac {9}{2}}}{9}-\frac {D a^{3} b^{2} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}+C \,a^{4} b \,d^{5} \sqrt {x d +c}-\frac {C \,b^{5} c d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{5} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {C a \,b^{4} d^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {D a \,b^{4} c d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {D \left (x d +c \right )^{\frac {11}{2}} b^{5}}{11}\right )}{b^{6}}-\frac {2 d^{3} \left (A \,a^{3} b^{3} d^{3}-3 A \,a^{2} b^{4} c \,d^{2}+3 A a \,b^{5} c^{2} d -A \,b^{6} c^{3}-B \,a^{4} d^{3} b^{2}+3 B \,a^{3} b^{3} c \,d^{2}-3 B \,a^{2} b^{4} c^{2} d +b^{5} B \,c^{3} a +a^{5} C \,d^{3} b -3 C \,a^{4} c \,d^{2} b^{2}+3 C \,a^{3} b^{3} c^{2} d -c^{3} C \,b^{4} a^{2}-D a^{6} d^{3}+3 D a^{5} c \,d^{2} b -3 D a^{4} c^{2} d \,b^{2}+D c^{3} b^{3} a^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(836\) |
| default | \(\frac {\frac {2 \left (-B \,a^{3} b^{2} d^{5} \sqrt {x d +c}+\frac {C \,a^{2} b^{3} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {D a^{3} b^{2} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {B a \,b^{4} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}+A \,a^{2} b^{3} d^{5} \sqrt {x d +c}+A \,b^{5} c^{2} d^{3} \sqrt {x d +c}+\frac {D a^{2} b^{3} d^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}-B a \,b^{4} c^{2} d^{3} \sqrt {x d +c}+2 D a^{4} b c \,d^{4} \sqrt {x d +c}+\frac {B \,b^{5} d^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {D b^{5} c^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{5} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}-D a^{5} d^{5} \sqrt {x d +c}+\frac {C \,b^{5} d \left (x d +c \right )^{\frac {9}{2}}}{9}-\frac {2 D b^{5} c \left (x d +c \right )^{\frac {9}{2}}}{9}-D a^{3} b^{2} c^{2} d^{3} \sqrt {x d +c}-2 C \,a^{3} b^{2} c \,d^{4} \sqrt {x d +c}+C \,a^{2} b^{3} c^{2} d^{3} \sqrt {x d +c}-\frac {B a \,b^{4} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-2 A a \,b^{4} c \,d^{4} \sqrt {x d +c}+2 B \,a^{2} b^{3} c \,d^{4} \sqrt {x d +c}-\frac {A a \,b^{4} d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{2} b^{3} d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {C \,a^{3} b^{2} d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {D a^{4} b \,d^{4} \left (x d +c \right )^{\frac {3}{2}}}{3}+\frac {C \,a^{2} b^{3} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {D a \,b^{4} d \left (x d +c \right )^{\frac {9}{2}}}{9}-\frac {D a^{3} b^{2} d^{3} \left (x d +c \right )^{\frac {5}{2}}}{5}+C \,a^{4} b \,d^{5} \sqrt {x d +c}-\frac {C \,b^{5} c d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{5} c \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-\frac {C a \,b^{4} d^{2} \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {D a \,b^{4} c d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {D \left (x d +c \right )^{\frac {11}{2}} b^{5}}{11}\right )}{b^{6}}-\frac {2 d^{3} \left (A \,a^{3} b^{3} d^{3}-3 A \,a^{2} b^{4} c \,d^{2}+3 A a \,b^{5} c^{2} d -A \,b^{6} c^{3}-B \,a^{4} d^{3} b^{2}+3 B \,a^{3} b^{3} c \,d^{2}-3 B \,a^{2} b^{4} c^{2} d +b^{5} B \,c^{3} a +a^{5} C \,d^{3} b -3 C \,a^{4} c \,d^{2} b^{2}+3 C \,a^{3} b^{3} c^{2} d -c^{3} C \,b^{4} a^{2}-D a^{6} d^{3}+3 D a^{5} c \,d^{2} b -3 D a^{4} c^{2} d \,b^{2}+D c^{3} b^{3} a^{3}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{6} \sqrt {\left (a d -b c \right ) b}}}{d^{3}}\) | \(836\) |
Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
2/((a*d-b*c)*b)^(1/2)*(-d^3*(a*d-b*c)^3*(A*b^3-B*a*b^2+C*a^2*b-D*a^3)*arct an(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*((1/5*x^2*(5/1 1*D*x^3+5/9*C*x^2+5/7*B*x+A)*b^5-1/3*x*(1/3*D*x^3+3/7*C*x^2+3/5*B*x+A)*a*b ^4+a^2*(1/5*C*x^2+1/3*B*x+1/7*D*x^3+A)*b^3-(1/5*D*x^2+1/3*C*x+B)*a^3*b^2+a ^4*(1/3*D*x+C)*b-D*a^5)*d^5-7/3*(-11/35*x*(115/363*D*x^3+95/231*C*x^2+45/7 7*B*x+A)*b^4+a*(19/147*D*x^3+9/49*C*x^2+11/35*B*x+A)*b^3-(9/49*D*x^2+11/35 *C*x+B)*a^2*b^2+a^3*(11/35*D*x+C)*b-D*a^4)*c*b*d^4+23/15*((565/5313*D*x^3+ 25/161*C*x^2+45/161*B*x+A)*b^3-(25/161*D*x^2+45/161*C*x+B)*a*b^2+a^2*(45/1 61*D*x+C)*b-a^3*D)*c^2*b^2*d^3+1/7*((1/33*D*x^2+1/9*C*x+B)*b^2-(1/9*D*x+C) *a*b+D*a^2)*c^3*b^3*d^2-2/63*((2/11*D*x+C)*b-D*a)*c^4*b^4*d+8/693*D*b^5*c^ 5)*(d*x+c)^(1/2))/d^3/b^6
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (294) = 588\).
Time = 0.10 (sec) , antiderivative size = 1397, normalized size of antiderivative = 4.25 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x, algorithm="fricas")
Output:
[-1/3465*(3465*((D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 - A*b^5)*c^2*d^3 - 2*(D*a ^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*c*d^4 + (D*a^5 - C*a^4*b + B*a^3*b ^2 - A*a^2*b^3)*d^5)*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*(315*D*b^5*d^5*x^5 + 40*D* b^5*c^5 + 110*(D*a*b^4 - C*b^5)*c^4*d + 495*(D*a^2*b^3 - C*a*b^4 + B*b^5)* c^3*d^2 - 5313*(D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 - A*b^5)*c^2*d^3 + 8085*(D *a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*c*d^4 - 3465*(D*a^5 - C*a^4*b + B*a^3*b^2 - A*a^2*b^3)*d^5 + 35*(23*D*b^5*c*d^4 - 11*(D*a*b^4 - C*b^5)*d^5 )*x^4 + 5*(113*D*b^5*c^2*d^3 - 209*(D*a*b^4 - C*b^5)*c*d^4 + 99*(D*a^2*b^3 - C*a*b^4 + B*b^5)*d^5)*x^3 + 3*(5*D*b^5*c^3*d^2 - 275*(D*a*b^4 - C*b^5)* c^2*d^3 + 495*(D*a^2*b^3 - C*a*b^4 + B*b^5)*c*d^4 - 231*(D*a^3*b^2 - C*a^2 *b^3 + B*a*b^4 - A*b^5)*d^5)*x^2 - (20*D*b^5*c^4*d + 55*(D*a*b^4 - C*b^5)* c^3*d^2 - 1485*(D*a^2*b^3 - C*a*b^4 + B*b^5)*c^2*d^3 + 2541*(D*a^3*b^2 - C *a^2*b^3 + B*a*b^4 - A*b^5)*c*d^4 - 1155*(D*a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*d^5)*x)*sqrt(d*x + c))/(b^6*d^3), 2/3465*(3465*((D*a^3*b^2 - C* a^2*b^3 + B*a*b^4 - A*b^5)*c^2*d^3 - 2*(D*a^4*b - C*a^3*b^2 + B*a^2*b^3 - A*a*b^4)*c*d^4 + (D*a^5 - C*a^4*b + B*a^3*b^2 - A*a^2*b^3)*d^5)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (31 5*D*b^5*d^5*x^5 + 40*D*b^5*c^5 + 110*(D*a*b^4 - C*b^5)*c^4*d + 495*(D*a^2* b^3 - C*a*b^4 + B*b^5)*c^3*d^2 - 5313*(D*a^3*b^2 - C*a^2*b^3 + B*a*b^4 ...
