3.3.34 \(\int (c x)^m (b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=73 \[ \frac {b^3 x^7 (c x)^m}{m+7}+\frac {3 b^2 c x^9 (c x)^m}{m+9}+\frac {3 b c^2 x^{11} (c x)^m}{m+11}+\frac {c^3 x^{13} (c x)^m}{m+13} \]

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Rubi [A]  time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1142, 1584, 270} \begin {gather*} \frac {3 b^2 c x^9 (c x)^m}{m+9}+\frac {b^3 x^7 (c x)^m}{m+7}+\frac {3 b c^2 x^{11} (c x)^m}{m+11}+\frac {c^3 x^{13} (c x)^m}{m+13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^2 + c*x^4)^3,x]

[Out]

(b^3*x^7*(c*x)^m)/(7 + m) + (3*b^2*c*x^9*(c*x)^m)/(9 + m) + (3*b*c^2*x^11*(c*x)^m)/(11 + m) + (c^3*x^13*(c*x)^
m)/(13 + m)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1142

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx &=\left (x^{-m} (c x)^m\right ) \operatorname {Subst}\left (\int x^m \left (b x^2+c x^4\right )^3 \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname {Subst}\left (\int x^{6+m} \left (b+c x^2\right )^3 \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname {Subst}\left (\int \left (b^3 x^{6+m}+3 b^2 c x^{8+m}+3 b c^2 x^{10+m}+c^3 x^{12+m}\right ) \, dx,x,x\right )\\ &=\frac {b^3 x^7 (c x)^m}{7+m}+\frac {3 b^2 c x^9 (c x)^m}{9+m}+\frac {3 b c^2 x^{11} (c x)^m}{11+m}+\frac {c^3 x^{13} (c x)^m}{13+m}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 59, normalized size = 0.81 \begin {gather*} x^7 (c x)^m \left (\frac {b^3}{m+7}+\frac {3 b^2 c x^2}{m+9}+\frac {3 b c^2 x^4}{m+11}+\frac {c^3 x^6}{m+13}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(b*x^2 + c*x^4)^3,x]

[Out]

x^7*(c*x)^m*(b^3/(7 + m) + (3*b^2*c*x^2)/(9 + m) + (3*b*c^2*x^4)/(11 + m) + (c^3*x^6)/(13 + m))

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IntegrateAlgebraic [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int (c x)^m \left (b x^2+c x^4\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c*x)^m*(b*x^2 + c*x^4)^3,x]

[Out]

Defer[IntegrateAlgebraic][(c*x)^m*(b*x^2 + c*x^4)^3, x]

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fricas [B]  time = 0.66, size = 161, normalized size = 2.21 \begin {gather*} \frac {{\left ({\left (c^{3} m^{3} + 27 \, c^{3} m^{2} + 239 \, c^{3} m + 693 \, c^{3}\right )} x^{13} + 3 \, {\left (b c^{2} m^{3} + 29 \, b c^{2} m^{2} + 271 \, b c^{2} m + 819 \, b c^{2}\right )} x^{11} + 3 \, {\left (b^{2} c m^{3} + 31 \, b^{2} c m^{2} + 311 \, b^{2} c m + 1001 \, b^{2} c\right )} x^{9} + {\left (b^{3} m^{3} + 33 \, b^{3} m^{2} + 359 \, b^{3} m + 1287 \, b^{3}\right )} x^{7}\right )} \left (c x\right )^{m}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

((c^3*m^3 + 27*c^3*m^2 + 239*c^3*m + 693*c^3)*x^13 + 3*(b*c^2*m^3 + 29*b*c^2*m^2 + 271*b*c^2*m + 819*b*c^2)*x^
11 + 3*(b^2*c*m^3 + 31*b^2*c*m^2 + 311*b^2*c*m + 1001*b^2*c)*x^9 + (b^3*m^3 + 33*b^3*m^2 + 359*b^3*m + 1287*b^
3)*x^7)*(c*x)^m/(m^4 + 40*m^3 + 590*m^2 + 3800*m + 9009)

