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

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

[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

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Rubi [A]  time = 0.057281, antiderivative size = 113, normalized size of antiderivative = 1., 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} \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{9} x^{9/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{7} x^{7/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{13} B c^2 x^{13/2} \]

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*}

Mathematica [A]  time = 0.0989268, size = 102, normalized size = 0.9 \[ \frac{2 x^{3/2} \left (3003 a^2 (5 A+3 B x)+286 a x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))+5 x^2 \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )\right )}{45045} \]

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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Maple [A]  time = 0.005, size = 102, normalized size = 0.9 \begin{align*}{\frac{6930\,B{c}^{2}{x}^{5}+8190\,A{c}^{2}{x}^{4}+16380\,B{x}^{4}bc+20020\,A{x}^{3}bc+20020\,aBc{x}^{3}+10010\,{b}^{2}B{x}^{3}+25740\,aAc{x}^{2}+12870\,A{b}^{2}{x}^{2}+25740\,B{x}^{2}ab+36036\,aAbx+18018\,{a}^{2}Bx+30030\,A{a}^{2}}{45045}{x}^{{\frac{3}{2}}}} \end{align*}

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 = 1.02068, size = 126, normalized size = 1.12 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.01035, size = 248, normalized size = 2.19 \begin{align*} \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{align*}

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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Sympy [A]  time = 6.40755, size = 121, normalized size = 1.07 \begin{align*} \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{align*}

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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Giac [A]  time = 1.29029, size = 139, normalized size = 1.23 \begin{align*} \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{align*}

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)