Time = 9.44 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {D \left (c + d x\right )^{\frac {11}{2}}}{11 b d^{2}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \left (C b d - D a d - 2 D b c\right )}{9 b^{2} d^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (B b^{2} d^{2} - C a b d^{2} - C b^{2} c d + D a^{2} d^{2} + D a b c d + D b^{2} c^{2}\right )}{7 b^{3} d^{2}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (A b^{3} d - B a b^{2} d + C a^{2} b d - D a^{3} d\right )}{5 b^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (- A a b^{3} d^{2} + A b^{4} c d + B a^{2} b^{2} d^{2} - B a b^{3} c d - C a^{3} b d^{2} + C a^{2} b^{2} c d + D a^{4} d^{2} - D a^{3} b c d\right )}{3 b^{5}} + \frac {\sqrt {c + d x} \left (A a^{2} b^{3} d^{3} - 2 A a b^{4} c d^{2} + A b^{5} c^{2} d - B a^{3} b^{2} d^{3} + 2 B a^{2} b^{3} c d^{2} - B a b^{4} c^{2} d + C a^{4} b d^{3} - 2 C a^{3} b^{2} c d^{2} + C a^{2} b^{3} c^{2} d - D a^{5} d^{3} + 2 D a^{4} b c d^{2} - D a^{3} b^{2} c^{2} d\right )}{b^{6}} + \frac {d \left (a d - b c\right )^{3} \left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{7} \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {D x^{3}}{3 b} + \frac {x^{2} \left (C b - D a\right )}{2 b^{2}} + \frac {x \left (B b^{2} - C a b + D a^{2}\right )}{b^{3}} - \frac {\left (- A b^{3} + B a b^{2} - C a^{2} b + D a^{3}\right ) \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{3}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**(5/2)*(D*x**3+C*x**2+B*x+A)/(b*x+a),x)
Output:
Piecewise((2*(D*(c + d*x)**(11/2)/(11*b*d**2) + (c + d*x)**(9/2)*(C*b*d - D*a*d - 2*D*b*c)/(9*b**2*d**2) + (c + d*x)**(7/2)*(B*b**2*d**2 - C*a*b*d** 2 - C*b**2*c*d + D*a**2*d**2 + D*a*b*c*d + D*b**2*c**2)/(7*b**3*d**2) + (c + d*x)**(5/2)*(A*b**3*d - B*a*b**2*d + C*a**2*b*d - D*a**3*d)/(5*b**4) + (c + d*x)**(3/2)*(-A*a*b**3*d**2 + A*b**4*c*d + B*a**2*b**2*d**2 - B*a*b** 3*c*d - C*a**3*b*d**2 + C*a**2*b**2*c*d + D*a**4*d**2 - D*a**3*b*c*d)/(3*b **5) + sqrt(c + d*x)*(A*a**2*b**3*d**3 - 2*A*a*b**4*c*d**2 + A*b**5*c**2*d - B*a**3*b**2*d**3 + 2*B*a**2*b**3*c*d**2 - B*a*b**4*c**2*d + C*a**4*b*d* *3 - 2*C*a**3*b**2*c*d**2 + C*a**2*b**3*c**2*d - D*a**5*d**3 + 2*D*a**4*b* c*d**2 - D*a**3*b**2*c**2*d)/b**6 + d*(a*d - b*c)**3*(-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)*atan(sqrt(c + d*x)/sqrt((a*d - b*c)/b))/(b**7*sqrt((a* d - b*c)/b)))/d, Ne(d, 0)), (c**(5/2)*(D*x**3/(3*b) + x**2*(C*b - D*a)/(2* b**2) + x*(B*b**2 - C*a*b + D*a**2)/b**3 - (-A*b**3 + B*a*b**2 - C*a**2*b + D*a**3)*Piecewise((x/a, Eq(b, 0)), (log(a + b*x)/b, True))/b**3), True))
Exception generated. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a),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. 863 vs. \(2 (294) = 588\).
