Integrand size = 23, antiderivative size = 109 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=-\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a (2 A b+a B)}{5 x^{5/2}}-\frac {2 \left (2 a b B+A \left (b^2+2 a c\right )\right )}{3 x^{3/2}}-\frac {2 \left (b^2 B+2 A b c+2 a B c\right )}{\sqrt {x}}+2 c (2 b B+A c) \sqrt {x}+\frac {2}{3} B c^2 x^{3/2} \] Output:
-2/7*a^2*A/x^(7/2)-2/5*a*(2*A*b+B*a)/x^(5/2)-2/3*(2*a*b*B+A*(2*a*c+b^2))/x ^(3/2)-2*(2*A*b*c+2*B*a*c+B*b^2)/x^(1/2)+2*c*(A*c+2*B*b)*x^(1/2)+2/3*B*c^2 *x^(3/2)
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=-\frac {2 \left (3 a^2 (5 A+7 B x)+14 a x (5 B x (b+3 c x)+A (3 b+5 c x))+35 x^2 \left (A \left (b^2+6 b c x-3 c^2 x^2\right )-B x \left (-3 b^2+6 b c x+c^2 x^2\right )\right )\right )}{105 x^{7/2}} \] Input:
Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^(9/2),x]
Output:
(-2*(3*a^2*(5*A + 7*B*x) + 14*a*x*(5*B*x*(b + 3*c*x) + A*(3*b + 5*c*x)) + 35*x^2*(A*(b^2 + 6*b*c*x - 3*c^2*x^2) - B*x*(-3*b^2 + 6*b*c*x + c^2*x^2))) )/(105*x^(7/2))
Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1195, 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.
\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {a^2 A}{x^{9/2}}+\frac {2 a B c+2 A b c+b^2 B}{x^{3/2}}+\frac {A \left (2 a c+b^2\right )+2 a b B}{x^{5/2}}+\frac {a (a B+2 A b)}{x^{7/2}}+\frac {c (A c+2 b B)}{\sqrt {x}}+B c^2 \sqrt {x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{3 x^{3/2}}-\frac {2 \left (2 a B c+2 A b c+b^2 B\right )}{\sqrt {x}}-\frac {2 a (a B+2 A b)}{5 x^{5/2}}+2 c \sqrt {x} (A c+2 b B)+\frac {2}{3} B c^2 x^{3/2}\) |
Input:
Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^(9/2),x]
Output:
(-2*a^2*A)/(7*x^(7/2)) - (2*a*(2*A*b + a*B))/(5*x^(5/2)) - (2*(2*a*b*B + A *(b^2 + 2*a*c)))/(3*x^(3/2)) - (2*(b^2*B + 2*A*b*c + 2*a*B*c))/Sqrt[x] + 2 *c*(2*b*B + A*c)*Sqrt[x] + (2*B*c^2*x^(3/2))/3
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {2 B \,c^{2} x^{\frac {3}{2}}}{3}+2 A \,c^{2} \sqrt {x}+4 B b c \sqrt {x}-\frac {2 \left (2 A b c +2 a B c +B \,b^{2}\right )}{\sqrt {x}}-\frac {2 \left (2 A a c +b^{2} A +2 a b B \right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{2} A}{7 x^{\frac {7}{2}}}-\frac {2 a \left (2 A b +B a \right )}{5 x^{\frac {5}{2}}}\) | \(93\) |
default | \(\frac {2 B \,c^{2} x^{\frac {3}{2}}}{3}+2 A \,c^{2} \sqrt {x}+4 B b c \sqrt {x}-\frac {2 \left (2 A b c +2 a B c +B \,b^{2}\right )}{\sqrt {x}}-\frac {2 \left (2 A a c +b^{2} A +2 a b B \right )}{3 x^{\frac {3}{2}}}-\frac {2 a^{2} A}{7 x^{\frac {7}{2}}}-\frac {2 a \left (2 A b +B a \right )}{5 x^{\frac {5}{2}}}\) | \(93\) |
gosper | \(-\frac {2 \left (-35 B \,c^{2} x^{5}-105 x^{4} A \,c^{2}-210 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+70 A a c \,x^{2}+35 x^{2} b^{2} A +70 B a \,x^{2} b +42 a b A x +21 a^{2} B x +15 a^{2} A \right )}{105 x^{\frac {7}{2}}}\) | \(102\) |
trager | \(-\frac {2 \left (-35 B \,c^{2} x^{5}-105 x^{4} A \,c^{2}-210 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+70 A a c \,x^{2}+35 x^{2} b^{2} A +70 B a \,x^{2} b +42 a b A x +21 a^{2} B x +15 a^{2} A \right )}{105 x^{\frac {7}{2}}}\) | \(102\) |
risch | \(-\frac {2 \left (-35 B \,c^{2} x^{5}-105 x^{4} A \,c^{2}-210 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+70 A a c \,x^{2}+35 x^{2} b^{2} A +70 B a \,x^{2} b +42 a b A x +21 a^{2} B x +15 a^{2} A \right )}{105 x^{\frac {7}{2}}}\) | \(102\) |
orering | \(-\frac {2 \left (-35 B \,c^{2} x^{5}-105 x^{4} A \,c^{2}-210 x^{4} B b c +210 x^{3} A b c +210 B a c \,x^{3}+105 x^{3} B \,b^{2}+70 A a c \,x^{2}+35 x^{2} b^{2} A +70 B a \,x^{2} b +42 a b A x +21 a^{2} B x +15 a^{2} A \right )}{105 x^{\frac {7}{2}}}\) | \(102\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^2/x^(9/2),x,method=_RETURNVERBOSE)
Output:
2/3*B*c^2*x^(3/2)+2*A*c^2*x^(1/2)+4*B*b*c*x^(1/2)-2*(2*A*b*c+2*B*a*c+B*b^2 )/x^(1/2)-2/3*(2*A*a*c+A*b^2+2*B*a*b)/x^(3/2)-2/7*a^2*A/x^(7/2)-2/5*a*(2*A *b+B*a)/x^(5/2)
