Integrand size = 32, antiderivative size = 371 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {(b c-a d) \left (b^3 (2 B c+5 A d)-a b^2 (4 c C+7 B d)-11 a^3 d D+3 a^2 b (3 C d+2 c D)\right ) \sqrt {c+d x}}{b^6}+\frac {\left (b^3 (2 B c+5 A d)-a b^2 (4 c C+7 B d)-11 a^3 d D+3 a^2 b (3 C d+2 c D)\right ) (c+d x)^{3/2}}{3 b^5}+\frac {2 \left (b^2 B-2 a b C+3 a^2 D\right ) (c+d x)^{5/2}}{5 b^4}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{5/2}}{b^4 (a+b x)}+\frac {2 (b C d-b c D-2 a d D) (c+d x)^{7/2}}{7 b^3 d^2}+\frac {2 D (c+d x)^{9/2}}{9 b^2 d^2}-\frac {(b c-a d)^{3/2} \left (b^3 (2 B c+5 A d)-a b^2 (4 c C+7 B d)-11 a^3 d D+3 a^2 b (3 C d+2 c D)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \] Output:
(-a*d+b*c)*(b^3*(5*A*d+2*B*c)-a*b^2*(7*B*d+4*C*c)-11*a^3*d*D+3*a^2*b*(3*C* d+2*D*c))*(d*x+c)^(1/2)/b^6+1/3*(b^3*(5*A*d+2*B*c)-a*b^2*(7*B*d+4*C*c)-11* a^3*d*D+3*a^2*b*(3*C*d+2*D*c))*(d*x+c)^(3/2)/b^5+2/5*(B*b^2-2*C*a*b+3*D*a^ 2)*(d*x+c)^(5/2)/b^4-(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(5/2)/b^4/(b*x+ a)+2/7*(C*b*d-2*D*a*d-D*b*c)*(d*x+c)^(7/2)/b^3/d^2+2/9*D*(d*x+c)^(9/2)/b^2 /d^2-(-a*d+b*c)^(3/2)*(b^3*(5*A*d+2*B*c)-a*b^2*(7*B*d+4*C*c)-11*a^3*d*D+3* a^2*b*(3*C*d+2*D*c))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(13 /2)
Time = 1.02 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.35 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {\sqrt {c+d x} \left (3465 a^5 d^4 D-105 a^4 b d^3 (27 C d+62 c D-22 d D x)+21 a^3 b^2 d^2 \left (153 c^2 D+c (240 C d-214 d D x)+d^2 \left (105 B-90 C x-22 D x^2\right )\right )+3 a^2 b^3 d \left (-60 c^3 D+c^2 (-749 C d+786 d D x)+2 c d^2 \left (-595 B+581 C x+141 D x^2\right )+d^3 \left (-525 A+490 B x+126 C x^2+66 D x^3\right )\right )-a b^4 \left (20 c^4 D+c^3 (-90 C d+170 d D x)+3 c^2 d^2 \left (-427 B+554 C x+130 D x^2\right )+2 d^4 x \left (525 A+147 B x+81 C x^2+55 D x^3\right )+2 c d^3 \left (-1050 A+1239 B x+327 C x^2+175 D x^3\right )\right )+b^5 \left (105 A d^2 \left (-3 c^2+14 c d x+2 d^2 x^2\right )+2 x \left (-10 c^4 D+5 c^3 d (9 C+D x)+3 c^2 d^2 (161 B+5 x (9 C+5 D x))+d^4 x^2 (63 B+5 x (9 C+7 D x))+c d^3 x (231 B+5 x (27 C+19 D x))\right )\right )\right )}{315 b^6 d^2 (a+b x)}+\frac {(-b c+a d)^{3/2} \left (b^3 (2 B c+5 A d)-a b^2 (4 c C+7 B d)-11 a^3 d D+3 a^2 b (3 C d+2 c D)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \] Input:
Integrate[((c + d*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]
Output:
