3.582 \(\int x^m \left (a+b x^3\right )^5 \, dx\)

Optimal. Leaf size=97 \[ \frac{a^5 x^{m+1}}{m+1}+\frac{5 a^4 b x^{m+4}}{m+4}+\frac{10 a^3 b^2 x^{m+7}}{m+7}+\frac{10 a^2 b^3 x^{m+10}}{m+10}+\frac{5 a b^4 x^{m+13}}{m+13}+\frac{b^5 x^{m+16}}{m+16} \]

[Out]

(a^5*x^(1 + m))/(1 + m) + (5*a^4*b*x^(4 + m))/(4 + m) + (10*a^3*b^2*x^(7 + m))/(
7 + m) + (10*a^2*b^3*x^(10 + m))/(10 + m) + (5*a*b^4*x^(13 + m))/(13 + m) + (b^5
*x^(16 + m))/(16 + m)

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Rubi [A]  time = 0.0959203, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^5 x^{m+1}}{m+1}+\frac{5 a^4 b x^{m+4}}{m+4}+\frac{10 a^3 b^2 x^{m+7}}{m+7}+\frac{10 a^2 b^3 x^{m+10}}{m+10}+\frac{5 a b^4 x^{m+13}}{m+13}+\frac{b^5 x^{m+16}}{m+16} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^3)^5,x]

[Out]

(a^5*x^(1 + m))/(1 + m) + (5*a^4*b*x^(4 + m))/(4 + m) + (10*a^3*b^2*x^(7 + m))/(
7 + m) + (10*a^2*b^3*x^(10 + m))/(10 + m) + (5*a*b^4*x^(13 + m))/(13 + m) + (b^5
*x^(16 + m))/(16 + m)

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Rubi in Sympy [A]  time = 16.1684, size = 87, normalized size = 0.9 \[ \frac{a^{5} x^{m + 1}}{m + 1} + \frac{5 a^{4} b x^{m + 4}}{m + 4} + \frac{10 a^{3} b^{2} x^{m + 7}}{m + 7} + \frac{10 a^{2} b^{3} x^{m + 10}}{m + 10} + \frac{5 a b^{4} x^{m + 13}}{m + 13} + \frac{b^{5} x^{m + 16}}{m + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**3+a)**5,x)

[Out]

a**5*x**(m + 1)/(m + 1) + 5*a**4*b*x**(m + 4)/(m + 4) + 10*a**3*b**2*x**(m + 7)/
(m + 7) + 10*a**2*b**3*x**(m + 10)/(m + 10) + 5*a*b**4*x**(m + 13)/(m + 13) + b*
*5*x**(m + 16)/(m + 16)

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Mathematica [A]  time = 0.0554959, size = 87, normalized size = 0.9 \[ x^m \left (\frac{a^5 x}{m+1}+\frac{5 a^4 b x^4}{m+4}+\frac{10 a^3 b^2 x^7}{m+7}+\frac{10 a^2 b^3 x^{10}}{m+10}+\frac{5 a b^4 x^{13}}{m+13}+\frac{b^5 x^{16}}{m+16}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^3)^5,x]

[Out]

x^m*((a^5*x)/(1 + m) + (5*a^4*b*x^4)/(4 + m) + (10*a^3*b^2*x^7)/(7 + m) + (10*a^
2*b^3*x^10)/(10 + m) + (5*a*b^4*x^13)/(13 + m) + (b^5*x^16)/(16 + m))

