Integrand size = 19, antiderivative size = 81 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {3 b^2 c (d x)^{9+m}}{d^9 (9+m)}+\frac {3 b c^2 (d x)^{11+m}}{d^{11} (11+m)}+\frac {c^3 (d x)^{13+m}}{d^{13} (13+m)} \] Output:
b^3*(d*x)^(7+m)/d^7/(7+m)+3*b^2*c*(d*x)^(9+m)/d^9/(9+m)+3*b*c^2*(d*x)^(11+ m)/d^11/(11+m)+c^3*(d*x)^(13+m)/d^13/(13+m)
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx=x^7 (d x)^m \left (\frac {b^3}{7+m}+\frac {3 b^2 c x^2}{9+m}+\frac {3 b c^2 x^4}{11+m}+\frac {c^3 x^6}{13+m}\right ) \] Input:
Integrate[(d*x)^m*(b*x^2 + c*x^4)^3,x]
Output:
x^7*(d*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))
Time = 0.40 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {9, 244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (b x^2+c x^4\right )^3 (d x)^m \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \frac {\int (d x)^{m+6} \left (c x^2+b\right )^3dx}{d^6}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \left (b^3 (d x)^{m+6}+\frac {3 b^2 c (d x)^{m+8}}{d^2}+\frac {3 b c^2 (d x)^{m+10}}{d^4}+\frac {c^3 (d x)^{m+12}}{d^6}\right )dx}{d^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {b^3 (d x)^{m+7}}{d (m+7)}+\frac {3 b^2 c (d x)^{m+9}}{d^3 (m+9)}+\frac {3 b c^2 (d x)^{m+11}}{d^5 (m+11)}+\frac {c^3 (d x)^{m+13}}{d^7 (m+13)}}{d^6}\) |
Input:
Int[(d*x)^m*(b*x^2 + c*x^4)^3,x]
Output:
((b^3*(d*x)^(7 + m))/(d*(7 + m)) + (3*b^2*c*(d*x)^(9 + m))/(d^3*(9 + m)) + (3*b*c^2*(d*x)^(11 + m))/(d^5*(11 + m)) + (c^3*(d*x)^(13 + m))/(d^7*(13 + m)))/d^6
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs. \(2(81)=162\).
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.23
method | result | size |
gosper | \(\frac {\left (d x \right )^{m} \left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 m \,x^{6} c^{3}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right ) x^{7}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\) | \(181\) |
risch | \(\frac {\left (d x \right )^{m} \left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 m \,x^{6} c^{3}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right ) x^{7}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\) | \(181\) |
orering | \(\frac {\left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 m \,x^{6} c^{3}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right ) x \left (d x \right )^{m} \left (c \,x^{4}+b \,x^{2}\right )^{3}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (c \,x^{2}+b \right )^{3}}\) | \(201\) |
parallelrisch | \(\frac {x^{13} \left (d x \right )^{m} c^{3} m^{3}+27 x^{13} \left (d x \right )^{m} c^{3} m^{2}+239 x^{13} \left (d x \right )^{m} c^{3} m +3 x^{11} \left (d x \right )^{m} b \,c^{2} m^{3}+693 x^{13} \left (d x \right )^{m} c^{3}+87 x^{11} \left (d x \right )^{m} b \,c^{2} m^{2}+813 x^{11} \left (d x \right )^{m} b \,c^{2} m +3 x^{9} \left (d x \right )^{m} b^{2} c \,m^{3}+2457 x^{11} \left (d x \right )^{m} b \,c^{2}+93 x^{9} \left (d x \right )^{m} b^{2} c \,m^{2}+933 x^{9} \left (d x \right )^{m} b^{2} c m +x^{7} \left (d x \right )^{m} b^{3} m^{3}+3003 x^{9} \left (d x \right )^{m} b^{2} c +33 x^{7} \left (d x \right )^{m} b^{3} m^{2}+359 x^{7} \left (d x \right )^{m} b^{3} m +1287 x^{7} \left (d x \right )^{m} b^{3}}{\left (13+m \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right )}\) | \(265\) |
