3.2.12 \(\int (d x)^m (b x+c x^2)^3 \, dx\) [112]

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

[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 = 0.05, antiderivative size = 81, 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 = {661, 45} \begin {gather*} \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)} \end {gather*}

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 45

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 661

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 )^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*}

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Mathematica [A]
time = 0.05, size = 57, normalized size = 0.70 \begin {gather*} x^4 (d x)^m \left (\frac {b^3}{4+m}+\frac {3 b^2 c x}{5+m}+\frac {3 b c^2 x^2}{6+m}+\frac {c^3 x^3}{7+m}\right ) \end {gather*}

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))

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Maple [A]
time = 0.38, size = 82, normalized size = 1.01

method	result	size
norman	\(\frac {b^{3} x^{4} {\mathrm e}^{m \ln \left (d x \right )}}{4+m}+\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (d x \right )}}{7+m}+\frac {3 b \,c^{2} x^{6} {\mathrm e}^{m \ln \left (d x \right )}}{6+m}+\frac {3 b^{2} c \,x^{5} {\mathrm e}^{m \ln \left (d x \right )}}{5+m}\)	\(82\)
gosper	\(\frac {\left (d x \right )^{m} \left (c^{3} m^{3} x^{3}+3 b \,c^{2} m^{3} x^{2}+15 c^{3} m^{2} x^{3}+3 b^{2} c \,m^{3} x +48 b \,c^{2} m^{2} x^{2}+74 m \,x^{3} c^{3}+b^{3} m^{3}+51 b^{2} c \,m^{2} x +249 b \,c^{2} m \,x^{2}+120 c^{3} x^{3}+18 b^{3} m^{2}+282 b^{2} c m x +420 b \,c^{2} x^{2}+107 b^{3} m +504 b^{2} c x +210 b^{3}\right ) x^{4}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\)	\(173\)
risch	\(\frac {\left (d x \right )^{m} \left (c^{3} m^{3} x^{3}+3 b \,c^{2} m^{3} x^{2}+15 c^{3} m^{2} x^{3}+3 b^{2} c \,m^{3} x +48 b \,c^{2} m^{2} x^{2}+74 m \,x^{3} c^{3}+b^{3} m^{3}+51 b^{2} c \,m^{2} x +249 b \,c^{2} m \,x^{2}+120 c^{3} x^{3}+18 b^{3} m^{2}+282 b^{2} c m x +420 b \,c^{2} x^{2}+107 b^{3} m +504 b^{2} c x +210 b^{3}\right ) x^{4}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right )}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 77, normalized size = 0.95 \begin {gather*} \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 {gather*}

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)

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Fricas [A]
time = 1.46, size = 161, normalized size = 1.99 \begin {gather*} \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 {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (73) = 146\).
time = 0.36, size = 711, normalized size = 8.78 \begin {gather*} \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} m^{3} x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {18 b^{3} m^{2} x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {107 b^{3} m x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {210 b^{3} x^{4} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {3 b^{2} c m^{3} x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {51 b^{2} c m^{2} x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {282 b^{2} c m x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {504 b^{2} c x^{5} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {3 b c^{2} m^{3} x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {48 b c^{2} m^{2} x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {249 b c^{2} m x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {420 b c^{2} x^{6} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {c^{3} m^{3} x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {15 c^{3} m^{2} x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {74 c^{3} m x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} + \frac {120 c^{3} x^{7} \left (d x\right )^{m}}{m^{4} + 22 m^{3} + 179 m^{2} + 638 m + 840} & \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)**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*m**
3*x**4*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 18*b**3*m**2*x**4*(d*x)**m/(m**4 + 22*m**3 + 179*m
**2 + 638*m + 840) + 107*b**3*m*x**4*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 210*b**3*x**4*(d*x)*
*m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 3*b**2*c*m**3*x**5*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m
+ 840) + 51*b**2*c*m**2*x**5*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 282*b**2*c*m*x**5*(d*x)**m/(
m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 504*b**2*c*x**5*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840)
+ 3*b*c**2*m**3*x**6*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 48*b*c**2*m**2*x**6*(d*x)**m/(m**4 +
22*m**3 + 179*m**2 + 638*m + 840) + 249*b*c**2*m*x**6*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 42
0*b*c**2*x**6*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + c**3*m**3*x**7*(d*x)**m/(m**4 + 22*m**3 + 1
79*m**2 + 638*m + 840) + 15*c**3*m**2*x**7*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 74*c**3*m*x**7
*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*m + 840) + 120*c**3*x**7*(d*x)**m/(m**4 + 22*m**3 + 179*m**2 + 638*
m + 840), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (81) = 162\).
time = 1.04, size = 264, normalized size = 3.26 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.28, size = 171, normalized size = 2.11 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b^3\,x^4\,\left (m^3+18\,m^2+107\,m+210\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {c^3\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,b\,c^2\,x^6\,\left (m^3+16\,m^2+83\,m+140\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,b^2\,c\,x^5\,\left (m^3+17\,m^2+94\,m+168\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(d*x)^m,x)

[Out]

(d*x)^m*((b^3*x^4*(107*m + 18*m^2 + m^3 + 210))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (c^3*x^7*(74*m + 15*m
^2 + m^3 + 120))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840) + (3*b*c^2*x^6*(83*m + 16*m^2 + m^3 + 140))/(638*m + 1
79*m^2 + 22*m^3 + m^4 + 840) + (3*b^2*c*x^5*(94*m + 17*m^2 + m^3 + 168))/(638*m + 179*m^2 + 22*m^3 + m^4 + 840
))

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