(Sqrt[c + d*x]*(3465*a^5*d^4*D - 105*a^4*b*d^3*(27*C*d + 62*c*D - 22*d*D*x ) + 21*a^3*b^2*d^2*(153*c^2*D + c*(240*C*d - 214*d*D*x) + d^2*(105*B - 90* C*x - 22*D*x^2)) + 3*a^2*b^3*d*(-60*c^3*D + c^2*(-749*C*d + 786*d*D*x) + 2 *c*d^2*(-595*B + 581*C*x + 141*D*x^2) + d^3*(-525*A + 490*B*x + 126*C*x^2 + 66*D*x^3)) - a*b^4*(20*c^4*D + c^3*(-90*C*d + 170*d*D*x) + 3*c^2*d^2*(-4 27*B + 554*C*x + 130*D*x^2) + 2*d^4*x*(525*A + 147*B*x + 81*C*x^2 + 55*D*x ^3) + 2*c*d^3*(-1050*A + 1239*B*x + 327*C*x^2 + 175*D*x^3)) + b^5*(105*A*d ^2*(-3*c^2 + 14*c*d*x + 2*d^2*x^2) + 2*x*(-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^2*(63*B + 5*x*(9*C + 7*D*x )) + c*d^3*x*(231*B + 5*x*(27*C + 19*D*x))))))/(315*b^6*d^2*(a + b*x)) + ( (-(b*c) + a*d)^(3/2)*(b^3*(2*B*c + 5*A*d) - a*b^2*(4*c*C + 7*B*d) - 11*a^3 *d*D + 3*a^2*b*(3*C*d + 2*c*D))*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/b^(13/2)
Time = 1.11 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.23, 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.
| \(\displaystyle \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle -\frac {\int -\frac {(c+d x)^{5/2} \left (2 \left (c-\frac {a d}{b}\right ) D x^2+\frac {2 (b c-a d) (b C-a D) x}{b^2}+\frac {-7 d D a^3+b (7 C d+2 c D) a^2-b^2 (2 c C+7 B d) a+b^3 (2 B c+5 A d)}{b^3}\right )}{2 (a+b x)}dx}{b c-a d}-\frac {(c+d x)^{7/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d x)^{5/2} \left (-\frac {7 d D a^3}{b^3}+\frac {(7 C d+2 c D) a^2}{b^2}-\frac {(2 c C+7 B d) a}{b}+2 \left (c-\frac {a d}{b}\right ) D x^2+2 B c+5 A d+\frac {2 (b c-a d) (b C-a D) x}{b^2}\right )}{a+b x}dx}{2 (b c-a d)}-\frac {(c+d x)^{7/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (-2 D c^3+2 C d c^2-2 B d^2 c-2 \left (c-\frac {a d}{b}\right ) D (c+d x)^2-d^3 \left (5 A-\frac {7 a \left (D a^2-b C a+b^2 B\right )}{b^3}\right )-\frac {2 (b c-a d) (b C d-a D d-2 b c D) (c+d x)}{b^2}\right )}{b c-a d-b (c+d x)}d\sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(c+d x)^{7/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\int \left (\frac {2 (b c-a d) D (c+d x)^4}{b^2}+\frac {2 (b c-a d) (b C d-2 a D d-b c D) (c+d x)^3}{b^3}+\frac {d^2 \left (-11 d D a^3+3 b (3 C d+2 c D) a^2-b^2 (4 c C+7 B d) a+b^3 (2 B c+5 A d)\right ) (c+d x)^2}{b^4}+\frac {d^2 (b c-a d) \left (-11 d D a^3+3 b (3 C d+2 c D) a^2-b^2 (4 c C+7 B d) a+b^3 (2 B c+5 A d)\right ) (c+d x)}{b^5}+\frac {d^2 (b c-a d)^2 \left (-11 d D a^3+3 b (3 C d+2 c D) a^2-b^2 (4 c C+7 B d) a+b^3 (2 B c+5 A d)\right )}{b^6}+\frac {-5 A c^3 d^3 b^6-2 B c^4 d^2 b^6+15 a A c^2 d^4 b^5+13 a B c^3 d^3 b^5+4 a c^4 C d^2 b^5-15 a^2 A c d^5 b^4-27 a^2 B c^2 d^4 b^4-21 a^2 c^3 C d^3 b^4-6 a^2 c^4 d^2 D b^4+5 a^3 A d^6 b^3+23 a^3 B c d^5 b^3+39 a^3 c^2 C d^4 b^3+29 a^3 c^3 d^3 D b^3-7 a^4 B d^6 b^2-31 a^4 c C d^5 b^2-51 a^4 c^2 d^4 D b^2+9 a^5 C d^6 b+39 a^5 c d^5 D b-11 a^6 d^6 D}{b^6 (b c-a d-b (c+d x))}\right )d\sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(c+d x)^{7/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {d^2 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right ) \left (-11 a^3 d D+3 a^2 b (2 c D+3 C d)-a b^2 (7 B d+4 c C)+b^3 (5 A d+2 B c)\right )}{b^{13/2}}+\frac {d^2 \sqrt {c+d x} (b c-a d)^2 \left (-11 a^3 d D+3 a^2 b (2 c D+3 C d)-a b^2 (7 B d+4 c C)+b^3 (5 A d+2 B c)\right )}{b^6}+\frac {d^2 (c+d x)^{3/2} (b c-a d) \left (-11 a^3 d D+3 a^2 b (2 c D+3 C d)-a b^2 (7 B d+4 c C)+b^3 (5 A d+2 B c)\right )}{3 b^5}+\frac {d^2 (c+d x)^{5/2} \left (-11 a^3 d D+3 a^2 b (2 c D+3 C d)-a b^2 (7 B d+4 c C)+b^3 (5 A d+2 B c)\right )}{5 b^4}+\frac {2 (c+d x)^{7/2} (b c-a d) (-2 a d D-b c D+b C d)}{7 b^3}+\frac {2 D (c+d x)^{9/2} (b c-a d)}{9 b^2}}{d^2 (b c-a d)}-\frac {(c+d x)^{7/2} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
Input:
Int[((c + d*x)^(5/2)*(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)^(7/2))/((b*c - a*d)*(a + b*x))) + ((d^2*(b*c - a*d)^2*(b^3*(2*B*c + 5*A*d) - a*b^2*(4*c*C + 7*B*d ) - 11*a^3*d*D + 3*a^2*b*(3*C*d + 2*c*D))*Sqrt[c + d*x])/b^6 + (d^2*(b*c - a*d)*(b^3*(2*B*c + 5*A*d) - a*b^2*(4*c*C + 7*B*d) - 11*a^3*d*D + 3*a^2*b* (3*C*d + 2*c*D))*(c + d*x)^(3/2))/(3*b^5) + (d^2*(b^3*(2*B*c + 5*A*d) - a* b^2*(4*c*C + 7*B*d) - 11*a^3*d*D + 3*a^2*b*(3*C*d + 2*c*D))*(c + d*x)^(5/2 ))/(5*b^4) + (2*(b*c - a*d)*(b*C*d - b*c*D - 2*a*d*D)*(c + d*x)^(7/2))/(7* b^3) + (2*(b*c - a*d)*D*(c + d*x)^(9/2))/(9*b^2) - (d^2*(b*c - a*d)^(5/2)* (b^3*(2*B*c + 5*A*d) - a*b^2*(4*c*C + 7*B*d) - 11*a^3*d*D + 3*a^2*b*(3*C*d + 2*c*D))*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/b^(13/2))/(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 = 0.93 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (-\left (\left (A d +\frac {2 B c}{5}\right ) b^{3}-\frac {7 \left (B d +\frac {4 C c}{7}\right ) a \,b^{2}}{5}+\frac {9 \left (C d +\frac {2 D c}{3}\right ) a^{2} b}{5}-\frac {11 a^{3} d D}{5}\right ) \left (a d -b c \right )^{2} d^{2} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (\frac {\left (-\frac {2 x^{2} \left (\frac {1}{3} D x^{3}+\frac {3}{7} C \,x^{2}+\frac {3}{5} B x +A \right ) d^{4}}{3}-\frac {14 x \left (\frac {19}{147} D x^{3}+\frac {9}{49} C \,x^{2}+\frac {11}{35} B x +A \right ) c \,d^{3}}{3}+\left (-\frac {10}{21} D x^{3}-\frac {6}{7} C \,x^{2}-\frac {46}{15} B x +A \right ) c^{2} d^{2}-\frac {2 x \left (\frac {D x}{9}+C \right ) c^{3} d}{7}+\frac {4 D c^{4} x}{63}\right ) b^{5}}{5}-\frac {4 \left (-\frac {x \left (\frac {11}{105} D x^{3}+\frac {27}{175} C \,x^{2}+\frac {7}{25} B x +A \right ) d^{4}}{2}+c \left (-\frac {1}{6} D x^{3}-\frac {109}{350} C \,x^{2}-\frac {59}{50} B x +A \right ) d^{3}+\frac {61 \left (-\frac {130}{427} D x^{2}-\frac {554}{427} C x +B \right ) c^{2} d^{2}}{100}+\frac {3 \left (-\frac {17 D x}{9}+C \right ) c^{3} d}{70}-\frac {D c^{4}}{105}\right ) a \,b^{4}}{3}+\left (\left (-\frac {6}{25} C \,x^{2}-\frac {14}{15} B x -\frac {22}{175} D x^{3}+A \right ) d^{3}+\frac {34 \left (-\frac {141}{595} D x^{2}-\frac {83}{85} C x +B \right ) c \,d^{2}}{15}+\frac {107 \left (-\frac {786 D x}{749}+C \right ) c^{2} d}{75}+\frac {4 D c^{3}}{35}\right ) d \,a^{2} b^{3}-\frac {7 d^{2} a^{3} \left (\left (-\frac {22}{105} D x^{2}-\frac {6}{7} C x +B \right ) d^{2}+\frac {16 \left (-\frac {107 D x}{120}+C \right ) c d}{7}+\frac {51 D c^{2}}{35}\right ) b^{2}}{5}+\frac {9 \left (\left (-\frac {22 D x}{27}+C \right ) d +\frac {62 D c}{27}\right ) d^{3} a^{4} b}{5}-\frac {11 D a^{5} d^{4}}{5}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {x d +c}\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{2} \left (b x +a \right ) b^{6}}\) | \(476\) |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {2 B a \,b^{3} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-C \,a^{2} b^{2} d^{3} \left (x d +c \right )^{\frac {3}{2}}+\frac {4 D a^{3} b \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}+2 A a \,b^{3} d^{4} \sqrt {x d +c}-2 A \,b^{4} c \,d^{3} \sqrt {x d +c}-3 B \,a^{2} b^{2} d^{4} \sqrt {x d +c}+4 C \,a^{3} b \,d^{4} \sqrt {x d +c}-6 C \,a^{2} b^{2} c \,d^{3} \sqrt {x d +c}+8 D a^{3} b c \,d^{3} \sqrt {x d +c}+\frac {2 D a \,b^{3} d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 C a \,b^{3} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {3 D a^{2} b^{2} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{4} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-5 D a^{4} d^{4} \sqrt {x d +c}+2 C a \,b^{3} c^{2} d^{2} \sqrt {x d +c}-3 D a^{2} b^{2} c^{2} d^{2} \sqrt {x d +c}-\frac {B \,b^{4} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-B \,b^{4} c^{2} d^{2} \sqrt {x d +c}+\frac {D b^{4} c \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {B \,b^{4} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}+4 B a \,b^{3} c \,d^{3} \sqrt {x d +c}+\frac {2 C a \,b^{3} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-D a^{2} b^{2} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}-\frac {C \,b^{4} d \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {D \left (x d +c \right )^{\frac {9}{2}} b^{4}}{9}\right )}{b^{6}}+\frac {2 d^{2} \left (\frac {\left (-\frac {1}{2} A \,a^{2} b^{3} d^{3}+A a \,b^{4} c \,d^{2}-\frac {1}{2} A \,b^{5} c^{2} d +\frac {1}{2} B \,a^{3} b^{2} d^{3}-B \,a^{2} b^{3} c \,d^{2}+\frac {1}{2} B a \,b^{4} c^{2} d -\frac {1}{2} C \,a^{4} b \,d^{3}+C \,a^{3} b^{2} c \,d^{2}-\frac {1}{2} C \,a^{2} b^{3} c^{2} d +\frac {1}{2} D a^{5} d^{3}-D a^{4} b c \,d^{2}+\frac {1}{2} D a^{3} b^{2} c^{2} d \right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {\left (5 A \,a^{2} b^{3} d^{3}-10 A a \,b^{4} c \,d^{2}+5 A \,b^{5} c^{2} d -7 B \,a^{3} b^{2} d^{3}+16 B \,a^{2} b^{3} c \,d^{2}-11 B a \,b^{4} c^{2} d +2 B \,b^{5} c^{3}+9 C \,a^{4} b \,d^{3}-22 C \,a^{3} b^{2} c \,d^{2}+17 C \,a^{2} b^{3} c^{2} d -4 C a \,b^{4} c^{3}-11 D a^{5} d^{3}+28 D a^{4} b c \,d^{2}-23 D a^{3} b^{2} c^{2} d +6 D a^{2} b^{3} c^{3}\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^{6}}}{d^{2}}\) | \(824\) |
| default | \(\frac {-\frac {2 \left (\frac {2 B a \,b^{3} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-C \,a^{2} b^{2} d^{3} \left (x d +c \right )^{\frac {3}{2}}+\frac {4 D a^{3} b \,d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}+2 A a \,b^{3} d^{4} \sqrt {x d +c}-2 A \,b^{4} c \,d^{3} \sqrt {x d +c}-3 B \,a^{2} b^{2} d^{4} \sqrt {x d +c}+4 C \,a^{3} b \,d^{4} \sqrt {x d +c}-6 C \,a^{2} b^{2} c \,d^{3} \sqrt {x d +c}+8 D a^{3} b c \,d^{3} \sqrt {x d +c}+\frac {2 D a \,b^{3} d \left (x d +c \right )^{\frac {7}{2}}}{7}+\frac {2 C a \,b^{3} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {3 D a^{2} b^{2} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}-\frac {A \,b^{4} d^{3} \left (x d +c \right )^{\frac {3}{2}}}{3}-5 D a^{4} d^{4} \sqrt {x d +c}+2 C a \,b^{3} c^{2} d^{2} \sqrt {x d +c}-3 D a^{2} b^{2} c^{2} d^{2} \sqrt {x d +c}-\frac {B \,b^{4} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-B \,b^{4} c^{2} d^{2} \sqrt {x d +c}+\frac {D b^{4} c \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {B \,b^{4} d^{2} \left (x d +c \right )^{\frac {5}{2}}}{5}+4 B a \,b^{3} c \,d^{3} \sqrt {x d +c}+\frac {2 C a \,b^{3} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}}{3}-D a^{2} b^{2} c \,d^{2} \left (x d +c \right )^{\frac {3}{2}}-\frac {C \,b^{4} d \left (x d +c \right )^{\frac {7}{2}}}{7}-\frac {D \left (x d +c \right )^{\frac {9}{2}} b^{4}}{9}\right )}{b^{6}}+\frac {2 d^{2} \left (\frac {\left (-\frac {1}{2} A \,a^{2} b^{3} d^{3}+A a \,b^{4} c \,d^{2}-\frac {1}{2} A \,b^{5} c^{2} d +\frac {1}{2} B \,a^{3} b^{2} d^{3}-B \,a^{2} b^{3} c \,d^{2}+\frac {1}{2} B a \,b^{4} c^{2} d -\frac {1}{2} C \,a^{4} b \,d^{3}+C \,a^{3} b^{2} c \,d^{2}-\frac {1}{2} C \,a^{2} b^{3} c^{2} d +\frac {1}{2} D a^{5} d^{3}-D a^{4} b c \,d^{2}+\frac {1}{2} D a^{3} b^{2} c^{2} d \right ) \sqrt {x d +c}}{\left (x d +c \right ) b +a d -b c}+\frac {\left (5 A \,a^{2} b^{3} d^{3}-10 A a \,b^{4} c \,d^{2}+5 A \,b^{5} c^{2} d -7 B \,a^{3} b^{2} d^{3}+16 B \,a^{2} b^{3} c \,d^{2}-11 B a \,b^{4} c^{2} d +2 B \,b^{5} c^{3}+9 C \,a^{4} b \,d^{3}-22 C \,a^{3} b^{2} c \,d^{2}+17 C \,a^{2} b^{3} c^{2} d -4 C a \,b^{4} c^{3}-11 D a^{5} d^{3}+28 D a^{4} b c \,d^{2}-23 D a^{3} b^{2} c^{2} d +6 D a^{2} b^{3} c^{3}\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^{6}}}{d^{2}}\) | \(824\) |
Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-5*(-((A*d+2/5*B*c)*b^3-7/5*(B*d+4/7*C*c)*a*b^2+9/5*(C*d+2/3*D*c)*a^2*b-11 /5*a^3*d*D)*(a*d-b*c)^2*d^2*(b*x+a)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^( 1/2))+(1/5*(-2/3*x^2*(1/3*D*x^3+3/7*C*x^2+3/5*B*x+A)*d^4-14/3*x*(19/147*D* x^3+9/49*C*x^2+11/35*B*x+A)*c*d^3+(-10/21*D*x^3-6/7*C*x^2-46/15*B*x+A)*c^2 *d^2-2/7*x*(1/9*D*x+C)*c^3*d+4/63*D*c^4*x)*b^5-4/3*(-1/2*x*(11/105*D*x^3+2 7/175*C*x^2+7/25*B*x+A)*d^4+c*(-1/6*D*x^3-109/350*C*x^2-59/50*B*x+A)*d^3+6 1/100*(-130/427*D*x^2-554/427*C*x+B)*c^2*d^2+3/70*(-17/9*D*x+C)*c^3*d-1/10 5*D*c^4)*a*b^4+((-6/25*C*x^2-14/15*B*x-22/175*D*x^3+A)*d^3+34/15*(-141/595 *D*x^2-83/85*C*x+B)*c*d^2+107/75*(-786/749*D*x+C)*c^2*d+4/35*D*c^3)*d*a^2* b^3-7/5*d^2*a^3*((-22/105*D*x^2-6/7*C*x+B)*d^2+16/7*(-107/120*D*x+C)*c*d+5 1/35*D*c^2)*b^2+9/5*((-22/27*D*x+C)*d+62/27*D*c)*d^3*a^4*b-11/5*D*a^5*d^4) *((a*d-b*c)*b)^(1/2)*(d*x+c)^(1/2))/((a*d-b*c)*b)^(1/2)/d^2/(b*x+a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 808 vs. \(2 (341) = 682\).
