3.27 \(\int x^m (A+B x) \left (b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=96 \[ \frac{A b^3 x^{m+4}}{m+4}+\frac{b^2 x^{m+5} (3 A c+b B)}{m+5}+\frac{c^2 x^{m+7} (A c+3 b B)}{m+7}+\frac{3 b c x^{m+6} (A c+b B)}{m+6}+\frac{B c^3 x^{m+8}}{m+8} \]

[Out]

(A*b^3*x^(4 + m))/(4 + m) + (b^2*(b*B + 3*A*c)*x^(5 + m))/(5 + m) + (3*b*c*(b*B
+ A*c)*x^(6 + m))/(6 + m) + (c^2*(3*b*B + A*c)*x^(7 + m))/(7 + m) + (B*c^3*x^(8
+ m))/(8 + m)

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Rubi [A]  time = 0.159285, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{A b^3 x^{m+4}}{m+4}+\frac{b^2 x^{m+5} (3 A c+b B)}{m+5}+\frac{c^2 x^{m+7} (A c+3 b B)}{m+7}+\frac{3 b c x^{m+6} (A c+b B)}{m+6}+\frac{B c^3 x^{m+8}}{m+8} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*x^(4 + m))/(4 + m) + (b^2*(b*B + 3*A*c)*x^(5 + m))/(5 + m) + (3*b*c*(b*B
+ A*c)*x^(6 + m))/(6 + m) + (c^2*(3*b*B + A*c)*x^(7 + m))/(7 + m) + (B*c^3*x^(8
+ m))/(8 + m)

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Rubi in Sympy [A]  time = 19.8472, size = 87, normalized size = 0.91 \[ \frac{A b^{3} x^{m + 4}}{m + 4} + \frac{B c^{3} x^{m + 8}}{m + 8} + \frac{b^{2} x^{m + 5} \left (3 A c + B b\right )}{m + 5} + \frac{3 b c x^{m + 6} \left (A c + B b\right )}{m + 6} + \frac{c^{2} x^{m + 7} \left (A c + 3 B b\right )}{m + 7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(B*x+A)*(c*x**2+b*x)**3,x)

[Out]

A*b**3*x**(m + 4)/(m + 4) + B*c**3*x**(m + 8)/(m + 8) + b**2*x**(m + 5)*(3*A*c +
B*b)/(m + 5) + 3*b*c*x**(m + 6)*(A*c + B*b)/(m + 6) + c**2*x**(m + 7)*(A*c + 3*
B*b)/(m + 7)

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Mathematica [A]  time = 0.127419, size = 87, normalized size = 0.91 \[ x^{m+4} \left (\frac{A b^3}{m+4}+\frac{b^2 x (3 A c+b B)}{m+5}+\frac{c^2 x^3 (A c+3 b B)}{m+7}+\frac{3 b c x^2 (A c+b B)}{m+6}+\frac{B c^3 x^4}{m+8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

x^(4 + m)*((A*b^3)/(4 + m) + (b^2*(b*B + 3*A*c)*x)/(5 + m) + (3*b*c*(b*B + A*c)*
x^2)/(6 + m) + (c^2*(3*b*B + A*c)*x^3)/(7 + m) + (B*c^3*x^4)/(8 + m))

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Maple [B]  time = 0.008, size = 454, normalized size = 4.7 \[{\frac{{x}^{4+m} \left ( B{c}^{3}{m}^{4}{x}^{4}+A{c}^{3}{m}^{4}{x}^{3}+3\,Bb{c}^{2}{m}^{4}{x}^{3}+22\,B{c}^{3}{m}^{3}{x}^{4}+3\,Ab{c}^{2}{m}^{4}{x}^{2}+23\,A{c}^{3}{m}^{3}{x}^{3}+3\,B{b}^{2}c{m}^{4}{x}^{2}+69\,Bb{c}^{2}{m}^{3}{x}^{3}+179\,B{c}^{3}{m}^{2}{x}^{4}+3\,A{b}^{2}c{m}^{4}x+72\,Ab{c}^{2}{m}^{3}{x}^{2}+194\,A{c}^{3}{m}^{2}{x}^{3}+B{b}^{3}{m}^{4}x+72\,B{b}^{2}c{m}^{3}{x}^{2}+582\,Bb{c}^{2}{m}^{2}{x}^{3}+638\,B{c}^{3}m{x}^{4}+A{b}^{3}{m}^{4}+75\,A{b}^{2}c{m}^{3}x+633\,Ab{c}^{2}{m}^{2}{x}^{2}+712\,A{c}^{3}m{x}^{3}+25\,B{b}^{3}{m}^{3}x+633\,B{b}^{2}c{m}^{2}{x}^{2}+2136\,Bb{c}^{2}m{x}^{3}+840\,B{c}^{3}{x}^{4}+26\,A{b}^{3}{m}^{3}+690\,A{b}^{2}c{m}^{2}x+2412\,Ab{c}^{2}m{x}^{2}+960\,A{c}^{3}{x}^{3}+230\,B{b}^{3}{m}^{2}x+2412\,B{b}^{2}cm{x}^{2}+2880\,Bb{c}^{2}{x}^{3}+251\,A{b}^{3}{m}^{2}+2760\,A{b}^{2}cmx+3360\,Ab{c}^{2}{x}^{2}+920\,B{b}^{3}mx+3360\,B{b}^{2}c{x}^{2}+1066\,A{b}^{3}m+4032\,A{b}^{2}cx+1344\,B{b}^{3}x+1680\,A{b}^{3} \right ) }{ \left ( 8+m \right ) \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(B*x+A)*(c*x^2+b*x)^3,x)

