3.248 \(\int (d x)^m (a+b x^3+c x^6)^2 \, dx\)
Optimal. Leaf size=101 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left (2 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac{2 a b (d x)^{m+4}}{d^4 (m+4)}+\frac{2 b c (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 (d x)^{m+13}}{d^{13} (m+13)} \]
[Out]
(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(4 + m))/(d^4*(4 + m)) + ((b^2 + 2*a*c)*(d*x)^(7 + m))/(d^7*(7
+ m)) + (2*b*c*(d*x)^(10 + m))/(d^10*(10 + m)) + (c^2*(d*x)^(13 + m))/(d^13*(13 + m))
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Rubi [A] time = 0.0609639, antiderivative size = 101, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{1353} \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left (2 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac{2 a b (d x)^{m+4}}{d^4 (m+4)}+\frac{2 b c (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 (d x)^{m+13}}{d^{13} (m+13)} \]
Antiderivative was successfully verified.
[In]
Int[(d*x)^m*(a + b*x^3 + c*x^6)^2,x]
[Out]
(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(4 + m))/(d^4*(4 + m)) + ((b^2 + 2*a*c)*(d*x)^(7 + m))/(d^7*(7
+ m)) + (2*b*c*(d*x)^(10 + m))/(d^10*(10 + m)) + (c^2*(d*x)^(13 + m))/(d^13*(13 + m))
Rule 1353
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d
*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && !Int
egerQ[Simplify[(m + 1)/n]]
Rubi steps
\begin{align*} \int (d x)^m \left (a+b x^3+c x^6\right )^2 \, dx &=\int \left (a^2 (d x)^m+\frac{2 a b (d x)^{3+m}}{d^3}+\frac{\left (b^2+2 a c\right ) (d x)^{6+m}}{d^6}+\frac{2 b c (d x)^{9+m}}{d^9}+\frac{c^2 (d x)^{12+m}}{d^{12}}\right ) \, dx\\ &=\frac{a^2 (d x)^{1+m}}{d (1+m)}+\frac{2 a b (d x)^{4+m}}{d^4 (4+m)}+\frac{\left (b^2+2 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac{2 b c (d x)^{10+m}}{d^{10} (10+m)}+\frac{c^2 (d x)^{13+m}}{d^{13} (13+m)}\\ \end{align*}
Mathematica [A] time = 0.0715919, size = 70, normalized size = 0.69 \[ x (d x)^m \left (\frac{a^2}{m+1}+\frac{x^6 \left (2 a c+b^2\right )}{m+7}+\frac{2 a b x^3}{m+4}+\frac{2 b c x^9}{m+10}+\frac{c^2 x^{12}}{m+13}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(d*x)^m*(a + b*x^3 + c*x^6)^2,x]
[Out]
x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^3)/(4 + m) + ((b^2 + 2*a*c)*x^6)/(7 + m) + (2*b*c*x^9)/(10 + m) + (c^2*x^12)
/(13 + m))
