### 3.112 $$\int (d x)^m (b x+c x^2)^3 \, dx$$

Optimal. Leaf size=81 $\frac{3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac{b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac{3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac{c^3 (d x)^{m+7}}{d^7 (m+7)}$

[Out]

(b^3*(d*x)^(4 + m))/(d^4*(4 + m)) + (3*b^2*c*(d*x)^(5 + m))/(d^5*(5 + m)) + (3*b*c^2*(d*x)^(6 + m))/(d^6*(6 +
m)) + (c^3*(d*x)^(7 + m))/(d^7*(7 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.0666968, antiderivative size = 81, 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} $\frac{3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac{b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac{3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac{c^3 (d x)^{m+7}}{d^7 (m+7)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d*x)^m*(b*x + c*x^2)^3,x]

[Out]

(b^3*(d*x)^(4 + m))/(d^4*(4 + m)) + (3*b^2*c*(d*x)^(5 + m))/(d^5*(5 + m)) + (3*b*c^2*(d*x)^(6 + m))/(d^6*(6 +
m)) + (c^3*(d*x)^(7 + m))/(d^7*(7 + 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 )^3 \, dx &=\frac{\int (d x)^{3+m} (b+c x)^3 \, dx}{d^3}\\ &=\frac{\int \left (b^3 (d x)^{3+m}+\frac{3 b^2 c (d x)^{4+m}}{d}+\frac{3 b c^2 (d x)^{5+m}}{d^2}+\frac{c^3 (d x)^{6+m}}{d^3}\right ) \, dx}{d^3}\\ &=\frac{b^3 (d x)^{4+m}}{d^4 (4+m)}+\frac{3 b^2 c (d x)^{5+m}}{d^5 (5+m)}+\frac{3 b c^2 (d x)^{6+m}}{d^6 (6+m)}+\frac{c^3 (d x)^{7+m}}{d^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.0360907, size = 57, normalized size = 0.7 $x^4 (d x)^m \left (\frac{3 b^2 c x}{m+5}+\frac{b^3}{m+4}+\frac{3 b c^2 x^2}{m+6}+\frac{c^3 x^3}{m+7}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d*x)^m*(b*x + c*x^2)^3,x]

[Out]

x^4*(d*x)^m*(b^3/(4 + m) + (3*b^2*c*x)/(5 + m) + (3*b*c^2*x^2)/(6 + m) + (c^3*x^3)/(7 + m))

________________________________________________________________________________________

Maple [B]  time = 0.058, size = 173, normalized size = 2.1 \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ({c}^{3}{m}^{3}{x}^{3}+3\,b{c}^{2}{m}^{3}{x}^{2}+15\,{c}^{3}{m}^{2}{x}^{3}+3\,{b}^{2}c{m}^{3}x+48\,b{c}^{2}{m}^{2}{x}^{2}+74\,{c}^{3}m{x}^{3}+{b}^{3}{m}^{3}+51\,{b}^{2}c{m}^{2}x+249\,b{c}^{2}m{x}^{2}+120\,{x}^{3}{c}^{3}+18\,{b}^{3}{m}^{2}+282\,{b}^{2}cmx+420\,b{x}^{2}{c}^{2}+107\,{b}^{3}m+504\,{b}^{2}xc+210\,{b}^{3} \right ){x}^{4}}{ \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(c*x^2+b*x)^3,x)

[Out]

(d*x)^m*(c^3*m^3*x^3+3*b*c^2*m^3*x^2+15*c^3*m^2*x^3+3*b^2*c*m^3*x+48*b*c^2*m^2*x^2+74*c^3*m*x^3+b^3*m^3+51*b^2
*c*m^2*x+249*b*c^2*m*x^2+120*c^3*x^3+18*b^3*m^2+282*b^2*c*m*x+420*b*c^2*x^2+107*b^3*m+504*b^2*c*x+210*b^3)*x^4
/(7+m)/(6+m)/(5+m)/(4+m)