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Maple [B]  time = 0.01, size = 432, normalized size = 4.5 \[{\frac{{x}^{1+m} \left ({b}^{5}{m}^{5}{x}^{15}+35\,{b}^{5}{m}^{4}{x}^{15}+445\,{b}^{5}{m}^{3}{x}^{15}+5\,a{b}^{4}{m}^{5}{x}^{12}+2485\,{b}^{5}{m}^{2}{x}^{15}+190\,a{b}^{4}{m}^{4}{x}^{12}+5714\,{b}^{5}m{x}^{15}+2555\,a{b}^{4}{m}^{3}{x}^{12}+3640\,{b}^{5}{x}^{15}+10\,{a}^{2}{b}^{3}{m}^{5}{x}^{9}+14810\,a{b}^{4}{m}^{2}{x}^{12}+410\,{a}^{2}{b}^{3}{m}^{4}{x}^{9}+34840\,a{b}^{4}m{x}^{12}+5950\,{a}^{2}{b}^{3}{m}^{3}{x}^{9}+22400\,a{b}^{4}{x}^{12}+10\,{a}^{3}{b}^{2}{m}^{5}{x}^{6}+36550\,{a}^{2}{b}^{3}{m}^{2}{x}^{9}+440\,{a}^{3}{b}^{2}{m}^{4}{x}^{6}+89240\,{a}^{2}{b}^{3}m{x}^{9}+6970\,{a}^{3}{b}^{2}{m}^{3}{x}^{6}+58240\,{a}^{2}{b}^{3}{x}^{9}+5\,{a}^{4}b{m}^{5}{x}^{3}+47260\,{a}^{3}{b}^{2}{m}^{2}{x}^{6}+235\,{a}^{4}b{m}^{4}{x}^{3}+123920\,{a}^{3}{b}^{2}m{x}^{6}+4085\,{a}^{4}b{m}^{3}{x}^{3}+83200\,{a}^{3}{b}^{2}{x}^{6}+{a}^{5}{m}^{5}+31685\,{a}^{4}b{m}^{2}{x}^{3}+50\,{a}^{5}{m}^{4}+100630\,{a}^{4}bm{x}^{3}+955\,{a}^{5}{m}^{3}+72800\,{a}^{4}b{x}^{3}+8650\,{a}^{5}{m}^{2}+36824\,{a}^{5}m+58240\,{a}^{5} \right ) }{ \left ( 1+m \right ) \left ( 4+m \right ) \left ( 7+m \right ) \left ( 10+m \right ) \left ( 13+m \right ) \left ( 16+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^3+a)^5,x)

[Out]

x^(1+m)*(b^5*m^5*x^15+35*b^5*m^4*x^15+445*b^5*m^3*x^15+5*a*b^4*m^5*x^12+2485*b^5
*m^2*x^15+190*a*b^4*m^4*x^12+5714*b^5*m*x^15+2555*a*b^4*m^3*x^12+3640*b^5*x^15+1
0*a^2*b^3*m^5*x^9+14810*a*b^4*m^2*x^12+410*a^2*b^3*m^4*x^9+34840*a*b^4*m*x^12+59
50*a^2*b^3*m^3*x^9+22400*a*b^4*x^12+10*a^3*b^2*m^5*x^6+36550*a^2*b^3*m^2*x^9+440
*a^3*b^2*m^4*x^6+89240*a^2*b^3*m*x^9+6970*a^3*b^2*m^3*x^6+58240*a^2*b^3*x^9+5*a^
4*b*m^5*x^3+47260*a^3*b^2*m^2*x^6+235*a^4*b*m^4*x^3+123920*a^3*b^2*m*x^6+4085*a^
4*b*m^3*x^3+83200*a^3*b^2*x^6+a^5*m^5+31685*a^4*b*m^2*x^3+50*a^5*m^4+100630*a^4*
b*m*x^3+955*a^5*m^3+72800*a^4*b*x^3+8650*a^5*m^2+36824*a^5*m+58240*a^5)/(1+m)/(4
+m)/(7+m)/(10+m)/(13+m)/(16+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((b*x^3 + a)^5*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245564, size = 495, normalized size = 5.1 \[ \frac{{\left ({\left (b^{5} m^{5} + 35 \, b^{5} m^{4} + 445 \, b^{5} m^{3} + 2485 \, b^{5} m^{2} + 5714 \, b^{5} m + 3640 \, b^{5}\right )} x^{16} + 5 \,{\left (a b^{4} m^{5} + 38 \, a b^{4} m^{4} + 511 \, a b^{4} m^{3} + 2962 \, a b^{4} m^{2} + 6968 \, a b^{4} m + 4480 \, a b^{4}\right )} x^{13} + 10 \,{\left (a^{2} b^{3} m^{5} + 41 \, a^{2} b^{3} m^{4} + 595 \, a^{2} b^{3} m^{3} + 3655 \, a^{2} b^{3} m^{2} + 8924 \, a^{2} b^{3} m + 5824 \, a^{2} b^{3}\right )} x^{10} + 10 \,{\left (a^{3} b^{2} m^{5} + 44 \, a^{3} b^{2} m^{4} + 697 \, a^{3} b^{2} m^{3} + 4726 \, a^{3} b^{2} m^{2} + 12392 \, a^{3} b^{2} m + 8320 \, a^{3} b^{2}\right )} x^{7} + 5 \,{\left (a^{4} b m^{5} + 47 \, a^{4} b m^{4} + 817 \, a^{4} b m^{3} + 6337 \, a^{4} b m^{2} + 20126 \, a^{4} b m + 14560 \, a^{4} b\right )} x^{4} +{\left (a^{5} m^{5} + 50 \, a^{5} m^{4} + 955 \, a^{5} m^{3} + 8650 \, a^{5} m^{2} + 36824 \, a^{5} m + 58240 \, a^{5}\right )} x\right )} x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^m,x, algorithm="fricas")