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Maple [B] time = 0.007, size = 301, normalized size = 3. \begin{align*}{\frac{ \left ({c}^{2}{m}^{4}{x}^{12}+22\,{c}^{2}{m}^{3}{x}^{12}+159\,{c}^{2}{m}^{2}{x}^{12}+2\,bc{m}^{4}{x}^{9}+418\,{c}^{2}m{x}^{12}+50\,bc{m}^{3}{x}^{9}+280\,{c}^{2}{x}^{12}+390\,bc{m}^{2}{x}^{9}+2\,ac{m}^{4}{x}^{6}+{b}^{2}{m}^{4}{x}^{6}+1070\,bcm{x}^{9}+56\,ac{m}^{3}{x}^{6}+28\,{b}^{2}{m}^{3}{x}^{6}+728\,bc{x}^{9}+498\,ac{m}^{2}{x}^{6}+249\,{b}^{2}{m}^{2}{x}^{6}+2\,ab{m}^{4}{x}^{3}+1484\,acm{x}^{6}+742\,{b}^{2}m{x}^{6}+62\,ab{m}^{3}{x}^{3}+1040\,ac{x}^{6}+520\,{b}^{2}{x}^{6}+642\,ab{m}^{2}{x}^{3}+{a}^{2}{m}^{4}+2402\,abm{x}^{3}+34\,{a}^{2}{m}^{3}+1820\,ab{x}^{3}+411\,{a}^{2}{m}^{2}+2074\,{a}^{2}m+3640\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 13+m \right ) \left ( 10+m \right ) \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^m*(c*x^6+b*x^3+a)^2,x)
[Out]
x*(c^2*m^4*x^12+22*c^2*m^3*x^12+159*c^2*m^2*x^12+2*b*c*m^4*x^9+418*c^2*m*x^12+50*b*c*m^3*x^9+280*c^2*x^12+390*
b*c*m^2*x^9+2*a*c*m^4*x^6+b^2*m^4*x^6+1070*b*c*m*x^9+56*a*c*m^3*x^6+28*b^2*m^3*x^6+728*b*c*x^9+498*a*c*m^2*x^6
+249*b^2*m^2*x^6+2*a*b*m^4*x^3+1484*a*c*m*x^6+742*b^2*m*x^6+62*a*b*m^3*x^3+1040*a*c*x^6+520*b^2*x^6+642*a*b*m^
2*x^3+a^2*m^4+2402*a*b*m*x^3+34*a^2*m^3+1820*a*b*x^3+411*a^2*m^2+2074*a^2*m+3640*a^2)*(d*x)^m/(13+m)/(10+m)/(7
+m)/(4+m)/(1+m)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(c*x^6+b*x^3+a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.58221, size = 599, normalized size = 5.93 \begin{align*} \frac{{\left ({\left (c^{2} m^{4} + 22 \, c^{2} m^{3} + 159 \, c^{2} m^{2} + 418 \, c^{2} m + 280 \, c^{2}\right )} x^{13} + 2 \,{\left (b c m^{4} + 25 \, b c m^{3} + 195 \, b c m^{2} + 535 \, b c m + 364 \, b c\right )} x^{10} +{\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 28 \,{\left (b^{2} + 2 \, a c\right )} m^{3} + 249 \,{\left (b^{2} + 2 \, a c\right )} m^{2} + 520 \, b^{2} + 1040 \, a c + 742 \,{\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \,{\left (a b m^{4} + 31 \, a b m^{3} + 321 \, a b m^{2} + 1201 \, a b m + 910 \, a b\right )} x^{4} +{\left (a^{2} m^{4} + 34 \, a^{2} m^{3} + 411 \, a^{2} m^{2} + 2074 \, a^{2} m + 3640 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(c*x^6+b*x^3+a)^2,x, algorithm="fricas")
[Out]
((c^2*m^4 + 22*c^2*m^3 + 159*c^2*m^2 + 418*c^2*m + 280*c^2)*x^13 + 2*(b*c*m^4 + 25*b*c*m^3 + 195*b*c*m^2 + 535
*b*c*m + 364*b*c)*x^10 + ((b^2 + 2*a*c)*m^4 + 28*(b^2 + 2*a*c)*m^3 + 249*(b^2 + 2*a*c)*m^2 + 520*b^2 + 1040*a*
c + 742*(b^2 + 2*a*c)*m)*x^7 + 2*(a*b*m^4 + 31*a*b*m^3 + 321*a*b*m^2 + 1201*a*b*m + 910*a*b)*x^4 + (a^2*m^4 +
34*a^2*m^3 + 411*a^2*m^2 + 2074*a^2*m + 3640*a^2)*x)*(d*x)^m/(m^5 + 35*m^4 + 445*m^3 + 2485*m^2 + 5714*m + 364
0)
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Sympy [A] time = 6.02886, size = 1510, normalized size = 14.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)**m*(c*x**6+b*x**3+a)**2,x)
[Out]
Piecewise(((-a**2/(12*x**12) - 2*a*b/(9*x**9) - a*c/(3*x**6) - b**2/(6*x**6) - 2*b*c/(3*x**3) + c**2*log(x))/d