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giac [B]  time = 0.18, size = 264, normalized size = 3.62 \begin {gather*} \frac {\left (c x\right )^{m} c^{3} m^{3} x^{13} + 27 \, \left (c x\right )^{m} c^{3} m^{2} x^{13} + 3 \, \left (c x\right )^{m} b c^{2} m^{3} x^{11} + 239 \, \left (c x\right )^{m} c^{3} m x^{13} + 87 \, \left (c x\right )^{m} b c^{2} m^{2} x^{11} + 693 \, \left (c x\right )^{m} c^{3} x^{13} + 3 \, \left (c x\right )^{m} b^{2} c m^{3} x^{9} + 813 \, \left (c x\right )^{m} b c^{2} m x^{11} + 93 \, \left (c x\right )^{m} b^{2} c m^{2} x^{9} + 2457 \, \left (c x\right )^{m} b c^{2} x^{11} + \left (c x\right )^{m} b^{3} m^{3} x^{7} + 933 \, \left (c x\right )^{m} b^{2} c m x^{9} + 33 \, \left (c x\right )^{m} b^{3} m^{2} x^{7} + 3003 \, \left (c x\right )^{m} b^{2} c x^{9} + 359 \, \left (c x\right )^{m} b^{3} m x^{7} + 1287 \, \left (c x\right )^{m} b^{3} x^{7}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

((c*x)^m*c^3*m^3*x^13 + 27*(c*x)^m*c^3*m^2*x^13 + 3*(c*x)^m*b*c^2*m^3*x^11 + 239*(c*x)^m*c^3*m*x^13 + 87*(c*x)
^m*b*c^2*m^2*x^11 + 693*(c*x)^m*c^3*x^13 + 3*(c*x)^m*b^2*c*m^3*x^9 + 813*(c*x)^m*b*c^2*m*x^11 + 93*(c*x)^m*b^2
*c*m^2*x^9 + 2457*(c*x)^m*b*c^2*x^11 + (c*x)^m*b^3*m^3*x^7 + 933*(c*x)^m*b^2*c*m*x^9 + 33*(c*x)^m*b^3*m^2*x^7
+ 3003*(c*x)^m*b^2*c*x^9 + 359*(c*x)^m*b^3*m*x^7 + 1287*(c*x)^m*b^3*x^7)/(m^4 + 40*m^3 + 590*m^2 + 3800*m + 90
09)

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maple [B]  time = 0.01, size = 181, normalized size = 2.48 \begin {gather*} \frac {\left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 c^{3} m \,x^{6}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right ) x^{7} \left (c x \right )^{m}}{\left (m +13\right ) \left (m +11\right ) \left (m +9\right ) \left (m +7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(c*x^4+b*x^2)^3,x)

[Out]

(c*x)^m*(c^3*m^3*x^6+27*c^3*m^2*x^6+3*b*c^2*m^3*x^4+239*c^3*m*x^6+87*b*c^2*m^2*x^4+693*c^3*x^6+3*b^2*c*m^3*x^2
+813*b*c^2*m*x^4+93*b^2*c*m^2*x^2+2457*b*c^2*x^4+b^3*m^3+933*b^2*c*m*x^2+33*b^3*m^2+3003*b^2*c*x^2+359*b^3*m+1
287*b^3)*x^7/(13+m)/(11+m)/(9+m)/(7+m)

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maxima [A]  time = 1.51, size = 76, normalized size = 1.04 \begin {gather*} \frac {c^{m + 3} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{m + 2} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c^{m + 1} x^{9} x^{m}}{m + 9} + \frac {b^{3} c^{m} x^{7} x^{m}}{m + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

c^(m + 3)*x^13*x^m/(m + 13) + 3*b*c^(m + 2)*x^11*x^m/(m + 11) + 3*b^2*c^(m + 1)*x^9*x^m/(m + 9) + b^3*c^m*x^7*
x^m/(m + 7)

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mupad [B]  time = 4.29, size = 171, normalized size = 2.34 \begin {gather*} {\left (c\,x\right )}^m\,\left (\frac {b^3\,x^7\,\left (m^3+33\,m^2+359\,m+1287\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {c^3\,x^{13}\,\left (m^3+27\,m^2+239\,m+693\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {3\,b\,c^2\,x^{11}\,\left (m^3+29\,m^2+271\,m+819\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {3\,b^2\,c\,x^9\,\left (m^3+31\,m^2+311\,m+1001\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^2 + c*x^4)^3,x)

[Out]

(c*x)^m*((b^3*x^7*(359*m + 33*m^2 + m^3 + 1287))/(3800*m + 590*m^2 + 40*m^3 + m^4 + 9009) + (c^3*x^13*(239*m +
27*m^2 + m^3 + 693))/(3800*m + 590*m^2 + 40*m^3 + m^4 + 9009) + (3*b*c^2*x^11*(271*m + 29*m^2 + m^3 + 819))/(
3800*m + 590*m^2 + 40*m^3 + m^4 + 9009) + (3*b^2*c*x^9*(311*m + 31*m^2 + m^3 + 1001))/(3800*m + 590*m^2 + 40*m
^3 + m^4 + 9009))