Time = 0.11 (sec) , antiderivative size = 1626, normalized size of antiderivative = 4.38 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="fricas ")
Output:
[-1/630*(315*(2*(3*D*a^3*b^2 - 2*C*a^2*b^3 + B*a*b^4)*c^2*d^2 - (17*D*a^4* b - 13*C*a^3*b^2 + 9*B*a^2*b^3 - 5*A*a*b^4)*c*d^3 + (11*D*a^5 - 9*C*a^4*b + 7*B*a^3*b^2 - 5*A*a^2*b^3)*d^4 + (2*(3*D*a^2*b^3 - 2*C*a*b^4 + B*b^5)*c^ 2*d^2 - (17*D*a^3*b^2 - 13*C*a^2*b^3 + 9*B*a*b^4 - 5*A*b^5)*c*d^3 + (11*D* a^4*b - 9*C*a^3*b^2 + 7*B*a^2*b^3 - 5*A*a*b^4)*d^4)*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*(70*D*b^5*d^4*x^5 - 20*D*a*b^4*c^4 - 90*(2*D*a^2*b^3 - C*a*b^4)*c^ 3*d + 21*(153*D*a^3*b^2 - 107*C*a^2*b^3 + 61*B*a*b^4 - 15*A*b^5)*c^2*d^2 - 210*(31*D*a^4*b - 24*C*a^3*b^2 + 17*B*a^2*b^3 - 10*A*a*b^4)*c*d^3 + 315*( 11*D*a^5 - 9*C*a^4*b + 7*B*a^3*b^2 - 5*A*a^2*b^3)*d^4 + 10*(19*D*b^5*c*d^3 - (11*D*a*b^4 - 9*C*b^5)*d^4)*x^4 + 2*(75*D*b^5*c^2*d^2 - 5*(35*D*a*b^4 - 27*C*b^5)*c*d^3 + 9*(11*D*a^2*b^3 - 9*C*a*b^4 + 7*B*b^5)*d^4)*x^3 + 2*(5* D*b^5*c^3*d - 15*(13*D*a*b^4 - 9*C*b^5)*c^2*d^2 + 3*(141*D*a^2*b^3 - 109*C *a*b^4 + 77*B*b^5)*c*d^3 - 21*(11*D*a^3*b^2 - 9*C*a^2*b^3 + 7*B*a*b^4 - 5* A*b^5)*d^4)*x^2 - 2*(10*D*b^5*c^4 + 5*(17*D*a*b^4 - 9*C*b^5)*c^3*d - 3*(39 3*D*a^2*b^3 - 277*C*a*b^4 + 161*B*b^5)*c^2*d^2 + 21*(107*D*a^3*b^2 - 83*C* a^2*b^3 + 59*B*a*b^4 - 35*A*b^5)*c*d^3 - 105*(11*D*a^4*b - 9*C*a^3*b^2 + 7 *B*a^2*b^3 - 5*A*a*b^4)*d^4)*x)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2), -1 /315*(315*(2*(3*D*a^3*b^2 - 2*C*a^2*b^3 + B*a*b^4)*c^2*d^2 - (17*D*a^4*b - 13*C*a^3*b^2 + 9*B*a^2*b^3 - 5*A*a*b^4)*c*d^3 + (11*D*a^5 - 9*C*a^4*b ...
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(5/2)*(D*x**3+C*x**2+B*x+A)/(b*x+a)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(5/2)*(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. 914 vs. \(2 (341) = 682\).
