3.14 \(\int x^m (A+B x) (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=71 \[ \frac{A b^2 x^{m+3}}{m+3}+\frac{b x^{m+4} (2 A c+b B)}{m+4}+\frac{c x^{m+5} (A c+2 b B)}{m+5}+\frac{B c^2 x^{m+6}}{m+6} \]

[Out]

(A*b^2*x^(3 + m))/(3 + m) + (b*(b*B + 2*A*c)*x^(4 + m))/(4 + m) + (c*(2*b*B + A*c)*x^(5 + m))/(5 + m) + (B*c^2
*x^(6 + m))/(6 + m)

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Rubi [A]  time = 0.0424106, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{A b^2 x^{m+3}}{m+3}+\frac{b x^{m+4} (2 A c+b B)}{m+4}+\frac{c x^{m+5} (A c+2 b B)}{m+5}+\frac{B c^2 x^{m+6}}{m+6} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(A*b^2*x^(3 + m))/(3 + m) + (b*(b*B + 2*A*c)*x^(4 + m))/(4 + m) + (c*(2*b*B + A*c)*x^(5 + m))/(5 + m) + (B*c^2
*x^(6 + m))/(6 + m)

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^m (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^{2+m}+b (b B+2 A c) x^{3+m}+c (2 b B+A c) x^{4+m}+B c^2 x^{5+m}\right ) \, dx\\ &=\frac{A b^2 x^{3+m}}{3+m}+\frac{b (b B+2 A c) x^{4+m}}{4+m}+\frac{c (2 b B+A c) x^{5+m}}{5+m}+\frac{B c^2 x^{6+m}}{6+m}\\ \end{align*}

Mathematica [A]  time = 0.0917928, size = 71, normalized size = 1. \[ \frac{x^{m+3} \left (\left (\frac{b^2}{m+3}+\frac{2 b c x}{m+4}+\frac{c^2 x^2}{m+5}\right ) (A c (m+6)-b B (m+3))+B (b+c x)^3\right )}{c (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(3 + m)*(B*(b + c*x)^3 + (-(b*B*(3 + m)) + A*c*(6 + m))*(b^2/(3 + m) + (2*b*c*x)/(4 + m) + (c^2*x^2)/(5 + m
))))/(c*(6 + m))

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Maple [B]  time = 0.012, size = 246, normalized size = 3.5 \begin{align*}{\frac{{x}^{3+m} \left ( B{c}^{2}{m}^{3}{x}^{3}+A{c}^{2}{m}^{3}{x}^{2}+2\,Bbc{m}^{3}{x}^{2}+12\,B{c}^{2}{m}^{2}{x}^{3}+2\,Abc{m}^{3}x+13\,A{c}^{2}{m}^{2}{x}^{2}+B{b}^{2}{m}^{3}x+26\,Bbc{m}^{2}{x}^{2}+47\,B{c}^{2}m{x}^{3}+A{b}^{2}{m}^{3}+28\,Abc{m}^{2}x+54\,A{c}^{2}m{x}^{2}+14\,B{b}^{2}{m}^{2}x+108\,Bbcm{x}^{2}+60\,B{c}^{2}{x}^{3}+15\,A{b}^{2}{m}^{2}+126\,Abcmx+72\,A{c}^{2}{x}^{2}+63\,B{b}^{2}mx+144\,Bbc{x}^{2}+74\,A{b}^{2}m+180\,Abcx+90\,{b}^{2}Bx+120\,A{b}^{2} \right ) }{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(3+m)*(B*c^2*m^3*x^3+A*c^2*m^3*x^2+2*B*b*c*m^3*x^2+12*B*c^2*m^2*x^3+2*A*b*c*m^3*x+13*A*c^2*m^2*x^2+B*b^2*m^3
*x+26*B*b*c*m^2*x^2+47*B*c^2*m*x^3+A*b^2*m^3+28*A*b*c*m^2*x+54*A*c^2*m*x^2+14*B*b^2*m^2*x+108*B*b*c*m*x^2+60*B
*c^2*x^3+15*A*b^2*m^2+126*A*b*c*m*x+72*A*c^2*x^2+63*B*b^2*m*x+144*B*b*c*x^2+74*A*b^2*m+180*A*b*c*x+90*B*b^2*x+
120*A*b^2)/(6+m)/(5+m)/(4+m)/(3+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.89503, size = 498, normalized size = 7.01 \begin{align*} \frac{{\left ({\left (B c^{2} m^{3} + 12 \, B c^{2} m^{2} + 47 \, B c^{2} m + 60 \, B c^{2}\right )} x^{6} +{\left ({\left (2 \, B b c + A c^{2}\right )} m^{3} + 144 \, B b c + 72 \, A c^{2} + 13 \,{\left (2 \, B b c + A c^{2}\right )} m^{2} + 54 \,{\left (2 \, B b c + A c^{2}\right )} m\right )} x^{5} +{\left ({\left (B b^{2} + 2 \, A b c\right )} m^{3} + 90 \, B b^{2} + 180 \, A b c + 14 \,{\left (B b^{2} + 2 \, A b c\right )} m^{2} + 63 \,{\left (B b^{2} + 2 \, A b c\right )} m\right )} x^{4} +{\left (A b^{2} m^{3} + 15 \, A b^{2} m^{2} + 74 \, A b^{2} m + 120 \, A b^{2}\right )} x^{3}\right )} x^{m}}{m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*c^2*m^3 + 12*B*c^2*m^2 + 47*B*c^2*m + 60*B*c^2)*x^6 + ((2*B*b*c + A*c^2)*m^3 + 144*B*b*c + 72*A*c^2 + 13*(
2*B*b*c + A*c^2)*m^2 + 54*(2*B*b*c + A*c^2)*m)*x^5 + ((B*b^2 + 2*A*b*c)*m^3 + 90*B*b^2 + 180*A*b*c + 14*(B*b^2
+ 2*A*b*c)*m^2 + 63*(B*b^2 + 2*A*b*c)*m)*x^4 + (A*b^2*m^3 + 15*A*b^2*m^2 + 74*A*b^2*m + 120*A*b^2)*x^3)*x^m/(
m^4 + 18*m^3 + 119*m^2 + 342*m + 360)

