3.248 \(\int (d x)^m \left (a+b x^3+c x^6\right )^2 \, dx\)

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 = 0.128446, 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 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left (2 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac{2 a b (d x)^{m+4}}{d^4 (m+4)}+\frac{2 b c (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 (d x)^{m+13}}{d^{13} (m+13)} \]

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))

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Rubi in Sympy [A]  time = 26.3899, size = 90, normalized size = 0.89 \[ \frac{a^{2} \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{2 a b \left (d x\right )^{m + 4}}{d^{4} \left (m + 4\right )} + \frac{2 b c \left (d x\right )^{m + 10}}{d^{10} \left (m + 10\right )} + \frac{c^{2} \left (d x\right )^{m + 13}}{d^{13} \left (m + 13\right )} + \frac{\left (d x\right )^{m + 7} \left (2 a c + b^{2}\right )}{d^{7} \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**6+b*x**3+a)**2,x)

[Out]

a**2*(d*x)**(m + 1)/(d*(m + 1)) + 2*a*b*(d*x)**(m + 4)/(d**4*(m + 4)) + 2*b*c*(d
*x)**(m + 10)/(d**10*(m + 10)) + c**2*(d*x)**(m + 13)/(d**13*(m + 13)) + (d*x)**
(m + 7)*(2*a*c + b**2)/(d**7*(m + 7))

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Mathematica [A]  time = 0.0842854, size = 70, normalized size = 0.69 \[ (d x)^m \left (\frac{a^2 x}{m+1}+\frac{x^7 \left (2 a c+b^2\right )}{m+7}+\frac{2 a b x^4}{m+4}+\frac{2 b c x^{10}}{m+10}+\frac{c^2 x^{13}}{m+13}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(a + b*x^3 + c*x^6)^2,x]

[Out]

(d*x)^m*((a^2*x)/(1 + m) + (2*a*b*x^4)/(4 + m) + ((b^2 + 2*a*c)*x^7)/(7 + m) + (
2*b*c*x^10)/(10 + m) + (c^2*x^13)/(13 + m))

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Maple [B]  time = 0.009, size = 301, normalized size = 3. \[{\frac{ \left ({c}^{2}{m}^{4}{x}^{12}+22\,{c}^{2}{m}^{3}{x}^{12}+159\,{c}^{2}{m}^{2}{x}^{12}+2\,bc{m}^{4}{x}^{9}+418\,{c}^{2}m{x}^{12}+50\,bc{m}^{3}{x}^{9}+280\,{c}^{2}{x}^{12}+390\,bc{m}^{2}{x}^{9}+2\,ac{m}^{4}{x}^{6}+{b}^{2}{m}^{4}{x}^{6}+1070\,bcm{x}^{9}+56\,ac{m}^{3}{x}^{6}+28\,{b}^{2}{m}^{3}{x}^{6}+728\,bc{x}^{9}+498\,ac{m}^{2}{x}^{6}+249\,{b}^{2}{m}^{2}{x}^{6}+2\,ab{m}^{4}{x}^{3}+1484\,acm{x}^{6}+742\,{b}^{2}m{x}^{6}+62\,ab{m}^{3}{x}^{3}+1040\,ac{x}^{6}+520\,{b}^{2}{x}^{6}+642\,ab{m}^{2}{x}^{3}+{a}^{2}{m}^{4}+2402\,abm{x}^{3}+34\,{a}^{2}{m}^{3}+1820\,ab{x}^{3}+411\,{a}^{2}{m}^{2}+2074\,{a}^{2}m+3640\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 13+m \right ) \left ( 10+m \right ) \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^2*(d*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301979, size = 325, normalized size = 3.22 \[ \frac{{\left ({\left (c^{2} m^{4} + 22 \, c^{2} m^{3} + 159 \, c^{2} m^{2} + 418 \, c^{2} m + 280 \, c^{2}\right )} x^{13} + 2 \,{\left (b c m^{4} + 25 \, b c m^{3} + 195 \, b c m^{2} + 535 \, b c m + 364 \, b c\right )} x^{10} +{\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 28 \,{\left (b^{2} + 2 \, a c\right )} m^{3} + 249 \,{\left (b^{2} + 2 \, a c\right )} m^{2} + 520 \, b^{2} + 1040 \, a c + 742 \,{\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \,{\left (a b m^{4} + 31 \, a b m^{3} + 321 \, a b m^{2} + 1201 \, a b m + 910 \, a b\right )} x^{4} +{\left (a^{2} m^{4} + 34 \, a^{2} m^{3} + 411 \, a^{2} m^{2} + 2074 \, a^{2} m + 3640 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 35 \, m^{4} + 445 \, m^{3} + 2485 \, m^{2} + 5714 \, m + 3640} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^2*(d*x)^m,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 + 3640)

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Sympy [A]  time = 18.5219, size = 1510, normalized size = 14.95 \[ \text{result too large to display} \]

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 + 36
40) + 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 + 57
14*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 + 3640) + 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 + 445*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 + 5714*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 + 2
485*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 + 445*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/XCAS [A]  time = 0.291641, size = 687, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^2*(d*x)^m,x, algorithm="giac")

[Out]

(c^2*m^4*x^13*e^(m*ln(d*x)) + 22*c^2*m^3*x^13*e^(m*ln(d*x)) + 159*c^2*m^2*x^13*e
^(m*ln(d*x)) + 2*b*c*m^4*x^10*e^(m*ln(d*x)) + 418*c^2*m*x^13*e^(m*ln(d*x)) + 50*
b*c*m^3*x^10*e^(m*ln(d*x)) + 280*c^2*x^13*e^(m*ln(d*x)) + 390*b*c*m^2*x^10*e^(m*
ln(d*x)) + b^2*m^4*x^7*e^(m*ln(d*x)) + 2*a*c*m^4*x^7*e^(m*ln(d*x)) + 1070*b*c*m*
x^10*e^(m*ln(d*x)) + 28*b^2*m^3*x^7*e^(m*ln(d*x)) + 56*a*c*m^3*x^7*e^(m*ln(d*x))
+ 728*b*c*x^10*e^(m*ln(d*x)) + 249*b^2*m^2*x^7*e^(m*ln(d*x)) + 498*a*c*m^2*x^7*
e^(m*ln(d*x)) + 2*a*b*m^4*x^4*e^(m*ln(d*x)) + 742*b^2*m*x^7*e^(m*ln(d*x)) + 1484
*a*c*m*x^7*e^(m*ln(d*x)) + 62*a*b*m^3*x^4*e^(m*ln(d*x)) + 520*b^2*x^7*e^(m*ln(d*
x)) + 1040*a*c*x^7*e^(m*ln(d*x)) + 642*a*b*m^2*x^4*e^(m*ln(d*x)) + a^2*m^4*x*e^(
m*ln(d*x)) + 2402*a*b*m*x^4*e^(m*ln(d*x)) + 34*a^2*m^3*x*e^(m*ln(d*x)) + 1820*a*
b*x^4*e^(m*ln(d*x)) + 411*a^2*m^2*x*e^(m*ln(d*x)) + 2074*a^2*m*x*e^(m*ln(d*x)) +
3640*a^2*x*e^(m*ln(d*x)))/(m^5 + 35*m^4 + 445*m^3 + 2485*m^2 + 5714*m + 3640)