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

Optimal. Leaf size=113 \[ \frac {2}{3} a^2 A x^{3/2}+\frac {2}{9} x^{9/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{7} x^{7/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{5} a x^{5/2} (a B+2 A b)+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{13} B c^2 x^{13/2} \]

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Rubi [A]  time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {765} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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 765

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*} \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\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*}

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Mathematica [A]  time = 0.09, size = 102, normalized size = 0.90 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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

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IntegrateAlgebraic [A]  time = 0.06, size = 131, normalized size = 1.16 \begin {gather*} \frac {2 \left (15015 a^2 A x^{3/2}+9009 a^2 B x^{5/2}+18018 a A b x^{5/2}+12870 a A c x^{7/2}+12870 a b B x^{7/2}+10010 a B c x^{9/2}+6435 A b^2 x^{7/2}+10010 A b c x^{9/2}+4095 A c^2 x^{11/2}+5005 b^2 B x^{9/2}+8190 b B c x^{11/2}+3465 B c^2 x^{13/2}\right )}{45045} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(15015*a^2*A*x^(3/2) + 18018*a*A*b*x^(5/2) + 9009*a^2*B*x^(5/2) + 6435*A*b^2*x^(7/2) + 12870*a*b*B*x^(7/2)
+ 12870*a*A*c*x^(7/2) + 5005*b^2*B*x^(9/2) + 10010*A*b*c*x^(9/2) + 10010*a*B*c*x^(9/2) + 8190*b*B*c*x^(11/2) +
4095*A*c^2*x^(11/2) + 3465*B*c^2*x^(13/2)))/45045

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fricas [A]  time = 0.42, size = 96, normalized size = 0.85 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.16, size = 103, normalized size = 0.91 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 102, normalized size = 0.90 \begin {gather*} \frac {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 B a 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 B \,a^{2} x +15015 A \,a^{2}\right ) x^{\frac {3}{2}}}{45045} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*x^(3/2)*(3465*B*c^2*x^5+4095*A*c^2*x^4+8190*B*b*c*x^4+10010*A*b*c*x^3+10010*B*a*c*x^3+5005*B*b^2*x^3+1
2870*A*a*c*x^2+6435*A*b^2*x^2+12870*B*a*b*x^2+18018*A*a*b*x+9009*B*a^2*x+15015*A*a^2)

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maxima [A]  time = 0.50, size = 93, normalized size = 0.82 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.04, size = 93, normalized size = 0.82 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 5.60, size = 121, normalized size = 1.07 \begin {gather*} \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}} \left (2 A b c + 2 B a c + B b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \left (2 A a c + A b^{2} + 2 B a b\right )}{7} + \frac {2 x^{\frac {5}{2}} \left (2 A a b + B a^{2}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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