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sympy [A]  time = 5.28, size = 758, normalized size = 10.38 \begin {gather*} \begin {cases} \frac {- \frac {b^{3}}{6 x^{6}} - \frac {3 b^{2} c}{4 x^{4}} - \frac {3 b c^{2}}{2 x^{2}} + c^{3} \log {\relax (x )}}{c^{13}} & \text {for}\: m = -13 \\\frac {- \frac {b^{3}}{4 x^{4}} - \frac {3 b^{2} c}{2 x^{2}} + 3 b c^{2} \log {\relax (x )} + \frac {c^{3} x^{2}}{2}}{c^{11}} & \text {for}\: m = -11 \\\frac {- \frac {b^{3}}{2 x^{2}} + 3 b^{2} c \log {\relax (x )} + \frac {3 b c^{2} x^{2}}{2} + \frac {c^{3} x^{4}}{4}}{c^{9}} & \text {for}\: m = -9 \\\frac {b^{3} \log {\relax (x )} + \frac {3 b^{2} c x^{2}}{2} + \frac {3 b c^{2} x^{4}}{4} + \frac {c^{3} x^{6}}{6}}{c^{7}} & \text {for}\: m = -7 \\\frac {b^{3} c^{m} m^{3} x^{7} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {33 b^{3} c^{m} m^{2} x^{7} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {359 b^{3} c^{m} m x^{7} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {1287 b^{3} c^{m} x^{7} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {3 b^{2} c c^{m} m^{3} x^{9} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {93 b^{2} c c^{m} m^{2} x^{9} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {933 b^{2} c c^{m} m x^{9} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {3003 b^{2} c c^{m} x^{9} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {3 b c^{2} c^{m} m^{3} x^{11} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {87 b c^{2} c^{m} m^{2} x^{11} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {813 b c^{2} c^{m} m x^{11} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {2457 b c^{2} c^{m} x^{11} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {c^{3} c^{m} m^{3} x^{13} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {27 c^{3} c^{m} m^{2} x^{13} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {239 c^{3} c^{m} m x^{13} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} + \frac {693 c^{3} c^{m} x^{13} x^{m}}{m^{4} + 40 m^{3} + 590 m^{2} + 3800 m + 9009} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(c*x**4+b*x**2)**3,x)

[Out]

Piecewise(((-b**3/(6*x**6) - 3*b**2*c/(4*x**4) - 3*b*c**2/(2*x**2) + c**3*log(x))/c**13, Eq(m, -13)), ((-b**3/
(4*x**4) - 3*b**2*c/(2*x**2) + 3*b*c**2*log(x) + c**3*x**2/2)/c**11, Eq(m, -11)), ((-b**3/(2*x**2) + 3*b**2*c*
log(x) + 3*b*c**2*x**2/2 + c**3*x**4/4)/c**9, Eq(m, -9)), ((b**3*log(x) + 3*b**2*c*x**2/2 + 3*b*c**2*x**4/4 +
c**3*x**6/6)/c**7, Eq(m, -7)), (b**3*c**m*m**3*x**7*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 33*b**3
*c**m*m**2*x**7*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 359*b**3*c**m*m*x**7*x**m/(m**4 + 40*m**3 +
590*m**2 + 3800*m + 9009) + 1287*b**3*c**m*x**7*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 3*b**2*c*c
**m*m**3*x**9*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 93*b**2*c*c**m*m**2*x**9*x**m/(m**4 + 40*m**3
+ 590*m**2 + 3800*m + 9009) + 933*b**2*c*c**m*m*x**9*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 3003*
b**2*c*c**m*x**9*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 3*b*c**2*c**m*m**3*x**11*x**m/(m**4 + 40*m
**3 + 590*m**2 + 3800*m + 9009) + 87*b*c**2*c**m*m**2*x**11*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) +
813*b*c**2*c**m*m*x**11*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 2457*b*c**2*c**m*x**11*x**m/(m**4
+ 40*m**3 + 590*m**2 + 3800*m + 9009) + c**3*c**m*m**3*x**13*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009)
+ 27*c**3*c**m*m**2*x**13*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 239*c**3*c**m*m*x**13*x**m/(m**4
+ 40*m**3 + 590*m**2 + 3800*m + 9009) + 693*c**3*c**m*x**13*x**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009),
True))

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