3.149 \(\int x^{7/2} (A+B x) (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(17/2))/17 + (2*B*c^2*x^(19/2))
/19

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Rubi [A]  time = 0.0333596, 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} \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(17/2))/17 + (2*B*c^2*x^(19/2))
/19

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 x^{7/2} (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^{11/2}+b (b B+2 A c) x^{13/2}+c (2 b B+A c) x^{15/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{13} A b^2 x^{13/2}+\frac{2}{15} b (b B+2 A c) x^{15/2}+\frac{2}{17} c (2 b B+A c) x^{17/2}+\frac{2}{19} B c^2 x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0192609, size = 55, normalized size = 0.87 \[ \frac{2 x^{13/2} \left (19 A \left (255 b^2+442 b c x+195 c^2 x^2\right )+13 B x \left (323 b^2+570 b c x+255 c^2 x^2\right )\right )}{62985} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*x^(13/2)*(19*A*(255*b^2 + 442*b*c*x + 195*c^2*x^2) + 13*B*x*(323*b^2 + 570*b*c*x + 255*c^2*x^2)))/62985

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Maple [A]  time = 0.004, size = 52, normalized size = 0.8 \begin{align*}{\frac{6630\,B{c}^{2}{x}^{3}+7410\,A{c}^{2}{x}^{2}+14820\,B{x}^{2}bc+16796\,Abcx+8398\,{b}^{2}Bx+9690\,A{b}^{2}}{62985}{x}^{{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)*(c*x^2+b*x)^2,x)

[Out]

2/62985*x^(13/2)*(3315*B*c^2*x^3+3705*A*c^2*x^2+7410*B*b*c*x^2+8398*A*b*c*x+4199*B*b^2*x+4845*A*b^2)

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Maxima [A]  time = 1.12262, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{17} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{15}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/19*B*c^2*x^(19/2) + 2/13*A*b^2*x^(13/2) + 2/17*(2*B*b*c + A*c^2)*x^(17/2) + 2/15*(B*b^2 + 2*A*b*c)*x^(15/2)

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Fricas [A]  time = 1.71771, size = 147, normalized size = 2.33 \begin{align*} \frac{2}{62985} \,{\left (3315 \, B c^{2} x^{9} + 4845 \, A b^{2} x^{6} + 3705 \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + 4199 \,{\left (B b^{2} + 2 \, A b c\right )} x^{7}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/62985*(3315*B*c^2*x^9 + 4845*A*b^2*x^6 + 3705*(2*B*b*c + A*c^2)*x^8 + 4199*(B*b^2 + 2*A*b*c)*x^7)*sqrt(x)

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Sympy [A]  time = 17.42, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A b c x^{\frac{15}{2}}}{15} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} + \frac{4 B b c x^{\frac{17}{2}}}{17} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**2*x**(13/2)/13 + 4*A*b*c*x**(15/2)/15 + 2*A*c**2*x**(17/2)/17 + 2*B*b**2*x**(15/2)/15 + 4*B*b*c*x**(17/
2)/17 + 2*B*c**2*x**(19/2)/19

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Giac [A]  time = 1.11315, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{4}{17} \, B b c x^{\frac{17}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, A b c x^{\frac{15}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/19*B*c^2*x^(19/2) + 4/17*B*b*c*x^(17/2) + 2/17*A*c^2*x^(17/2) + 2/15*B*b^2*x^(15/2) + 4/15*A*b*c*x^(15/2) +
2/13*A*b^2*x^(13/2)