Input:
int((d*x)^m*(c*x^4+b*x^2)^3,x,method=_RETURNVERBOSE)
Output:
(d*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+128 7*b^3)*x^7/(13+m)/(11+m)/(9+m)/(7+m)
Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.99 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {{\left ({\left (c^{3} m^{3} + 27 \, c^{3} m^{2} + 239 \, c^{3} m + 693 \, c^{3}\right )} x^{13} + 3 \, {\left (b c^{2} m^{3} + 29 \, b c^{2} m^{2} + 271 \, b c^{2} m + 819 \, b c^{2}\right )} x^{11} + 3 \, {\left (b^{2} c m^{3} + 31 \, b^{2} c m^{2} + 311 \, b^{2} c m + 1001 \, b^{2} c\right )} x^{9} + {\left (b^{3} m^{3} + 33 \, b^{3} m^{2} + 359 \, b^{3} m + 1287 \, b^{3}\right )} x^{7}\right )} \left (d x\right )^{m}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \] Input:
integrate((d*x)^m*(c*x^4+b*x^2)^3,x, algorithm="fricas")
Output:
((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)*(d*x)^m/(m^4 + 40*m^3 + 590*m^2 + 3800*m + 9009)
Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (73) = 146\).
Time = 0.67 (sec) , antiderivative size = 731, normalized size of antiderivative = 9.02 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((d*x)**m*(c*x**4+b*x**2)**3,x)
Output:
Piecewise(((-b**3/(6*x**6) - 3*b**2*c/(4*x**4) - 3*b*c**2/(2*x**2) + c**3* log(x))/d**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)/d**11, Eq(m, -11)), ((-b**3/(2*x**2) + 3*b**2*c*lo g(x) + 3*b*c**2*x**2/2 + c**3*x**4/4)/d**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)/d**7, Eq(m, -7)), (b**3*m** 3*x**7*(d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 33*b**3*m**2 *x**7*(d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 359*b**3*m*x* *7*(d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 1287*b**3*x**7*( d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 3*b**2*c*m**3*x**9*( d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 93*b**2*c*m**2*x**9* (d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 933*b**2*c*m*x**9*( d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 3003*b**2*c*x**9*(d* x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 3*b*c**2*m**3*x**11*(d *x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 87*b*c**2*m**2*x**11* (d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 813*b*c**2*m*x**11* (d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 2457*b*c**2*x**11*( d*x)**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + c**3*m**3*x**13*(d*x )**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 27*c**3*m**2*x**13*(d*x )**m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 239*c**3*m*x**13*(d*x)* *m/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009) + 693*c**3*x**13*(d*x)**...
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {c^{3} d^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c d^{m} x^{9} x^{m}}{m + 9} + \frac {b^{3} d^{m} x^{7} x^{m}}{m + 7} \] Input:
integrate((d*x)^m*(c*x^4+b*x^2)^3,x, algorithm="maxima")
Output:
c^3*d^m*x^13*x^m/(m + 13) + 3*b*c^2*d^m*x^11*x^m/(m + 11) + 3*b^2*c*d^m*x^ 9*x^m/(m + 9) + b^3*d^m*x^7*x^m/(m + 7)
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (81) = 162\).