[Out]

x^(4+m)*(B*c^3*m^4*x^4+A*c^3*m^4*x^3+3*B*b*c^2*m^4*x^3+22*B*c^3*m^3*x^4+3*A*b*c^
2*m^4*x^2+23*A*c^3*m^3*x^3+3*B*b^2*c*m^4*x^2+69*B*b*c^2*m^3*x^3+179*B*c^3*m^2*x^
4+3*A*b^2*c*m^4*x+72*A*b*c^2*m^3*x^2+194*A*c^3*m^2*x^3+B*b^3*m^4*x+72*B*b^2*c*m^
3*x^2+582*B*b*c^2*m^2*x^3+638*B*c^3*m*x^4+A*b^3*m^4+75*A*b^2*c*m^3*x+633*A*b*c^2
*m^2*x^2+712*A*c^3*m*x^3+25*B*b^3*m^3*x+633*B*b^2*c*m^2*x^2+2136*B*b*c^2*m*x^3+8
40*B*c^3*x^4+26*A*b^3*m^3+690*A*b^2*c*m^2*x+2412*A*b*c^2*m*x^2+960*A*c^3*x^3+230
*B*b^3*m^2*x+2412*B*b^2*c*m*x^2+2880*B*b*c^2*x^3+251*A*b^3*m^2+2760*A*b^2*c*m*x+
3360*A*b*c^2*x^2+920*B*b^3*m*x+3360*B*b^2*c*x^2+1066*A*b^3*m+4032*A*b^2*c*x+1344
*B*b^3*x+1680*A*b^3)/(8+m)/(7+m)/(6+m)/(5+m)/(4+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^2 + b*x)^3*(B*x + A)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299127, size = 514, normalized size = 5.35 \[ \frac{{\left ({\left (B c^{3} m^{4} + 22 \, B c^{3} m^{3} + 179 \, B c^{3} m^{2} + 638 \, B c^{3} m + 840 \, B c^{3}\right )} x^{8} +{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} m^{4} + 2880 \, B b c^{2} + 960 \, A c^{3} + 23 \,{\left (3 \, B b c^{2} + A c^{3}\right )} m^{3} + 194 \,{\left (3 \, B b c^{2} + A c^{3}\right )} m^{2} + 712 \,{\left (3 \, B b c^{2} + A c^{3}\right )} m\right )} x^{7} + 3 \,{\left ({\left (B b^{2} c + A b c^{2}\right )} m^{4} + 1120 \, B b^{2} c + 1120 \, A b c^{2} + 24 \,{\left (B b^{2} c + A b c^{2}\right )} m^{3} + 211 \,{\left (B b^{2} c + A b c^{2}\right )} m^{2} + 804 \,{\left (B b^{2} c + A b c^{2}\right )} m\right )} x^{6} +{\left ({\left (B b^{3} + 3 \, A b^{2} c\right )} m^{4} + 1344 \, B b^{3} + 4032 \, A b^{2} c + 25 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} m^{3} + 230 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} m^{2} + 920 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} m\right )} x^{5} +{\left (A b^{3} m^{4} + 26 \, A b^{3} m^{3} + 251 \, A b^{3} m^{2} + 1066 \, A b^{3} m + 1680 \, A b^{3}\right )} x^{4}\right )} x^{m}}{m^{5} + 30 \, m^{4} + 355 \, m^{3} + 2070 \, m^{2} + 5944 \, m + 6720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(B*x + A)*x^m,x, algorithm="fricas")

[Out]

