\(\int \sqrt {x} (A+B x) (a+b x+c x^2)^2 \, dx\) [993]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{7} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{7/2}+\frac {2}{9} \left (b^2 B+2 A b c+2 a B c\right ) x^{9/2}+\frac {2}{11} c (2 b B+A c) x^{11/2}+\frac {2}{13} B c^2 x^{13/2} \]

[Out]

2/3*a^2*A*x^(3/2)+2/5*a*(2*A*b+B*a)*x^(5/2)+2/7*(2*a*b*B+A*(2*a*c+b^2))*x^(7/2)+2/9*(2*A*b*c+2*B*a*c+B*b^2)*x^
(9/2)+2/11*c*(A*c+2*B*b)*x^(11/2)+2/13*B*c^2*x^(13/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {779} \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{9} x^{9/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{7} x^{7/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{5} a x^{5/2} (a B+2 A b)+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{13} B c^2 x^{13/2} \]

[In]

Int[Sqrt[x]*(A + B*x)*(a + b*x + c*x^2)^2,x]

[Out]

(2*a^2*A*x^(3/2))/3 + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*(2*a*b*B + A*(b^2 + 2*a*c))*x^(7/2))/7 + (2*(b^2*B +
2*A*b*c + 2*a*B*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(11/2))/11 + (2*B*c^2*x^(13/2))/13

Rule 779

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A \sqrt {x}+a (2 A b+a B) x^{3/2}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{5/2}+\left (b^2 B+2 A b c+2 a B c\right ) x^{7/2}+c (2 b B+A c) x^{9/2}+B c^2 x^{11/2}\right ) \, dx \\ & = \frac {2}{3} a^2 A x^{3/2}+\frac {2}{5} a (2 A b+a B) x^{5/2}+\frac {2}{7} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{7/2}+\frac {2}{9} \left (b^2 B+2 A b c+2 a B c\right ) x^{9/2}+\frac {2}{11} c (2 b B+A c) x^{11/2}+\frac {2}{13} B c^2 x^{13/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2 x^{3/2} \left (3003 a^2 (5 A+3 B x)+286 a x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))+5 x^2 \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )\right )}{45045} \]

[In]

Integrate[Sqrt[x]*(A + B*x)*(a + b*x + c*x^2)^2,x]

[Out]

(2*x^(3/2)*(3003*a^2*(5*A + 3*B*x) + 286*a*x*(9*A*(7*b + 5*c*x) + 5*B*x*(9*b + 7*c*x)) + 5*x^2*(13*A*(99*b^2 +
154*b*c*x + 63*c^2*x^2) + 7*B*x*(143*b^2 + 234*b*c*x + 99*c^2*x^2))))/45045

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {2 B \,c^{2} x^{\frac {13}{2}}}{13}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\)	\(94\)
default	\(\frac {2 B \,c^{2} x^{\frac {13}{2}}}{13}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (2 A b a +B \,a^{2}\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\)	\(94\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5}+4095 A \,c^{2} x^{4}+8190 x^{4} B b c +10010 x^{3} A b c +10010 a B c \,x^{3}+5005 B \,b^{2} x^{3}+12870 a A c \,x^{2}+6435 A \,b^{2} x^{2}+12870 B a b \,x^{2}+18018 a A b x +9009 a^{2} B x +15015 A \,a^{2}\right )}{45045}\)	\(102\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5}+4095 A \,c^{2} x^{4}+8190 x^{4} B b c +10010 x^{3} A b c +10010 a B c \,x^{3}+5005 B \,b^{2} x^{3}+12870 a A c \,x^{2}+6435 A \,b^{2} x^{2}+12870 B a b \,x^{2}+18018 a A b x +9009 a^{2} B x +15015 A \,a^{2}\right )}{45045}\)	\(102\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5}+4095 A \,c^{2} x^{4}+8190 x^{4} B b c +10010 x^{3} A b c +10010 a B c \,x^{3}+5005 B \,b^{2} x^{3}+12870 a A c \,x^{2}+6435 A \,b^{2} x^{2}+12870 B a b \,x^{2}+18018 a A b x +9009 a^{2} B x +15015 A \,a^{2}\right )}{45045}\)	\(102\)

[In]

int((B*x+A)*(c*x^2+b*x+a)^2*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/13*B*c^2*x^(13/2)+2/11*(A*c^2+2*B*b*c)*x^(11/2)+2/9*(2*A*b*c+B*(2*a*c+b^2))*x^(9/2)+2/7*(2*a*b*B+A*(2*a*c+b^
2))*x^(7/2)+2/5*(2*A*a*b+B*a^2)*x^(5/2)+2/3*a^2*A*x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{45045} \, {\left (3465 \, B c^{2} x^{6} + 4095 \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + 5005 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{4} + 15015 \, A a^{2} x + 6435 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + 9009 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt {x} \]