Time = 0.15 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.62 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x, algorithm="giac")
Output:
-2*(D*a^3*b^3*c^3 - C*a^2*b^4*c^3 + B*a*b^5*c^3 - A*b^6*c^3 - 3*D*a^4*b^2* c^2*d + 3*C*a^3*b^3*c^2*d - 3*B*a^2*b^4*c^2*d + 3*A*a*b^5*c^2*d + 3*D*a^5* b*c*d^2 - 3*C*a^4*b^2*c*d^2 + 3*B*a^3*b^3*c*d^2 - 3*A*a^2*b^4*c*d^2 - D*a^ 6*d^3 + C*a^5*b*d^3 - B*a^4*b^2*d^3 + A*a^3*b^3*d^3)*arctan(sqrt(d*x + c)* b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^6) + 2/3465*(315*(d*x + c) ^(11/2)*D*b^10*d^30 - 770*(d*x + c)^(9/2)*D*b^10*c*d^30 + 495*(d*x + c)^(7 /2)*D*b^10*c^2*d^30 - 385*(d*x + c)^(9/2)*D*a*b^9*d^31 + 385*(d*x + c)^(9/ 2)*C*b^10*d^31 + 495*(d*x + c)^(7/2)*D*a*b^9*c*d^31 - 495*(d*x + c)^(7/2)* C*b^10*c*d^31 + 495*(d*x + c)^(7/2)*D*a^2*b^8*d^32 - 495*(d*x + c)^(7/2)*C *a*b^9*d^32 + 495*(d*x + c)^(7/2)*B*b^10*d^32 - 693*(d*x + c)^(5/2)*D*a^3* b^7*d^33 + 693*(d*x + c)^(5/2)*C*a^2*b^8*d^33 - 693*(d*x + c)^(5/2)*B*a*b^ 9*d^33 + 693*(d*x + c)^(5/2)*A*b^10*d^33 - 1155*(d*x + c)^(3/2)*D*a^3*b^7* c*d^33 + 1155*(d*x + c)^(3/2)*C*a^2*b^8*c*d^33 - 1155*(d*x + c)^(3/2)*B*a* b^9*c*d^33 + 1155*(d*x + c)^(3/2)*A*b^10*c*d^33 - 3465*sqrt(d*x + c)*D*a^3 *b^7*c^2*d^33 + 3465*sqrt(d*x + c)*C*a^2*b^8*c^2*d^33 - 3465*sqrt(d*x + c) *B*a*b^9*c^2*d^33 + 3465*sqrt(d*x + c)*A*b^10*c^2*d^33 + 1155*(d*x + c)^(3 /2)*D*a^4*b^6*d^34 - 1155*(d*x + c)^(3/2)*C*a^3*b^7*d^34 + 1155*(d*x + c)^ (3/2)*B*a^2*b^8*d^34 - 1155*(d*x + c)^(3/2)*A*a*b^9*d^34 + 6930*sqrt(d*x + c)*D*a^4*b^6*c*d^34 - 6930*sqrt(d*x + c)*C*a^3*b^7*c*d^34 + 6930*sqrt(d*x + c)*B*a^2*b^8*c*d^34 - 6930*sqrt(d*x + c)*A*a*b^9*c*d^34 - 3465*sqrt(...
