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

Optimal. Leaf size=63 \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{13} B c^2 x^{13/2} \]

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*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 = 0.0296317, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{13} B c^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*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 (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^{5/2}+b (b B+2 A c) x^{7/2}+c (2 b B+A c) x^{9/2}+B c^2 x^{11/2}\right ) \, dx\\ &=\frac{2}{7} A b^2 x^{7/2}+\frac{2}{9} b (b B+2 A c) 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 = 0.0174325, size = 55, normalized size = 0.87 \[ \frac{2 x^{7/2} \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )}{9009} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*(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)))/9009

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Maple [A]  time = 0.004, size = 52, normalized size = 0.8 \begin{align*}{\frac{1386\,B{c}^{2}{x}^{3}+1638\,A{c}^{2}{x}^{2}+3276\,B{x}^{2}bc+4004\,Abcx+2002\,{b}^{2}Bx+2574\,A{b}^{2}}{9009}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*x^(7/2)*(693*B*c^2*x^3+819*A*c^2*x^2+1638*B*b*c*x^2+2002*A*b*c*x+1001*B*b^2*x+1287*A*b^2)

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Maxima [A]  time = 1.04582, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{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 \, A b c\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.81605, size = 143, normalized size = 2.27 \begin{align*} \frac{2}{9009} \,{\left (693 \, B c^{2} x^{6} + 1287 \, A b^{2} x^{3} + 819 \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + 1001 \,{\left (B b^{2} + 2 \, A b c\right )} x^{4}\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)^2*x^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*B*c^2*x^6 + 1287*A*b^2*x^3 + 819*(2*B*b*c + A*c^2)*x^5 + 1001*(B*b^2 + 2*A*b*c)*x^4)*sqrt(x)

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Sympy [A]  time = 2.59119, size = 66, normalized size = 1.05 \begin{align*} \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \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 + B b^{2}\right )}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**2*x**(7/2)/7 + 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 + B*b**2
)/9

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Giac [A]  time = 1.12982, size = 72, normalized size = 1.14 \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} \, A b c x^{\frac{9}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^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*A*b*c*x^(9/2) + 2/7*
A*b^2*x^(7/2)