________________________________________________________________________________________

Maxima [A]  time = 1.17176, size = 104, normalized size = 1.28 \begin{align*} \frac{c^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac{3 \, b c^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac{3 \, b^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac{b^{3} d^{m} x^{4} x^{m}}{m + 4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

c^3*d^m*x^7*x^m/(m + 7) + 3*b*c^2*d^m*x^6*x^m/(m + 6) + 3*b^2*c*d^m*x^5*x^m/(m + 5) + b^3*d^m*x^4*x^m/(m + 4)

________________________________________________________________________________________

Fricas [A]  time = 2.00345, size = 363, normalized size = 4.48 \begin{align*} \frac{{\left ({\left (c^{3} m^{3} + 15 \, c^{3} m^{2} + 74 \, c^{3} m + 120 \, c^{3}\right )} x^{7} + 3 \,{\left (b c^{2} m^{3} + 16 \, b c^{2} m^{2} + 83 \, b c^{2} m + 140 \, b c^{2}\right )} x^{6} + 3 \,{\left (b^{2} c m^{3} + 17 \, b^{2} c m^{2} + 94 \, b^{2} c m + 168 \, b^{2} c\right )} x^{5} +{\left (b^{3} m^{3} + 18 \, b^{3} m^{2} + 107 \, b^{3} m + 210 \, b^{3}\right )} x^{4}\right )} \left (d x\right )^{m}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

((c^3*m^3 + 15*c^3*m^2 + 74*c^3*m + 120*c^3)*x^7 + 3*(b*c^2*m^3 + 16*b*c^2*m^2 + 83*b*c^2*m + 140*b*c^2)*x^6 +
3*(b^2*c*m^3 + 17*b^2*c*m^2 + 94*b^2*c*m + 168*b^2*c)*x^5 + (b^3*m^3 + 18*b^3*m^2 + 107*b^3*m + 210*b^3)*x^4)
*(d*x)^m/(m^4 + 22*m^3 + 179*m^2 + 638*m + 840)

________________________________________________________________________________________

Sympy [A]  time = 3.83979, size = 738, normalized size = 9.11 \begin{align*} \begin{cases} \frac{- \frac{b^{3}}{3 x^{3}} - \frac{3 b^{2} c}{2 x^{2}} - \frac{3 b c^{2}}{x} + c^{3} \log{\left (x \right )}}{d^{7}} & \text{for}\: m = -7 \\\frac{- \frac{b^{3}}{2 x^{2}} - \frac{3 b^{2} c}{x} + 3 b c^{2} \log{\left (x \right )} + c^{3} x}{d^{6}} & \text{for}\: m = -6 \\\frac{- \frac{b^{3}}{x} + 3 b^{2} c \log{\left (x \right )} + 3 b c^{2} x + \frac{c^{3} x^{2}}{2}}{d^{5}} & \text{for}\: m = -5 \\\frac{b^{3} \log{\left (x \right )} + 3 b^{2} c x + \frac{3 b c^{2} x^{2}}{2} + \frac{c^{3} x^{3}}{3}}{d^{4}} & \text{for}\: m = -4 \\\frac{b^{3} d^{m} m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{18 b^{3} d^{m} m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{107 b^{3} d^{m} m x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{210 b^{3} d^{m} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{3 b^{2} c d^{m} m^{3} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{51 b^{2} c d^{m} m^{2} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{282 b^{2} c d^{m} m x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{504 b^{2} c d^{m} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{3 b c^{2} d^{m} m^{3} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{48 b c^{2} d^{m} m^{2} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{249 b c^{2} d^{m} m x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{420 b c^{2} d^{m} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{c^{3} d^{m} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{15 c^{3} d^{m} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{74 c^{3} d^{m} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{120 c^{3} d^{m} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} & \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)**3,x)

[Out]

