Integrand size = 22, antiderivative size = 85 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2}{3} A b^3 x^{3/2}+\frac {2}{5} b^2 (b B+3 A c) x^{5/2}+\frac {6}{7} b c (b B+A c) x^{7/2}+\frac {2}{9} c^2 (3 b B+A c) x^{9/2}+\frac {2}{11} B c^3 x^{11/2} \] Output:
2/3*A*b^3*x^(3/2)+2/5*b^2*(3*A*c+B*b)*x^(5/2)+6/7*b*c*(A*c+B*b)*x^(7/2)+2/ 9*c^2*(A*c+3*B*b)*x^(9/2)+2/11*B*c^3*x^(11/2)
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2 x^{3/2} \left (11 A \left (105 b^3+189 b^2 c x+135 b c^2 x^2+35 c^3 x^3\right )+3 B x \left (231 b^3+495 b^2 c x+385 b c^2 x^2+105 c^3 x^3\right )\right )}{3465} \] Input:
Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]
Output:
(2*x^(3/2)*(11*A*(105*b^3 + 189*b^2*c*x + 135*b*c^2*x^2 + 35*c^3*x^3) + 3* B*x*(231*b^3 + 495*b^2*c*x + 385*b*c^2*x^2 + 105*c^3*x^3)))/3465
Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 85, 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.
\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \sqrt {x} (A+B x) (b+c x)^3dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (A b^3 \sqrt {x}+b^2 x^{3/2} (3 A c+b B)+c^2 x^{7/2} (A c+3 b B)+3 b c x^{5/2} (A c+b B)+B c^3 x^{9/2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} A b^3 x^{3/2}+\frac {2}{5} b^2 x^{5/2} (3 A c+b B)+\frac {2}{9} c^2 x^{9/2} (A c+3 b B)+\frac {6}{7} b c x^{7/2} (A c+b B)+\frac {2}{11} B c^3 x^{11/2}\) |
Input:
Int[((A + B*x)*(b*x + c*x^2)^3)/x^(5/2),x]
Output:
(2*A*b^3*x^(3/2))/3 + (2*b^2*(b*B + 3*A*c)*x^(5/2))/5 + (6*b*c*(b*B + A*c) *x^(7/2))/7 + (2*c^2*(3*b*B + A*c)*x^(9/2))/9 + (2*B*c^3*x^(11/2))/11
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Time = 0.70 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (315 B \,c^{3} x^{4}+385 A \,c^{3} x^{3}+1155 x^{3} B b \,c^{2}+1485 A b \,c^{2} x^{2}+1485 x^{2} B \,b^{2} c +2079 A \,b^{2} c x +693 x B \,b^{3}+1155 A \,b^{3}\right )}{3465}\) | \(76\) |
derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 A \,b^{2} c +B \,b^{3}\right ) x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} x^{\frac {3}{2}}}{3}\) | \(76\) |
default | \(\frac {2 B \,c^{3} x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 A \,b^{2} c +B \,b^{3}\right ) x^{\frac {5}{2}}}{5}+\frac {2 A \,b^{3} x^{\frac {3}{2}}}{3}\) | \(76\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (315 B \,c^{3} x^{4}+385 A \,c^{3} x^{3}+1155 x^{3} B b \,c^{2}+1485 A b \,c^{2} x^{2}+1485 x^{2} B \,b^{2} c +2079 A \,b^{2} c x +693 x B \,b^{3}+1155 A \,b^{3}\right )}{3465}\) | \(76\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (315 B \,c^{3} x^{4}+385 A \,c^{3} x^{3}+1155 x^{3} B b \,c^{2}+1485 A b \,c^{2} x^{2}+1485 x^{2} B \,b^{2} c +2079 A \,b^{2} c x +693 x B \,b^{3}+1155 A \,b^{3}\right )}{3465}\) | \(76\) |
orering | \(\frac {2 \left (315 B \,c^{3} x^{4}+385 A \,c^{3} x^{3}+1155 x^{3} B b \,c^{2}+1485 A b \,c^{2} x^{2}+1485 x^{2} B \,b^{2} c +2079 A \,b^{2} c x +693 x B \,b^{3}+1155 A \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{3}}{3465 x^{\frac {3}{2}} \left (c x +b \right )^{3}}\) | \(94\) |
Input:
int((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3465*x^(3/2)*(315*B*c^3*x^4+385*A*c^3*x^3+1155*B*b*c^2*x^3+1485*A*b*c^2* x^2+1485*B*b^2*c*x^2+2079*A*b^2*c*x+693*B*b^3*x+1155*A*b^3)