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{a+b\,x} \,d x \] Input:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x),x)
Output:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x), x)
Time = 0.14 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {-\frac {4 \sqrt {d x +c}\, a \,b^{5} c^{2} d^{3} x^{2}}{3}-\frac {8 \sqrt {d x +c}\, a \,b^{5} c \,d^{4} x^{3}}{9}-2 \sqrt {d x +c}\, a^{5} b \,d^{5}+\frac {2 \sqrt {d x +c}\, b^{7} c^{3} d}{7}+\frac {2 \sqrt {d x +c}\, b^{7} d^{4} x^{3}}{7}+\frac {2 \sqrt {d x +c}\, b^{6} d^{5} x^{5}}{11}+2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{5} d^{5}+\frac {20 \sqrt {d x +c}\, a^{4} b^{2} c \,d^{4}}{3}+\frac {2 \sqrt {d x +c}\, a^{4} b^{2} d^{5} x}{3}-\frac {116 \sqrt {d x +c}\, a^{3} b^{3} c^{2} d^{3}}{15}-\frac {2 \sqrt {d x +c}\, a^{3} b^{3} d^{5} x^{2}}{5}+\frac {352 \sqrt {d x +c}\, a^{2} b^{4} c^{3} d^{2}}{105}+\frac {2 \sqrt {d x +c}\, a^{2} b^{4} d^{5} x^{3}}{7}-\frac {2 \sqrt {d x +c}\, a \,b^{5} c^{4} d}{9}-\frac {2 \sqrt {d x +c}\, a \,b^{5} d^{5} x^{4}}{9}+\frac {6 \sqrt {d x +c}\, b^{7} c^{2} d^{2} x}{7}+\frac {6 \sqrt {d x +c}\, b^{7} c \,d^{3} x^{2}}{7}+\frac {2 \sqrt {d x +c}\, b^{6} c^{4} d x}{99}+\frac {16 \sqrt {d x +c}\, b^{6} c^{3} d^{2} x^{2}}{33}+\frac {92 \sqrt {d x +c}\, b^{6} c^{2} d^{3} x^{3}}{99}+\frac {68 \sqrt {d x +c}\, b^{6} c \,d^{4} x^{4}}{99}-\frac {4 \sqrt {d x +c}\, b^{6} c^{5}}{99}-6 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{4} b c \,d^{4}+6 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{3} b^{2} c^{2} d^{3}-2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b^{3} c^{3} d^{2}-\frac {32 \sqrt {d x +c}\, a^{3} b^{3} c \,d^{4} x}{15}+\frac {244 \sqrt {d x +c}\, a^{2} b^{4} c^{2} d^{3} x}{105}+\frac {44 \sqrt {d x +c}\, a^{2} b^{4} c \,d^{4} x^{2}}{35}-\frac {8 \sqrt {d x +c}\, a \,b^{5} c^{3} d^{2} x}{9}}{b^{7} d^{2}} \] Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a),x)
Output:
(2*(3465*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**5*d**5 - 10395*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/ (sqrt(b)*sqrt(a*d - b*c)))*a**4*b*c*d**4 + 10395*sqrt(b)*sqrt(a*d - b*c)*a tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*c**2*d**3 - 346 5*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)) )*a**2*b**3*c**3*d**2 - 3465*sqrt(c + d*x)*a**5*b*d**5 + 11550*sqrt(c + d* x)*a**4*b**2*c*d**4 + 1155*sqrt(c + d*x)*a**4*b**2*d**5*x - 13398*sqrt(c + d*x)*a**3*b**3*c**2*d**3 - 3696*sqrt(c + d*x)*a**3*b**3*c*d**4*x - 693*sq rt(c + d*x)*a**3*b**3*d**5*x**2 + 5808*sqrt(c + d*x)*a**2*b**4*c**3*d**2 + 4026*sqrt(c + d*x)*a**2*b**4*c**2*d**3*x + 2178*sqrt(c + d*x)*a**2*b**4*c *d**4*x**2 + 495*sqrt(c + d*x)*a**2*b**4*d**5*x**3 - 385*sqrt(c + d*x)*a*b **5*c**4*d - 1540*sqrt(c + d*x)*a*b**5*c**3*d**2*x - 2310*sqrt(c + d*x)*a* b**5*c**2*d**3*x**2 - 1540*sqrt(c + d*x)*a*b**5*c*d**4*x**3 - 385*sqrt(c + d*x)*a*b**5*d**5*x**4 + 495*sqrt(c + d*x)*b**7*c**3*d + 1485*sqrt(c + d*x )*b**7*c**2*d**2*x + 1485*sqrt(c + d*x)*b**7*c*d**3*x**2 + 495*sqrt(c + d* x)*b**7*d**4*x**3 - 70*sqrt(c + d*x)*b**6*c**5 + 35*sqrt(c + d*x)*b**6*c** 4*d*x + 840*sqrt(c + d*x)*b**6*c**3*d**2*x**2 + 1610*sqrt(c + d*x)*b**6*c* *2*d**3*x**3 + 1190*sqrt(c + d*x)*b**6*c*d**4*x**4 + 315*sqrt(c + d*x)*b** 6*d**5*x**5))/(3465*b**7*d**2)