Piecewise(((-b**3/(3*x**3) - 3*b**2*c/(2*x**2) - 3*b*c**2/x + c**3*log(x))/d**7, Eq(m, -7)), ((-b**3/(2*x**2)
- 3*b**2*c/x + 3*b*c**2*log(x) + c**3*x)/d**6, Eq(m, -6)), ((-b**3/x + 3*b**2*c*log(x) + 3*b*c**2*x + c**3*x**
2/2)/d**5, Eq(m, -5)), ((b**3*log(x) + 3*b**2*c*x + 3*b*c**2*x**2/2 + c**3*x**3/3)/d**4, Eq(m, -4)), (b**3*d**
m*m**3*x**4*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 18*b**3*d**m*m**2*x**4*x**m/(m**4 + 22*m**3 + 179
*m**2 + 638*m + 840) + 107*b**3*d**m*m*x**4*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 210*b**3*d**m*x**
4*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 3*b**2*c*d**m*m**3*x**5*x**m/(m**4 + 22*m**3 + 179*m**2 + 6
38*m + 840) + 51*b**2*c*d**m*m**2*x**5*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 282*b**2*c*d**m*m*x**5
*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 504*b**2*c*d**m*x**5*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m
+ 840) + 3*b*c**2*d**m*m**3*x**6*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 48*b*c**2*d**m*m**2*x**6*x*
*m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 249*b*c**2*d**m*m*x**6*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m
+ 840) + 420*b*c**2*d**m*x**6*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + c**3*d**m*m**3*x**7*x**m/(m**4
+ 22*m**3 + 179*m**2 + 638*m + 840) + 15*c**3*d**m*m**2*x**7*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) +
74*c**3*d**m*m*x**7*x**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 120*c**3*d**m*x**7*x**m/(m**4 + 22*m**3 +
179*m**2 + 638*m + 840), True))

________________________________________________________________________________________

Giac [B]  time = 1.27941, size = 356, normalized size = 4.4 \begin{align*} \frac{\left (d x\right )^{m} c^{3} m^{3} x^{7} + 3 \, \left (d x\right )^{m} b c^{2} m^{3} x^{6} + 15 \, \left (d x\right )^{m} c^{3} m^{2} x^{7} + 3 \, \left (d x\right )^{m} b^{2} c m^{3} x^{5} + 48 \, \left (d x\right )^{m} b c^{2} m^{2} x^{6} + 74 \, \left (d x\right )^{m} c^{3} m x^{7} + \left (d x\right )^{m} b^{3} m^{3} x^{4} + 51 \, \left (d x\right )^{m} b^{2} c m^{2} x^{5} + 249 \, \left (d x\right )^{m} b c^{2} m x^{6} + 120 \, \left (d x\right )^{m} c^{3} x^{7} + 18 \, \left (d x\right )^{m} b^{3} m^{2} x^{4} + 282 \, \left (d x\right )^{m} b^{2} c m x^{5} + 420 \, \left (d x\right )^{m} b c^{2} x^{6} + 107 \, \left (d x\right )^{m} b^{3} m x^{4} + 504 \, \left (d x\right )^{m} b^{2} c x^{5} + 210 \, \left (d x\right )^{m} b^{3} x^{4}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

((d*x)^m*c^3*m^3*x^7 + 3*(d*x)^m*b*c^2*m^3*x^6 + 15*(d*x)^m*c^3*m^2*x^7 + 3*(d*x)^m*b^2*c*m^3*x^5 + 48*(d*x)^m
*b*c^2*m^2*x^6 + 74*(d*x)^m*c^3*m*x^7 + (d*x)^m*b^3*m^3*x^4 + 51*(d*x)^m*b^2*c*m^2*x^5 + 249*(d*x)^m*b*c^2*m*x
^6 + 120*(d*x)^m*c^3*x^7 + 18*(d*x)^m*b^3*m^2*x^4 + 282*(d*x)^m*b^2*c*m*x^5 + 420*(d*x)^m*b*c^2*x^6 + 107*(d*x
)^m*b^3*m*x^4 + 504*(d*x)^m*b^2*c*x^5 + 210*(d*x)^m*b^3*x^4)/(m^4 + 22*m^3 + 179*m^2 + 638*m + 840)