Time = 0.12 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.26 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {\left (d x\right )^{m} c^{3} m^{3} x^{13} + 27 \, \left (d x\right )^{m} c^{3} m^{2} x^{13} + 3 \, \left (d x\right )^{m} b c^{2} m^{3} x^{11} + 239 \, \left (d x\right )^{m} c^{3} m x^{13} + 87 \, \left (d x\right )^{m} b c^{2} m^{2} x^{11} + 693 \, \left (d x\right )^{m} c^{3} x^{13} + 3 \, \left (d x\right )^{m} b^{2} c m^{3} x^{9} + 813 \, \left (d x\right )^{m} b c^{2} m x^{11} + 93 \, \left (d x\right )^{m} b^{2} c m^{2} x^{9} + 2457 \, \left (d x\right )^{m} b c^{2} x^{11} + \left (d x\right )^{m} b^{3} m^{3} x^{7} + 933 \, \left (d x\right )^{m} b^{2} c m x^{9} + 33 \, \left (d x\right )^{m} b^{3} m^{2} x^{7} + 3003 \, \left (d x\right )^{m} b^{2} c x^{9} + 359 \, \left (d x\right )^{m} b^{3} m x^{7} + 1287 \, \left (d x\right )^{m} b^{3} x^{7}}{m^{4} + 40 \, m^{3} + 590 \, m^{2} + 3800 \, m + 9009} \] Input:
integrate((d*x)^m*(c*x^4+b*x^2)^3,x, algorithm="giac")
Output:
((d*x)^m*c^3*m^3*x^13 + 27*(d*x)^m*c^3*m^2*x^13 + 3*(d*x)^m*b*c^2*m^3*x^11 + 239*(d*x)^m*c^3*m*x^13 + 87*(d*x)^m*b*c^2*m^2*x^11 + 693*(d*x)^m*c^3*x^ 13 + 3*(d*x)^m*b^2*c*m^3*x^9 + 813*(d*x)^m*b*c^2*m*x^11 + 93*(d*x)^m*b^2*c *m^2*x^9 + 2457*(d*x)^m*b*c^2*x^11 + (d*x)^m*b^3*m^3*x^7 + 933*(d*x)^m*b^2 *c*m*x^9 + 33*(d*x)^m*b^3*m^2*x^7 + 3003*(d*x)^m*b^2*c*x^9 + 359*(d*x)^m*b ^3*m*x^7 + 1287*(d*x)^m*b^3*x^7)/(m^4 + 40*m^3 + 590*m^2 + 3800*m + 9009)
Time = 17.80 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.11 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx={\left (d\,x\right )}^m\,\left (\frac {b^3\,x^7\,\left (m^3+33\,m^2+359\,m+1287\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {c^3\,x^{13}\,\left (m^3+27\,m^2+239\,m+693\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {3\,b\,c^2\,x^{11}\,\left (m^3+29\,m^2+271\,m+819\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}+\frac {3\,b^2\,c\,x^9\,\left (m^3+31\,m^2+311\,m+1001\right )}{m^4+40\,m^3+590\,m^2+3800\,m+9009}\right ) \] Input:
int((d*x)^m*(b*x^2 + c*x^4)^3,x)
Output:
(d*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))/(38 00*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))
Time = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.23 \[ \int (d x)^m \left (b x^2+c x^4\right )^3 \, dx=\frac {x^{m} d^{m} x^{7} \left (c^{3} m^{3} x^{6}+27 c^{3} m^{2} x^{6}+3 b \,c^{2} m^{3} x^{4}+239 c^{3} m \,x^{6}+87 b \,c^{2} m^{2} x^{4}+693 c^{3} x^{6}+3 b^{2} c \,m^{3} x^{2}+813 b \,c^{2} m \,x^{4}+93 b^{2} c \,m^{2} x^{2}+2457 b \,c^{2} x^{4}+b^{3} m^{3}+933 b^{2} c m \,x^{2}+33 b^{3} m^{2}+3003 b^{2} c \,x^{2}+359 b^{3} m +1287 b^{3}\right )}{m^{4}+40 m^{3}+590 m^{2}+3800 m +9009} \] Input:
int((d*x)^m*(c*x^4+b*x^2)^3,x)
Output:
(x**m*d**m*x**7*(b**3*m**3 + 33*b**3*m**2 + 359*b**3*m + 1287*b**3 + 3*b** 2*c*m**3*x**2 + 93*b**2*c*m**2*x**2 + 933*b**2*c*m*x**2 + 3003*b**2*c*x**2 + 3*b*c**2*m**3*x**4 + 87*b*c**2*m**2*x**4 + 813*b*c**2*m*x**4 + 2457*b*c **2*x**4 + c**3*m**3*x**6 + 27*c**3*m**2*x**6 + 239*c**3*m*x**6 + 693*c**3 *x**6))/(m**4 + 40*m**3 + 590*m**2 + 3800*m + 9009)