Time = 0.14 (sec) , antiderivative size = 914, normalized size of antiderivative = 2.46 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="giac")
Output:
(6*D*a^2*b^3*c^3 - 4*C*a*b^4*c^3 + 2*B*b^5*c^3 - 23*D*a^3*b^2*c^2*d + 17*C *a^2*b^3*c^2*d - 11*B*a*b^4*c^2*d + 5*A*b^5*c^2*d + 28*D*a^4*b*c*d^2 - 22* C*a^3*b^2*c*d^2 + 16*B*a^2*b^3*c*d^2 - 10*A*a*b^4*c*d^2 - 11*D*a^5*d^3 + 9 *C*a^4*b*d^3 - 7*B*a^3*b^2*d^3 + 5*A*a^2*b^3*d^3)*arctan(sqrt(d*x + c)*b/s qrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^6) + (sqrt(d*x + c)*D*a^3*b^2 *c^2*d - sqrt(d*x + c)*C*a^2*b^3*c^2*d + sqrt(d*x + c)*B*a*b^4*c^2*d - sqr t(d*x + c)*A*b^5*c^2*d - 2*sqrt(d*x + c)*D*a^4*b*c*d^2 + 2*sqrt(d*x + c)*C *a^3*b^2*c*d^2 - 2*sqrt(d*x + c)*B*a^2*b^3*c*d^2 + 2*sqrt(d*x + c)*A*a*b^4 *c*d^2 + sqrt(d*x + c)*D*a^5*d^3 - sqrt(d*x + c)*C*a^4*b*d^3 + sqrt(d*x + c)*B*a^3*b^2*d^3 - sqrt(d*x + c)*A*a^2*b^3*d^3)/(((d*x + c)*b - b*c + a*d) *b^6) + 2/315*(35*(d*x + c)^(9/2)*D*b^16*d^16 - 45*(d*x + c)^(7/2)*D*b^16* c*d^16 - 90*(d*x + c)^(7/2)*D*a*b^15*d^17 + 45*(d*x + c)^(7/2)*C*b^16*d^17 + 189*(d*x + c)^(5/2)*D*a^2*b^14*d^18 - 126*(d*x + c)^(5/2)*C*a*b^15*d^18 + 63*(d*x + c)^(5/2)*B*b^16*d^18 + 315*(d*x + c)^(3/2)*D*a^2*b^14*c*d^18 - 210*(d*x + c)^(3/2)*C*a*b^15*c*d^18 + 105*(d*x + c)^(3/2)*B*b^16*c*d^18 + 945*sqrt(d*x + c)*D*a^2*b^14*c^2*d^18 - 630*sqrt(d*x + c)*C*a*b^15*c^2*d ^18 + 315*sqrt(d*x + c)*B*b^16*c^2*d^18 - 420*(d*x + c)^(3/2)*D*a^3*b^13*d ^19 + 315*(d*x + c)^(3/2)*C*a^2*b^14*d^19 - 210*(d*x + c)^(3/2)*B*a*b^15*d ^19 + 105*(d*x + c)^(3/2)*A*b^16*d^19 - 2520*sqrt(d*x + c)*D*a^3*b^13*c*d^ 19 + 1890*sqrt(d*x + c)*C*a^2*b^14*c*d^19 - 1260*sqrt(d*x + c)*B*a*b^15...
Timed out. \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^2} \,d x \] Input:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^2,x)
Output:
int(((c + d*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^2, x)
Time = 0.14 (sec) , antiderivative size = 1188, normalized size of antiderivative = 3.20 \[ \int \frac {(c+d x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x)
Output:
( - 3465*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**5*d**4 + 8190*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**3 - 3465*sqrt(b)*sqrt(a*d - b*c)*ata n((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**4*b*d**4*x - 630*sqrt(b) *sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b* *3*d**3 - 5985*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqr t(a*d - b*c)))*a**3*b**2*c**2*d**2 + 8190*sqrt(b)*sqrt(a*d - b*c)*atan((sq rt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**3*b**2*c*d**3*x + 1260*sqrt(b )*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b **4*c*d**2 - 630*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s qrt(a*d - b*c)))*a**2*b**4*d**3*x + 1260*sqrt(b)*sqrt(a*d - b*c)*atan((sqr t(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3*c**3*d - 5985*sqrt(b)*s qrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a**2*b**3 *c**2*d**2*x - 630*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b) *sqrt(a*d - b*c)))*a*b**5*c**2*d + 1260*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt (c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**5*c*d**2*x + 1260*sqrt(b)*sqr t(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**4*c**3 *d*x - 630*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a* d - b*c)))*b**6*c**2*d*x + 3465*sqrt(c + d*x)*a**5*b*d**4 - 9345*sqrt(c + d*x)*a**4*b**2*c*d**3 + 2310*sqrt(c + d*x)*a**4*b**2*d**4*x + 630*sqrt(...