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=\frac {2 \, {\left (35 \, B c^{2} x^{5} + 105 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} - 105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 15 \, A a^{2} - 35 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 21 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2/x^(9/2),x, algorithm="fricas")
Output:
2/105*(35*B*c^2*x^5 + 105*(2*B*b*c + A*c^2)*x^4 - 105*(B*b^2 + 2*(B*a + A* b)*c)*x^3 - 15*A*a^2 - 35*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 - 21*(B*a^2 + 2* A*a*b)*x)/x^(7/2)
Time = 0.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=- \frac {2 A a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 A a b}{5 x^{\frac {5}{2}}} - \frac {4 A a c}{3 x^{\frac {3}{2}}} - \frac {2 A b^{2}}{3 x^{\frac {3}{2}}} - \frac {4 A b c}{\sqrt {x}} + 2 A c^{2} \sqrt {x} - \frac {2 B a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 B a b}{3 x^{\frac {3}{2}}} - \frac {4 B a c}{\sqrt {x}} - \frac {2 B b^{2}}{\sqrt {x}} + 4 B b c \sqrt {x} + \frac {2 B c^{2} x^{\frac {3}{2}}}{3} \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**2/x**(9/2),x)
Output:
-2*A*a**2/(7*x**(7/2)) - 4*A*a*b/(5*x**(5/2)) - 4*A*a*c/(3*x**(3/2)) - 2*A *b**2/(3*x**(3/2)) - 4*A*b*c/sqrt(x) + 2*A*c**2*sqrt(x) - 2*B*a**2/(5*x**( 5/2)) - 4*B*a*b/(3*x**(3/2)) - 4*B*a*c/sqrt(x) - 2*B*b**2/sqrt(x) + 4*B*b* c*sqrt(x) + 2*B*c**2*x**(3/2)/3
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=\frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 2 \, {\left (2 \, B b c + A c^{2}\right )} \sqrt {x} - \frac {2 \, {\left (105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 15 \, A a^{2} + 35 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2/x^(9/2),x, algorithm="maxima")
Output:
2/3*B*c^2*x^(3/2) + 2*(2*B*b*c + A*c^2)*sqrt(x) - 2/105*(105*(B*b^2 + 2*(B *a + A*b)*c)*x^3 + 15*A*a^2 + 35*(2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 21*(B*a ^2 + 2*A*a*b)*x)/x^(7/2)
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=\frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 4 \, B b c \sqrt {x} + 2 \, A c^{2} \sqrt {x} - \frac {2 \, {\left (105 \, B b^{2} x^{3} + 210 \, B a c x^{3} + 210 \, A b c x^{3} + 70 \, B a b x^{2} + 35 \, A b^{2} x^{2} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 42 \, A a b x + 15 \, A a^{2}\right )}}{105 \, x^{\frac {7}{2}}} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2/x^(9/2),x, algorithm="giac")
Output:
2/3*B*c^2*x^(3/2) + 4*B*b*c*sqrt(x) + 2*A*c^2*sqrt(x) - 2/105*(105*B*b^2*x ^3 + 210*B*a*c*x^3 + 210*A*b*c*x^3 + 70*B*a*b*x^2 + 35*A*b^2*x^2 + 70*A*a* c*x^2 + 21*B*a^2*x + 42*A*a*b*x + 15*A*a^2)/x^(7/2)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=\sqrt {x}\,\left (2\,A\,c^2+4\,B\,b\,c\right )-\frac {\frac {2\,A\,a^2}{7}+x^2\,\left (\frac {2\,A\,b^2}{3}+\frac {4\,B\,a\,b}{3}+\frac {4\,A\,a\,c}{3}\right )+x^3\,\left (2\,B\,b^2+4\,A\,c\,b+4\,B\,a\,c\right )+x\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )}{x^{7/2}}+\frac {2\,B\,c^2\,x^{3/2}}{3} \] Input:
int(((A + B*x)*(a + b*x + c*x^2)^2)/x^(9/2),x)
Output:
x^(1/2)*(2*A*c^2 + 4*B*b*c) - ((2*A*a^2)/7 + x^2*((2*A*b^2)/3 + (4*A*a*c)/ 3 + (4*B*a*b)/3) + x^3*(2*B*b^2 + 4*A*b*c + 4*B*a*c) + x*((2*B*a^2)/5 + (4 *A*a*b)/5))/x^(7/2) + (2*B*c^2*x^(3/2))/3
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx=\frac {\frac {2}{3} b \,c^{2} x^{5}+2 a \,c^{2} x^{4}+4 b^{2} c \,x^{4}-8 a b c \,x^{3}-2 b^{3} x^{3}-\frac {4}{3} a^{2} c \,x^{2}-2 a \,b^{2} x^{2}-\frac {6}{5} a^{2} b x -\frac {2}{7} a^{3}}{\sqrt {x}\, x^{3}} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^2/x^(9/2),x)
Output:
(2*( - 15*a**3 - 63*a**2*b*x - 70*a**2*c*x**2 - 105*a*b**2*x**2 - 420*a*b* c*x**3 + 105*a*c**2*x**4 - 105*b**3*x**3 + 210*b**2*c*x**4 + 35*b*c**2*x** 5))/(105*sqrt(x)*x**3)