[Out]

((b^5*m^5 + 35*b^5*m^4 + 445*b^5*m^3 + 2485*b^5*m^2 + 5714*b^5*m + 3640*b^5)*x^1
6 + 5*(a*b^4*m^5 + 38*a*b^4*m^4 + 511*a*b^4*m^3 + 2962*a*b^4*m^2 + 6968*a*b^4*m
+ 4480*a*b^4)*x^13 + 10*(a^2*b^3*m^5 + 41*a^2*b^3*m^4 + 595*a^2*b^3*m^3 + 3655*a
^2*b^3*m^2 + 8924*a^2*b^3*m + 5824*a^2*b^3)*x^10 + 10*(a^3*b^2*m^5 + 44*a^3*b^2*
m^4 + 697*a^3*b^2*m^3 + 4726*a^3*b^2*m^2 + 12392*a^3*b^2*m + 8320*a^3*b^2)*x^7 +
5*(a^4*b*m^5 + 47*a^4*b*m^4 + 817*a^4*b*m^3 + 6337*a^4*b*m^2 + 20126*a^4*b*m +
14560*a^4*b)*x^4 + (a^5*m^5 + 50*a^5*m^4 + 955*a^5*m^3 + 8650*a^5*m^2 + 36824*a^
5*m + 58240*a^5)*x)*x^m/(m^6 + 51*m^5 + 1005*m^4 + 9605*m^3 + 45474*m^2 + 95064*
m + 58240)

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Sympy [A]  time = 41.5975, size = 2006, normalized size = 20.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**3+a)**5,x)

[Out]