**13, Eq(m, -13)), ((-a**2/(9*x**9) - a*b/(3*x**6) - 2*a*c/(3*x**3) - b**2/(3*x**3) + 2*b*c*log(x) + c**2*x**3
/3)/d**10, Eq(m, -10)), ((-a**2/(6*x**6) - 2*a*b/(3*x**3) + 2*a*c*log(x) + b**2*log(x) + 2*b*c*x**3/3 + c**2*x
**6/6)/d**7, Eq(m, -7)), ((-a**2/(3*x**3) + 2*a*b*log(x) + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**6/3 + c**2*x**9
/9)/d**4, Eq(m, -4)), ((a**2*log(x) + 2*a*b*x**3/3 + a*c*x**6/3 + b**2*x**6/6 + 2*b*c*x**9/9 + c**2*x**12/12)/
d, Eq(m, -1)), (a**2*d**m*m**4*x*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 34*a**2*d**m*m
**3*x*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 411*a**2*d**m*m**2*x*x**m/(m**5 + 35*m**4
+ 445*m**3 + 2485*m**2 + 5714*m + 3640) + 2074*a**2*d**m*m*x*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 57
14*m + 3640) + 3640*a**2*d**m*x*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 2*a*b*d**m*m**4
*x**4*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 62*a*b*d**m*m**3*x**4*x**m/(m**5 + 35*m**
4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 642*a*b*d**m*m**2*x**4*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2
+ 5714*m + 3640) + 2402*a*b*d**m*m*x**4*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1820*a
*b*d**m*x**4*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 2*a*c*d**m*m**4*x**7*x**m/(m**5 +
35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 56*a*c*d**m*m**3*x**7*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485
*m**2 + 5714*m + 3640) + 498*a*c*d**m*m**2*x**7*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) +
1484*a*c*d**m*m*x**7*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1040*a*c*d**m*x**7*x**m/(
m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + b**2*d**m*m**4*x**7*x**m/(m**5 + 35*m**4 + 445*m**3 +
2485*m**2 + 5714*m + 3640) + 28*b**2*d**m*m**3*x**7*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 36
40) + 249*b**2*d**m*m**2*x**7*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 742*b**2*d**m*m*x
**7*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 520*b**2*d**m*x**7*x**m/(m**5 + 35*m**4 + 4
45*m**3 + 2485*m**2 + 5714*m + 3640) + 2*b*c*d**m*m**4*x**10*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 571
4*m + 3640) + 50*b*c*d**m*m**3*x**10*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 390*b*c*d*
*m*m**2*x**10*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 1070*b*c*d**m*m*x**10*x**m/(m**5
+ 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 728*b*c*d**m*x**10*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*
m**2 + 5714*m + 3640) + c**2*d**m*m**4*x**13*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 22
