3.2.13 \(\int (d x)^m (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=58 \[ \frac {b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac {2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac {c^2 (d x)^{m+5}}{d^5 (m+5)} \]

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Rubi [A]  time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {647, 43} \begin {gather*} \frac {b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac {2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac {c^2 (d x)^{m+5}}{d^5 (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(b*x + c*x^2)^2,x]

[Out]

(b^2*(d*x)^(3 + m))/(d^3*(3 + m)) + (2*b*c*(d*x)^(4 + m))/(d^4*(4 + m)) + (c^2*(d*x)^(5 + m))/(d^5*(5 + m))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 647

Int[((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e^p, Int[(e*x)^(m + p)*(b + c*x)
^p, x], x] /; FreeQ[{b, c, e, m}, x] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 41, normalized size = 0.71 \begin {gather*} x^3 (d x)^m \left (\frac {b^2}{m+3}+\frac {2 b c x}{m+4}+\frac {c^2 x^2}{m+5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(b*x + c*x^2)^2,x]

[Out]

x^3*(d*x)^m*(b^2/(3 + m) + (2*b*c*x)/(4 + m) + (c^2*x^2)/(5 + m))

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IntegrateAlgebraic [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (b x+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 89, normalized size = 1.53 \begin {gather*} \frac {{\left ({\left (c^{2} m^{2} + 7 \, c^{2} m + 12 \, c^{2}\right )} x^{5} + 2 \, {\left (b c m^{2} + 8 \, b c m + 15 \, b c\right )} x^{4} + {\left (b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}\right )} x^{3}\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c^2*m^2 + 7*c^2*m + 12*c^2)*x^5 + 2*(b*c*m^2 + 8*b*c*m + 15*b*c)*x^4 + (b^2*m^2 + 9*b^2*m + 20*b^2)*x^3)*(d*
x)^m/(m^3 + 12*m^2 + 47*m + 60)

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giac [B]  time = 0.17, size = 141, normalized size = 2.43 \begin {gather*} \frac {\left (d x\right )^{m} c^{2} m^{2} x^{5} + 2 \, \left (d x\right )^{m} b c m^{2} x^{4} + 7 \, \left (d x\right )^{m} c^{2} m x^{5} + \left (d x\right )^{m} b^{2} m^{2} x^{3} + 16 \, \left (d x\right )^{m} b c m x^{4} + 12 \, \left (d x\right )^{m} c^{2} x^{5} + 9 \, \left (d x\right )^{m} b^{2} m x^{3} + 30 \, \left (d x\right )^{m} b c x^{4} + 20 \, \left (d x\right )^{m} b^{2} x^{3}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*x)^m*c^2*m^2*x^5 + 2*(d*x)^m*b*c*m^2*x^4 + 7*(d*x)^m*c^2*m*x^5 + (d*x)^m*b^2*m^2*x^3 + 16*(d*x)^m*b*c*m*x^
4 + 12*(d*x)^m*c^2*x^5 + 9*(d*x)^m*b^2*m*x^3 + 30*(d*x)^m*b*c*x^4 + 20*(d*x)^m*b^2*x^3)/(m^3 + 12*m^2 + 47*m +
60)

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maple [A]  time = 0.04, size = 90, normalized size = 1.55 \begin {gather*} \frac {\left (c^{2} m^{2} x^{2}+2 b c \,m^{2} x +7 c^{2} m \,x^{2}+b^{2} m^{2}+16 b c m x +12 c^{2} x^{2}+9 b^{2} m +30 b c x +20 b^{2}\right ) x^{3} \left (d x \right )^{m}}{\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((d*x)^m*(c*x^2+b*x)^2,x)

[Out]

(d*x)^m*(c^2*m^2*x^2+2*b*c*m^2*x+7*c^2*m*x^2+b^2*m^2+16*b*c*m*x+12*c^2*x^2+9*b^2*m+30*b*c*x+20*b^2)*x^3/(m+5)/
(m+4)/(m+3)

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maxima [A]  time = 1.53, size = 55, normalized size = 0.95 \begin {gather*} \frac {c^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, b c d^{m} x^{4} x^{m}}{m + 4} + \frac {b^{2} d^{m} x^{3} x^{m}}{m + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*d^m*x^5*x^m/(m + 5) + 2*b*c*d^m*x^4*x^m/(m + 4) + b^2*d^m*x^3*x^m/(m + 3)

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mupad [B]  time = 0.23, size = 97, normalized size = 1.67 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b^2\,x^3\,\left (m^2+9\,m+20\right )}{m^3+12\,m^2+47\,m+60}+\frac {c^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {2\,b\,c\,x^4\,\left (m^2+8\,m+15\right )}{m^3+12\,m^2+47\,m+60}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(d*x)^m,x)

[Out]

(d*x)^m*((b^2*x^3*(9*m + m^2 + 20))/(47*m + 12*m^2 + m^3 + 60) + (c^2*x^5*(7*m + m^2 + 12))/(47*m + 12*m^2 + m
^3 + 60) + (2*b*c*x^4*(8*m + m^2 + 15))/(47*m + 12*m^2 + m^3 + 60))

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sympy [A]  time = 0.99, size = 345, normalized size = 5.95 \begin {gather*} \begin {cases} \frac {- \frac {b^{2}}{2 x^{2}} - \frac {2 b c}{x} + c^{2} \log {\relax (x )}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {b^{2}}{x} + 2 b c \log {\relax (x )} + c^{2} x}{d^{4}} & \text {for}\: m = -4 \\\frac {b^{2} \log {\relax (x )} + 2 b c x + \frac {c^{2} x^{2}}{2}}{d^{3}} & \text {for}\: m = -3 \\\frac {b^{2} d^{m} m^{2} x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {9 b^{2} d^{m} m x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {20 b^{2} d^{m} x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {2 b c d^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {16 b c d^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {30 b c d^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {c^{2} d^{m} m^{2} x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {7 c^{2} d^{m} m x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac {12 c^{2} d^{m} x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**2+b*x)**2,x)

[Out]

Piecewise(((-b**2/(2*x**2) - 2*b*c/x + c**2*log(x))/d**5, Eq(m, -5)), ((-b**2/x + 2*b*c*log(x) + c**2*x)/d**4,
Eq(m, -4)), ((b**2*log(x) + 2*b*c*x + c**2*x**2/2)/d**3, Eq(m, -3)), (b**2*d**m*m**2*x**3*x**m/(m**3 + 12*m**
2 + 47*m + 60) + 9*b**2*d**m*m*x**3*x**m/(m**3 + 12*m**2 + 47*m + 60) + 20*b**2*d**m*x**3*x**m/(m**3 + 12*m**2
+ 47*m + 60) + 2*b*c*d**m*m**2*x**4*x**m/(m**3 + 12*m**2 + 47*m + 60) + 16*b*c*d**m*m*x**4*x**m/(m**3 + 12*m*
*2 + 47*m + 60) + 30*b*c*d**m*x**4*x**m/(m**3 + 12*m**2 + 47*m + 60) + c**2*d**m*m**2*x**5*x**m/(m**3 + 12*m**
2 + 47*m + 60) + 7*c**2*d**m*m*x**5*x**m/(m**3 + 12*m**2 + 47*m + 60) + 12*c**2*d**m*x**5*x**m/(m**3 + 12*m**2
+ 47*m + 60), True))

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