### 3.113 $$\int (d x)^m (b x+c x^2)^2 \, dx$$

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.0408282, antiderivative size = 58, normalized size of antiderivative = 1., 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} $\frac{b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac{c^2 (d x)^{m+5}}{d^5 (m+5)}$

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 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]

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])

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*}

Mathematica [A]  time = 0.0306284, size = 41, normalized size = 0.71 $x^3 (d x)^m \left (\frac{b^2}{m+3}+\frac{2 b c x}{m+4}+\frac{c^2 x^2}{m+5}\right )$

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))

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 90, normalized size = 1.6 \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ({c}^{2}{m}^{2}{x}^{2}+2\,bc{m}^{2}x+7\,{c}^{2}m{x}^{2}+{b}^{2}{m}^{2}+16\,bcmx+12\,{c}^{2}{x}^{2}+9\,{b}^{2}m+30\,bcx+20\,{b}^{2} \right ){x}^{3}}{ \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((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/(5+m)/
(4+m)/(3+m)

________________________________________________________________________________________

Maxima [A]  time = 1.16304, size = 74, normalized size = 1.28 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.1322, size = 193, normalized size = 3.33 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 2.60889, size = 345, normalized size = 5.95 \begin{align*} \begin{cases} \frac{- \frac{b^{2}}{2 x^{2}} - \frac{2 b c}{x} + c^{2} \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{b^{2}}{x} + 2 b c \log{\left (x \right )} + c^{2} x}{d^{4}} & \text{for}\: m = -4 \\\frac{b^{2} \log{\left (x \right )} + 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{align*}

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))

________________________________________________________________________________________

Giac [B]  time = 1.29118, size = 190, normalized size = 3.28 \begin{align*} \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{align*}

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)