((B*c^3*m^4 + 22*B*c^3*m^3 + 179*B*c^3*m^2 + 638*B*c^3*m + 840*B*c^3)*x^8 + ((3*
B*b*c^2 + A*c^3)*m^4 + 2880*B*b*c^2 + 960*A*c^3 + 23*(3*B*b*c^2 + A*c^3)*m^3 + 1
94*(3*B*b*c^2 + A*c^3)*m^2 + 712*(3*B*b*c^2 + A*c^3)*m)*x^7 + 3*((B*b^2*c + A*b*
c^2)*m^4 + 1120*B*b^2*c + 1120*A*b*c^2 + 24*(B*b^2*c + A*b*c^2)*m^3 + 211*(B*b^2
*c + A*b*c^2)*m^2 + 804*(B*b^2*c + A*b*c^2)*m)*x^6 + ((B*b^3 + 3*A*b^2*c)*m^4 +
1344*B*b^3 + 4032*A*b^2*c + 25*(B*b^3 + 3*A*b^2*c)*m^3 + 230*(B*b^3 + 3*A*b^2*c)
*m^2 + 920*(B*b^3 + 3*A*b^2*c)*m)*x^5 + (A*b^3*m^4 + 26*A*b^3*m^3 + 251*A*b^3*m^
2 + 1066*A*b^3*m + 1680*A*b^3)*x^4)*x^m/(m^5 + 30*m^4 + 355*m^3 + 2070*m^2 + 594
4*m + 6720)

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Sympy [A]  time = 8.2457, size = 2026, normalized size = 21.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(B*x+A)*(c*x**2+b*x)**3,x)

[Out]

Piecewise((-A*b**3/(4*x**4) - A*b**2*c/x**3 - 3*A*b*c**2/(2*x**2) - A*c**3/x - B
*b**3/(3*x**3) - 3*B*b**2*c/(2*x**2) - 3*B*b*c**2/x + B*c**3*log(x), Eq(m, -8)),
(-A*b**3/(3*x**3) - 3*A*b**2*c/(2*x**2) - 3*A*b*c**2/x + A*c**3*log(x) - B*b**3
/(2*x**2) - 3*B*b**2*c/x + 3*B*b*c**2*log(x) + B*c**3*x, Eq(m, -7)), (-A*b**3/(2
*x**2) - 3*A*b**2*c/x + 3*A*b*c**2*log(x) + A*c**3*x - B*b**3/x + 3*B*b**2*c*log
(x) + 3*B*b*c**2*x + B*c**3*x**2/2, Eq(m, -6)), (-A*b**3/x + 3*A*b**2*c*log(x) +
3*A*b*c**2*x + A*c**3*x**2/2 + B*b**3*log(x) + 3*B*b**2*c*x + 3*B*b*c**2*x**2/2
+ B*c**3*x**3/3, Eq(m, -5)), (A*b**3*log(x) + 3*A*b**2*c*x + 3*A*b*c**2*x**2/2
+ A*c**3*x**3/3 + B*b**3*x + 3*B*b**2*c*x**2/2 + B*b*c**2*x**3 + B*c**3*x**4/4,
Eq(m, -4)), (A*b**3*m**4*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944
*m + 6720) + 26*A*b**3*m**3*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5
944*m + 6720) + 251*A*b**3*m**2*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2
+ 5944*m + 6720) + 1066*A*b**3*m*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m*
*2 + 5944*m + 6720) + 1680*A*b**3*x**4*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m*
*2 + 5944*m + 6720) + 3*A*b**2*c*m**4*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 207
0*m**2 + 5944*m + 6720) + 75*A*b**2*c*m**3*x**5*x**m/(m**5 + 30*m**4 + 355*m**3
+ 2070*m**2 + 5944*m + 6720) + 690*A*b**2*c*m**2*x**5*x**m/(m**5 + 30*m**4 + 355
*m**3 + 2070*m**2 + 5944*m + 6720) + 2760*A*b**2*c*m*x**5*x**m/(m**5 + 30*m**4 +
355*m**3 + 2070*m**2 + 5944*m + 6720) + 4032*A*b**2*c*x**5*x**m/(m**5 + 30*m**4
+ 355*m**3 + 2070*m**2 + 5944*m + 6720) + 3*A*b*c**2*m**4*x**6*x**m/(m**5 + 30*
m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 72*A*b*c**2*m**3*x**6*x**m/(m**5
+ 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 633*A*b*c**2*m**2*x**6*x**m/
(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 2412*A*b*c**2*m*x**6*x
**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 3360*A*b*c**2*x**6
*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + A*c**3*m**4*x**7
*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 23*A*c**3*m**3*x
**7*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 194*A*c**3*m*
*2*x**7*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 712*A*c**
3*m*x**7*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 960*A*c*
*3*x**7*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + B*b**3*m*
*4*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 25*B*b**3
*m**3*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 230*B*
b**3*m**2*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 92
0*B*b**3*m*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 1
344*B*b**3*x**5*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 3
*B*b**2*c*m**4*x**6*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720)
+ 72*B*b**2*c*m**3*x**6*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 5944*m +
6720) + 633*B*b**2*c*m**2*x**6*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 + 594
4*m + 6720) + 2412*B*b**2*c*m*x**6*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2 +
5944*m + 6720) + 3360*B*b**2*c*x**6*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*m**2
+ 5944*m + 6720) + 3*B*b*c**2*m**4*x**7*x**m/(m**5 + 30*m**4 + 355*m**3 + 2070*
m**2 + 5944*m + 6720) + 69*B*b*c**2*m**3*x**7*x**m/(m**5 + 30*m**4 + 355*m**3 +
2070*m**2 + 5944*m + 6720) + 582*B*b*c**2*m**2*x**7*x**m/(m**5 + 30*m**4 + 355*m
**3 + 2070*m**2 + 5944*m + 6720) + 2136*B*b*c**2*m*x**7*x**m/(m**5 + 30*m**4 + 3
55*m**3 + 2070*m**2 + 5944*m + 6720) + 2880*B*b*c**2*x**7*x**m/(m**5 + 30*m**4 +
355*m**3 + 2070*m**2 + 5944*m + 6720) + B*c**3*m**4*x**8*x**m/(m**5 + 30*m**4 +
355*m**3 + 2070*m**2 + 5944*m + 6720) + 22*B*c**3*m**3*x**8*x**m/(m**5 + 30*m**
4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 179*B*c**3*m**2*x**8*x**m/(m**5 + 30
*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 638*B*c**3*m*x**8*x**m/(m**5 + 3
0*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720) + 840*B*c**3*x**8*x**m/(m**5 + 30
*m**4 + 355*m**3 + 2070*m**2 + 5944*m + 6720), True))