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Sympy [A]  time = 1.67619, size = 1027, normalized size = 14.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-A*b**2/(3*x**3) - A*b*c/x**2 - A*c**2/x - B*b**2/(2*x**2) - 2*B*b*c/x + B*c**2*log(x), Eq(m, -6)),
(-A*b**2/(2*x**2) - 2*A*b*c/x + A*c**2*log(x) - B*b**2/x + 2*B*b*c*log(x) + B*c**2*x, Eq(m, -5)), (-A*b**2/x
+ 2*A*b*c*log(x) + A*c**2*x + B*b**2*log(x) + 2*B*b*c*x + B*c**2*x**2/2, Eq(m, -4)), (A*b**2*log(x) + 2*A*b*c*
x + A*c**2*x**2/2 + B*b**2*x + B*b*c*x**2 + B*c**2*x**3/3, Eq(m, -3)), (A*b**2*m**3*x**3*x**m/(m**4 + 18*m**3
+ 119*m**2 + 342*m + 360) + 15*A*b**2*m**2*x**3*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 74*A*b**2*m*x
**3*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 120*A*b**2*x**3*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m +
360) + 2*A*b*c*m**3*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 28*A*b*c*m**2*x**4*x**m/(m**4 + 18*
m**3 + 119*m**2 + 342*m + 360) + 126*A*b*c*m*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 180*A*b*c*x
**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + A*c**2*m**3*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m
+ 360) + 13*A*c**2*m**2*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 54*A*c**2*m*x**5*x**m/(m**4 + 18
*m**3 + 119*m**2 + 342*m + 360) + 72*A*c**2*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + B*b**2*m**3*
x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 14*B*b**2*m**2*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 34
2*m + 360) + 63*B*b**2*m*x**4*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 90*B*b**2*x**4*x**m/(m**4 + 18*
m**3 + 119*m**2 + 342*m + 360) + 2*B*b*c*m**3*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 26*B*b*c*m
**2*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 108*B*b*c*m*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 3
42*m + 360) + 144*B*b*c*x**5*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + B*c**2*m**3*x**6*x**m/(m**4 + 18
*m**3 + 119*m**2 + 342*m + 360) + 12*B*c**2*m**2*x**6*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 47*B*c*
*2*m*x**6*x**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 60*B*c**2*x**6*x**m/(m**4 + 18*m**3 + 119*m**2 + 34
2*m + 360), True))

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Giac [B]  time = 1.13678, size = 459, normalized size = 6.46 \begin{align*} \frac{B c^{2} m^{3} x^{6} x^{m} + 2 \, B b c m^{3} x^{5} x^{m} + A c^{2} m^{3} x^{5} x^{m} + 12 \, B c^{2} m^{2} x^{6} x^{m} + B b^{2} m^{3} x^{4} x^{m} + 2 \, A b c m^{3} x^{4} x^{m} + 26 \, B b c m^{2} x^{5} x^{m} + 13 \, A c^{2} m^{2} x^{5} x^{m} + 47 \, B c^{2} m x^{6} x^{m} + A b^{2} m^{3} x^{3} x^{m} + 14 \, B b^{2} m^{2} x^{4} x^{m} + 28 \, A b c m^{2} x^{4} x^{m} + 108 \, B b c m x^{5} x^{m} + 54 \, A c^{2} m x^{5} x^{m} + 60 \, B c^{2} x^{6} x^{m} + 15 \, A b^{2} m^{2} x^{3} x^{m} + 63 \, B b^{2} m x^{4} x^{m} + 126 \, A b c m x^{4} x^{m} + 144 \, B b c x^{5} x^{m} + 72 \, A c^{2} x^{5} x^{m} + 74 \, A b^{2} m x^{3} x^{m} + 90 \, B b^{2} x^{4} x^{m} + 180 \, A b c x^{4} x^{m} + 120 \, A b^{2} x^{3} x^{m}}{m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*c^2*m^3*x^6*x^m + 2*B*b*c*m^3*x^5*x^m + A*c^2*m^3*x^5*x^m + 12*B*c^2*m^2*x^6*x^m + B*b^2*m^3*x^4*x^m + 2*A*
b*c*m^3*x^4*x^m + 26*B*b*c*m^2*x^5*x^m + 13*A*c^2*m^2*x^5*x^m + 47*B*c^2*m*x^6*x^m + A*b^2*m^3*x^3*x^m + 14*B*
b^2*m^2*x^4*x^m + 28*A*b*c*m^2*x^4*x^m + 108*B*b*c*m*x^5*x^m + 54*A*c^2*m*x^5*x^m + 60*B*c^2*x^6*x^m + 15*A*b^
2*m^2*x^3*x^m + 63*B*b^2*m*x^4*x^m + 126*A*b*c*m*x^4*x^m + 144*B*b*c*x^5*x^m + 72*A*c^2*x^5*x^m + 74*A*b^2*m*x
^3*x^m + 90*B*b^2*x^4*x^m + 180*A*b*c*x^4*x^m + 120*A*b^2*x^3*x^m)/(m^4 + 18*m^3 + 119*m^2 + 342*m + 360)