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2*x^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*B*c^2*x^6 + 4095*(2*B*b*c + A*c^2)*x^5 + 5005*(B*b^2 + 2*(B*a + A*b)*c)*x^4 + 15015*A*a^2*x + 64
35*(2*B*a*b + A*b^2 + 2*A*a*c)*x^3 + 9009*(B*a^2 + 2*A*a*b)*x^2)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.07 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2 A a^{2} x^{\frac {3}{2}}}{3} + \frac {2 B c^{2} x^{\frac {13}{2}}}{13} + \frac {2 x^{\frac {11}{2}} \left (A c^{2} + 2 B b c\right )}{11} + \frac {2 x^{\frac {9}{2}} \cdot \left (2 A b c + 2 B a c + B b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \cdot \left (2 A a c + A b^{2} + 2 B a b\right )}{7} + \frac {2 x^{\frac {5}{2}} \cdot \left (2 A a b + B a^{2}\right )}{5} \]

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**2*x**(1/2),x)

[Out]

2*A*a**2*x**(3/2)/3 + 2*B*c**2*x**(13/2)/13 + 2*x**(11/2)*(A*c**2 + 2*B*b*c)/11 + 2*x**(9/2)*(2*A*b*c + 2*B*a*
c + B*b**2)/9 + 2*x**(7/2)*(2*A*a*c + A*b**2 + 2*B*a*b)/7 + 2*x**(5/2)*(2*A*a*b + B*a**2)/5

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{13} \, B c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} + \frac {2}{7} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {5}{2}} \]

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2*x^(1/2),x, algorithm="maxima")

[Out]

2/13*B*c^2*x^(13/2) + 2/11*(2*B*b*c + A*c^2)*x^(11/2) + 2/9*(B*b^2 + 2*(B*a + A*b)*c)*x^(9/2) + 2/3*A*a^2*x^(3
/2) + 2/7*(2*B*a*b + A*b^2 + 2*A*a*c)*x^(7/2) + 2/5*(B*a^2 + 2*A*a*b)*x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{13} \, B c^{2} x^{\frac {13}{2}} + \frac {4}{11} \, B b c x^{\frac {11}{2}} + \frac {2}{11} \, A c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, B a c x^{\frac {9}{2}} + \frac {4}{9} \, A b c x^{\frac {9}{2}} + \frac {4}{7} \, B a b x^{\frac {7}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A a c x^{\frac {7}{2}} + \frac {2}{5} \, B a^{2} x^{\frac {5}{2}} + \frac {4}{5} \, A a b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} \]

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2*x^(1/2),x, algorithm="giac")

[Out]

2/13*B*c^2*x^(13/2) + 4/11*B*b*c*x^(11/2) + 2/11*A*c^2*x^(11/2) + 2/9*B*b^2*x^(9/2) + 4/9*B*a*c*x^(9/2) + 4/9*
A*b*c*x^(9/2) + 4/7*B*a*b*x^(7/2) + 2/7*A*b^2*x^(7/2) + 4/7*A*a*c*x^(7/2) + 2/5*B*a^2*x^(5/2) + 4/5*A*a*b*x^(5
/2) + 2/3*A*a^2*x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx=x^{5/2}\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )+x^{11/2}\,\left (\frac {2\,A\,c^2}{11}+\frac {4\,B\,b\,c}{11}\right )+x^{7/2}\,\left (\frac {2\,A\,b^2}{7}+\frac {4\,B\,a\,b}{7}+\frac {4\,A\,a\,c}{7}\right )+x^{9/2}\,\left (\frac {2\,B\,b^2}{9}+\frac {4\,A\,c\,b}{9}+\frac {4\,B\,a\,c}{9}\right )+\frac {2\,A\,a^2\,x^{3/2}}{3}+\frac {2\,B\,c^2\,x^{13/2}}{13} \]

[In]

int(x^(1/2)*(A + B*x)*(a + b*x + c*x^2)^2,x)

[Out]

x^(5/2)*((2*B*a^2)/5 + (4*A*a*b)/5) + x^(11/2)*((2*A*c^2)/11 + (4*B*b*c)/11) + x^(7/2)*((2*A*b^2)/7 + (4*A*a*c
)/7 + (4*B*a*b)/7) + x^(9/2)*((2*B*b^2)/9 + (4*A*b*c)/9 + (4*B*a*c)/9) + (2*A*a^2*x^(3/2))/3 + (2*B*c^2*x^(13/
2))/13