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GIAC/XCAS [A]  time = 0.279804, size = 922, normalized size = 9.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*c^3*m^4*x^8*e^(m*ln(x)) + 3*B*b*c^2*m^4*x^7*e^(m*ln(x)) + A*c^3*m^4*x^7*e^(m*
ln(x)) + 22*B*c^3*m^3*x^8*e^(m*ln(x)) + 3*B*b^2*c*m^4*x^6*e^(m*ln(x)) + 3*A*b*c^
2*m^4*x^6*e^(m*ln(x)) + 69*B*b*c^2*m^3*x^7*e^(m*ln(x)) + 23*A*c^3*m^3*x^7*e^(m*l
n(x)) + 179*B*c^3*m^2*x^8*e^(m*ln(x)) + B*b^3*m^4*x^5*e^(m*ln(x)) + 3*A*b^2*c*m^
4*x^5*e^(m*ln(x)) + 72*B*b^2*c*m^3*x^6*e^(m*ln(x)) + 72*A*b*c^2*m^3*x^6*e^(m*ln(
x)) + 582*B*b*c^2*m^2*x^7*e^(m*ln(x)) + 194*A*c^3*m^2*x^7*e^(m*ln(x)) + 638*B*c^
3*m*x^8*e^(m*ln(x)) + A*b^3*m^4*x^4*e^(m*ln(x)) + 25*B*b^3*m^3*x^5*e^(m*ln(x)) +
75*A*b^2*c*m^3*x^5*e^(m*ln(x)) + 633*B*b^2*c*m^2*x^6*e^(m*ln(x)) + 633*A*b*c^2*
m^2*x^6*e^(m*ln(x)) + 2136*B*b*c^2*m*x^7*e^(m*ln(x)) + 712*A*c^3*m*x^7*e^(m*ln(x
)) + 840*B*c^3*x^8*e^(m*ln(x)) + 26*A*b^3*m^3*x^4*e^(m*ln(x)) + 230*B*b^3*m^2*x^
5*e^(m*ln(x)) + 690*A*b^2*c*m^2*x^5*e^(m*ln(x)) + 2412*B*b^2*c*m*x^6*e^(m*ln(x))
+ 2412*A*b*c^2*m*x^6*e^(m*ln(x)) + 2880*B*b*c^2*x^7*e^(m*ln(x)) + 960*A*c^3*x^7
*e^(m*ln(x)) + 251*A*b^3*m^2*x^4*e^(m*ln(x)) + 920*B*b^3*m*x^5*e^(m*ln(x)) + 276
0*A*b^2*c*m*x^5*e^(m*ln(x)) + 3360*B*b^2*c*x^6*e^(m*ln(x)) + 3360*A*b*c^2*x^6*e^
(m*ln(x)) + 1066*A*b^3*m*x^4*e^(m*ln(x)) + 1344*B*b^3*x^5*e^(m*ln(x)) + 4032*A*b
^2*c*x^5*e^(m*ln(x)) + 1680*A*b^3*x^4*e^(m*ln(x)))/(m^5 + 30*m^4 + 355*m^3 + 207
0*m^2 + 5944*m + 6720)