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

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

[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))

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Rubi [A]  time = 0.142724, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b^3 (d x)^{m+4}}{d^4 (m+4)}+\frac{3 b^2 c (d x)^{m+5}}{d^5 (m+5)}+\frac{3 b c^2 (d x)^{m+6}}{d^6 (m+6)}+\frac{c^3 (d x)^{m+7}}{d^7 (m+7)} \]

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))

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Rubi in Sympy [A]  time = 24.4434, size = 73, normalized size = 0.9 \[ \frac{b^{3} \left (d x\right )^{m + 4}}{d^{4} \left (m + 4\right )} + \frac{3 b^{2} c \left (d x\right )^{m + 5}}{d^{5} \left (m + 5\right )} + \frac{3 b c^{2} \left (d x\right )^{m + 6}}{d^{6} \left (m + 6\right )} + \frac{c^{3} \left (d x\right )^{m + 7}}{d^{7} \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2+b*x)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0485184, size = 59, normalized size = 0.73 \[ (d x)^m \left (\frac{b^3 x^4}{m+4}+\frac{3 b^2 c x^5}{m+5}+\frac{3 b c^2 x^6}{m+6}+\frac{c^3 x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.008, size = 173, normalized size = 2.1 \[{\frac{ \left ( dx \right ) ^{m} \left ({c}^{3}{m}^{3}{x}^{3}+3\,b{c}^{2}{m}^{3}{x}^{2}+15\,{c}^{3}{m}^{2}{x}^{3}+3\,{b}^{2}c{m}^{3}x+48\,b{c}^{2}{m}^{2}{x}^{2}+74\,{c}^{3}m{x}^{3}+{b}^{3}{m}^{3}+51\,{b}^{2}c{m}^{2}x+249\,b{c}^{2}m{x}^{2}+120\,{x}^{3}{c}^{3}+18\,{b}^{3}{m}^{2}+282\,{b}^{2}cmx+420\,b{x}^{2}{c}^{2}+107\,{b}^{3}m+504\,{b}^{2}xc+210\,{b}^{3} \right ){x}^{4}}{ \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) }} \]

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+28
2*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)

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Maxima [A]  time = 0.715734, size = 104, normalized size = 1.28 \[ \frac{c^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac{3 \, b c^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac{3 \, b^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac{b^{3} d^{m} x^{4} x^{m}}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(d*x)^m,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)

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Fricas [A]  time = 0.235702, size = 217, normalized size = 2.68 \[ \frac{{\left ({\left (c^{3} m^{3} + 15 \, c^{3} m^{2} + 74 \, c^{3} m + 120 \, c^{3}\right )} x^{7} + 3 \,{\left (b c^{2} m^{3} + 16 \, b c^{2} m^{2} + 83 \, b c^{2} m + 140 \, b c^{2}\right )} x^{6} + 3 \,{\left (b^{2} c m^{3} + 17 \, b^{2} c m^{2} + 94 \, b^{2} c m + 168 \, b^{2} c\right )} x^{5} +{\left (b^{3} m^{3} + 18 \, b^{3} m^{2} + 107 \, b^{3} m + 210 \, b^{3}\right )} x^{4}\right )} \left (d x\right )^{m}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(d*x)^m,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)

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Sympy [A]  time = 4.87656, size = 738, normalized size = 9.11 \[ \begin{cases} \frac{- \frac{b^{3}}{3 x^{3}} - \frac{3 b^{2} c}{2 x^{2}} - \frac{3 b c^{2}}{x} + c^{3} \log{\left (x \right )}}{d^{7}} & \text{for}\: m = -7 \\\frac{- \frac{b^{3}}{2 x^{2}} - \frac{3 b^{2} c}{x} + 3 b c^{2} \log{\left (x \right )} + c^{3} x}{d^{6}} & \text{for}\: m = -6 \\\frac{- \frac{b^{3}}{x} + 3 b^{2} c \log{\left (x \right )} + 3 b c^{2} x + \frac{c^{3} x^{2}}{2}}{d^{5}} & \text{for}\: m = -5 \\\frac{b^{3} \log{\left (x \right )} + 3 b^{2} c x + \frac{3 b c^{2} x^{2}}{2} + \frac{c^{3} x^{3}}{3}}{d^{4}} & \text{for}\: m = -4 \\\frac{b^{3} d^{m} m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{18 b^{3} d^{m} m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{107 b^{3} d^{m} m x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{210 b^{3} d^{m} x^{4} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{3 b^{2} c d^{m} m^{3} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{51 b^{2} c d^{m} m^{2} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{282 b^{2} c d^{m} m x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{504 b^{2} c d^{m} x^{5} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{3 b c^{2} d^{m} m^{3} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{48 b c^{2} d^{m} m^{2} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{249 b c^{2} d^{m} m x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{420 b c^{2} d^{m} x^{6} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{c^{3} d^{m} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{15 c^{3} d^{m} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{74 c^{3} d^{m} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac{120 c^{3} d^{m} x^{7} x^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} & \text{otherwise} \end{cases} \]

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 + 638*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))

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GIAC/XCAS [A]  time = 0.212355, size = 400, normalized size = 4.94 \[ \frac{c^{3} m^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 3 \, b c^{2} m^{3} x^{6} e^{\left (m{\rm ln}\left (d x\right )\right )} + 15 \, c^{3} m^{2} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 3 \, b^{2} c m^{3} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 48 \, b c^{2} m^{2} x^{6} e^{\left (m{\rm ln}\left (d x\right )\right )} + 74 \, c^{3} m x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + b^{3} m^{3} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 51 \, b^{2} c m^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 249 \, b c^{2} m x^{6} e^{\left (m{\rm ln}\left (d x\right )\right )} + 120 \, c^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 18 \, b^{3} m^{2} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 282 \, b^{2} c m x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 420 \, b c^{2} x^{6} e^{\left (m{\rm ln}\left (d x\right )\right )} + 107 \, b^{3} m x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 504 \, b^{2} c x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 210 \, b^{3} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{4} + 22 \, m^{3} + 179 \, m^{2} + 638 \, m + 840} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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