*c**2*d**m*m**3*x**13*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 159*c**2*d**m*m**2*x**13*
x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m + 3640) + 418*c**2*d**m*m*x**13*x**m/(m**5 + 35*m**4 + 44
5*m**3 + 2485*m**2 + 5714*m + 3640) + 280*c**2*d**m*x**13*x**m/(m**5 + 35*m**4 + 445*m**3 + 2485*m**2 + 5714*m
+ 3640), True))
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Giac [B] time = 1.16794, size = 606, normalized size = 6. \begin{align*} \frac{\left (d x\right )^{m} c^{2} m^{4} x^{13} + 22 \, \left (d x\right )^{m} c^{2} m^{3} x^{13} + 159 \, \left (d x\right )^{m} c^{2} m^{2} x^{13} + 2 \, \left (d x\right )^{m} b c m^{4} x^{10} + 418 \, \left (d x\right )^{m} c^{2} m x^{13} + 50 \, \left (d x\right )^{m} b c m^{3} x^{10} + 280 \, \left (d x\right )^{m} c^{2} x^{13} + 390 \, \left (d x\right )^{m} b c m^{2} x^{10} + \left (d x\right )^{m} b^{2} m^{4} x^{7} + 2 \, \left (d x\right )^{m} a c m^{4} x^{7} + 1070 \, \left (d x\right )^{m} b c m x^{10} + 28 \, \left (d x\right )^{m} b^{2} m^{3} x^{7} + 56 \, \left (d x\right )^{m} a c m^{3} x^{7} + 728 \, \left (d x\right )^{m} b c x^{10} + 249 \, \left (d x\right )^{m} b^{2} m^{2} x^{7} + 498 \, \left (d x\right )^{m} a c m^{2} x^{7} + 2 \, \left (d x\right )^{m} a b m^{4} x^{4} + 742 \, \left (d x\right )^{m} b^{2} m x^{7} + 1484 \, \left (d x\right )^{m} a c m x^{7} + 62 \, \left (d x\right )^{m} a b m^{3} x^{4} + 520 \, \left (d x\right )^{m} b^{2} x^{7} + 1040 \, \left (d x\right )^{m} a c x^{7} + 642 \, \left (d x\right )^{m} a b m^{2} x^{4} + \left (d x\right )^{m} a^{2} m^{4} x + 2402 \, \left (d x\right )^{m} a b m x^{4} + 34 \, \left (d x\right )^{m} a^{2} m^{3} x + 1820 \, \left (d x\right )^{m} a b x^{4} + 411 \, \left (d x\right )^{m} a^{2} m^{2} x + 2074 \, \left (d x\right )^{m} a^{2} m x + 3640 \, \left (d x\right )^{m} a^{2} x}{m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(c*x^6+b*x^3+a)^2,x, algorithm="giac")
[Out]
((d*x)^m*c^2*m^4*x^13 + 22*(d*x)^m*c^2*m^3*x^13 + 159*(d*x)^m*c^2*m^2*x^13 + 2*(d*x)^m*b*c*m^4*x^10 + 418*(d*x
)^m*c^2*m*x^13 + 50*(d*x)^m*b*c*m^3*x^10 + 280*(d*x)^m*c^2*x^13 + 390*(d*x)^m*b*c*m^2*x^10 + (d*x)^m*b^2*m^4*x
^7 + 2*(d*x)^m*a*c*m^4*x^7 + 1070*(d*x)^m*b*c*m*x^10 + 28*(d*x)^m*b^2*m^3*x^7 + 56*(d*x)^m*a*c*m^3*x^7 + 728*(
d*x)^m*b*c*x^10 + 249*(d*x)^m*b^2*m^2*x^7 + 498*(d*x)^m*a*c*m^2*x^7 + 2*(d*x)^m*a*b*m^4*x^4 + 742*(d*x)^m*b^2*
m*x^7 + 1484*(d*x)^m*a*c*m*x^7 + 62*(d*x)^m*a*b*m^3*x^4 + 520*(d*x)^m*b^2*x^7 + 1040*(d*x)^m*a*c*x^7 + 642*(d*
x)^m*a*b*m^2*x^4 + (d*x)^m*a^2*m^4*x + 2402*(d*x)^m*a*b*m*x^4 + 34*(d*x)^m*a^2*m^3*x + 1820*(d*x)^m*a*b*x^4 +
411*(d*x)^m*a^2*m^2*x + 2074*(d*x)^m*a^2*m*x + 3640*(d*x)^m*a^2*x)/(m^5 + 35*m^4 + 445*m^3 + 2485*m^2 + 5714*m
+ 3640)