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2}{3465} \, {\left (315 \, B c^{3} x^{5} + 1155 \, A b^{3} x + 385 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 1485 \, {\left (B b^{2} c + A b c^{2}\right )} x^{3} + 693 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}\right )} \sqrt {x} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x, algorithm="fricas")
Output:
2/3465*(315*B*c^3*x^5 + 1155*A*b^3*x + 385*(3*B*b*c^2 + A*c^3)*x^4 + 1485* (B*b^2*c + A*b*c^2)*x^3 + 693*(B*b^3 + 3*A*b^2*c)*x^2)*sqrt(x)
Time = 0.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2 A b^{3} x^{\frac {3}{2}}}{3} + \frac {6 A b^{2} c x^{\frac {5}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {7}{2}}}{7} + \frac {2 A c^{3} x^{\frac {9}{2}}}{9} + \frac {2 B b^{3} x^{\frac {5}{2}}}{5} + \frac {6 B b^{2} c x^{\frac {7}{2}}}{7} + \frac {2 B b c^{2} x^{\frac {9}{2}}}{3} + \frac {2 B c^{3} x^{\frac {11}{2}}}{11} \] Input:
integrate((B*x+A)*(c*x**2+b*x)**3/x**(5/2),x)
Output:
2*A*b**3*x**(3/2)/3 + 6*A*b**2*c*x**(5/2)/5 + 6*A*b*c**2*x**(7/2)/7 + 2*A* c**3*x**(9/2)/9 + 2*B*b**3*x**(5/2)/5 + 6*B*b**2*c*x**(7/2)/7 + 2*B*b*c**2 *x**(9/2)/3 + 2*B*c**3*x**(11/2)/11
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2}{11} \, B c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} + \frac {2}{9} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {5}{2}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x, algorithm="maxima")
Output:
2/11*B*c^3*x^(11/2) + 2/3*A*b^3*x^(3/2) + 2/9*(3*B*b*c^2 + A*c^3)*x^(9/2) + 6/7*(B*b^2*c + A*b*c^2)*x^(7/2) + 2/5*(B*b^3 + 3*A*b^2*c)*x^(5/2)
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2}{11} \, B c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B b c^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A c^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B b^{2} c x^{\frac {7}{2}} + \frac {6}{7} \, A b c^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A b^{2} c x^{\frac {5}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x, algorithm="giac")
Output:
2/11*B*c^3*x^(11/2) + 2/3*B*b*c^2*x^(9/2) + 2/9*A*c^3*x^(9/2) + 6/7*B*b^2* c*x^(7/2) + 6/7*A*b*c^2*x^(7/2) + 2/5*B*b^3*x^(5/2) + 6/5*A*b^2*c*x^(5/2) + 2/3*A*b^3*x^(3/2)
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=x^{5/2}\,\left (\frac {2\,B\,b^3}{5}+\frac {6\,A\,c\,b^2}{5}\right )+x^{9/2}\,\left (\frac {2\,A\,c^3}{9}+\frac {2\,B\,b\,c^2}{3}\right )+\frac {2\,A\,b^3\,x^{3/2}}{3}+\frac {2\,B\,c^3\,x^{11/2}}{11}+\frac {6\,b\,c\,x^{7/2}\,\left (A\,c+B\,b\right )}{7} \] Input:
int(((b*x + c*x^2)^3*(A + B*x))/x^(5/2),x)
Output:
x^(5/2)*((2*B*b^3)/5 + (6*A*b^2*c)/5) + x^(9/2)*((2*A*c^3)/9 + (2*B*b*c^2) /3) + (2*A*b^3*x^(3/2))/3 + (2*B*c^3*x^(11/2))/11 + (6*b*c*x^(7/2)*(A*c + B*b))/7
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx=\frac {2 \sqrt {x}\, x \left (315 b \,c^{3} x^{4}+385 a \,c^{3} x^{3}+1155 b^{2} c^{2} x^{3}+1485 a b \,c^{2} x^{2}+1485 b^{3} c \,x^{2}+2079 a \,b^{2} c x +693 b^{4} x +1155 a \,b^{3}\right )}{3465} \] Input:
int((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x)
Output:
(2*sqrt(x)*x*(1155*a*b**3 + 2079*a*b**2*c*x + 1485*a*b*c**2*x**2 + 385*a*c **3*x**3 + 693*b**4*x + 1485*b**3*c*x**2 + 1155*b**2*c**2*x**3 + 315*b*c** 3*x**4))/3465