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

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

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.10, size = 71, normalized size = 1.00 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int x^m (A+B x) \left (b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 217, normalized size = 3.06 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.17, size = 340, normalized size = 4.79 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.06, size = 246, normalized size = 3.46 \begin {gather*} \frac {\left (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}\right ) x^{m +3}}{\left (m +6\right ) \left (m +5\right ) \left (m +4\right ) \left (m +3\right )} \end {gather*}

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^(m+3)*(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)/(m+6)/(m+5)/(m+4)/(m+3)

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maxima [A]  time = 0.86, size = 91, normalized size = 1.28 \begin {gather*} \frac {B c^{2} x^{m + 6}}{m + 6} + \frac {2 \, B b c x^{m + 5}}{m + 5} + \frac {A c^{2} x^{m + 5}}{m + 5} + \frac {B b^{2} x^{m + 4}}{m + 4} + \frac {2 \, A b c x^{m + 4}}{m + 4} + \frac {A b^{2} x^{m + 3}}{m + 3} \end {gather*}

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]

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

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mupad [B]  time = 1.20, size = 179, normalized size = 2.52 \begin {gather*} x^m\,\left (\frac {A\,b^2\,x^3\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {B\,c^2\,x^6\,\left (m^3+12\,m^2+47\,m+60\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {b\,x^4\,\left (2\,A\,c+B\,b\right )\,\left (m^3+14\,m^2+63\,m+90\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {c\,x^5\,\left (A\,c+2\,B\,b\right )\,\left (m^3+13\,m^2+54\,m+72\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x + c*x^2)^2*(A + B*x),x)

[Out]

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

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sympy [A]  time = 1.55, size = 1027, normalized size = 14.46 \begin {gather*} \begin {cases} - \frac {A b^{2}}{3 x^{3}} - \frac {A b c}{x^{2}} - \frac {A c^{2}}{x} - \frac {B b^{2}}{2 x^{2}} - \frac {2 B b c}{x} + B c^{2} \log {\relax (x )} & \text {for}\: m = -6 \\- \frac {A b^{2}}{2 x^{2}} - \frac {2 A b c}{x} + A c^{2} \log {\relax (x )} - \frac {B b^{2}}{x} + 2 B b c \log {\relax (x )} + B c^{2} x & \text {for}\: m = -5 \\- \frac {A b^{2}}{x} + 2 A b c \log {\relax (x )} + A c^{2} x + B b^{2} \log {\relax (x )} + 2 B b c x + \frac {B c^{2} x^{2}}{2} & \text {for}\: m = -4 \\A b^{2} \log {\relax (x )} + 2 A b c x + \frac {A c^{2} x^{2}}{2} + B b^{2} x + B b c x^{2} + \frac {B c^{2} x^{3}}{3} & \text {for}\: m = -3 \\\frac {A b^{2} m^{3} x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {15 A b^{2} m^{2} x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {74 A b^{2} m x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {120 A b^{2} x^{3} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {2 A b c m^{3} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {28 A b c m^{2} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {126 A b c m x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {180 A b c x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {A c^{2} m^{3} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {13 A c^{2} m^{2} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {54 A c^{2} m x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {72 A c^{2} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {B b^{2} m^{3} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {14 B b^{2} m^{2} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {63 B b^{2} m x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {90 B b^{2} x^{4} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {2 B b c m^{3} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {26 B b c m^{2} x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {108 B b c m x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {144 B b c x^{5} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {B c^{2} m^{3} x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {12 B c^{2} m^{2} x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {47 B c^{2} m x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} + \frac {60 B c^{2} x^{6} x^{m}}{m^{4} + 18 m^{3} + 119 m^{2} + 342 m + 360} & \text {otherwise} \end {cases} \end {gather*}

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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