Piecewise((-a**5/(15*x**15) - 5*a**4*b/(12*x**12) - 10*a**3*b**2/(9*x**9) - 5*a*
*2*b**3/(3*x**6) - 5*a*b**4/(3*x**3) + b**5*log(x), Eq(m, -16)), (-a**5/(12*x**1
2) - 5*a**4*b/(9*x**9) - 5*a**3*b**2/(3*x**6) - 10*a**2*b**3/(3*x**3) + 5*a*b**4
*log(x) + b**5*x**3/3, Eq(m, -13)), (-a**5/(9*x**9) - 5*a**4*b/(6*x**6) - 10*a**
3*b**2/(3*x**3) + 10*a**2*b**3*log(x) + 5*a*b**4*x**3/3 + b**5*x**6/6, Eq(m, -10
)), (-a**5/(6*x**6) - 5*a**4*b/(3*x**3) + 10*a**3*b**2*log(x) + 10*a**2*b**3*x**
3/3 + 5*a*b**4*x**6/6 + b**5*x**9/9, Eq(m, -7)), (-a**5/(3*x**3) + 5*a**4*b*log(
x) + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**6/3 + 5*a*b**4*x**9/9 + b**5*x**12/12,
Eq(m, -4)), (a**5*log(x) + 5*a**4*b*x**3/3 + 5*a**3*b**2*x**6/3 + 10*a**2*b**3*
x**9/9 + 5*a*b**4*x**12/12 + b**5*x**15/15, Eq(m, -1)), (a**5*m**5*x*x**m/(m**6
+ 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 50*a**5*m**4
*x*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240)
+ 955*a**5*m**3*x*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95
064*m + 58240) + 8650*a**5*m**2*x*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 +
45474*m**2 + 95064*m + 58240) + 36824*a**5*m*x*x**m/(m**6 + 51*m**5 + 1005*m**4
+ 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 58240*a**5*x*x**m/(m**6 + 51*m**5
+ 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 5*a**4*b*m**5*x**4*x*
*m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 235
*a**4*b*m**4*x**4*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95
064*m + 58240) + 4085*a**4*b*m**3*x**4*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m
**3 + 45474*m**2 + 95064*m + 58240) + 31685*a**4*b*m**2*x**4*x**m/(m**6 + 51*m**
5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 100630*a**4*b*m*x**4
*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) +
72800*a**4*b*x**4*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95
064*m + 58240) + 10*a**3*b**2*m**5*x**7*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*
m**3 + 45474*m**2 + 95064*m + 58240) + 440*a**3*b**2*m**4*x**7*x**m/(m**6 + 51*m
**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 6970*a**3*b**2*m**
3*x**7*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 582
40) + 47260*a**3*b**2*m**2*x**7*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 4
5474*m**2 + 95064*m + 58240) + 123920*a**3*b**2*m*x**7*x**m/(m**6 + 51*m**5 + 10
05*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 83200*a**3*b**2*x**7*x**m/
(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 10*a**
2*b**3*m**5*x**10*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95
064*m + 58240) + 410*a**2*b**3*m**4*x**10*x**m/(m**6 + 51*m**5 + 1005*m**4 + 960
5*m**3 + 45474*m**2 + 95064*m + 58240) + 5950*a**2*b**3*m**3*x**10*x**m/(m**6 +
51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 36550*a**2*b**
3*m**2*x**10*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m
+ 58240) + 89240*a**2*b**3*m*x**10*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3
+ 45474*m**2 + 95064*m + 58240) + 58240*a**2*b**3*x**10*x**m/(m**6 + 51*m**5 +
1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 5*a*b**4*m**5*x**13*x**m
/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 190*a
*b**4*m**4*x**13*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 950
64*m + 58240) + 2555*a*b**4*m**3*x**13*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m
**3 + 45474*m**2 + 95064*m + 58240) + 14810*a*b**4*m**2*x**13*x**m/(m**6 + 51*m*
*5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 34840*a*b**4*m*x**1
3*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) +
22400*a*b**4*x**13*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 +
95064*m + 58240) + b**5*m**5*x**16*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3
+ 45474*m**2 + 95064*m + 58240) + 35*b**5*m**4*x**16*x**m/(m**6 + 51*m**5 + 1005
*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 445*b**5*m**3*x**16*x**m/(m*
*6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 2485*b**5
*m**2*x**16*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m
+ 58240) + 5714*b**5*m*x**16*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 4547
4*m**2 + 95064*m + 58240) + 3640*b**5*x**16*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9
605*m**3 + 45474*m**2 + 95064*m + 58240), True))

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GIAC/XCAS [A]  time = 0.227736, size = 826, normalized size = 8.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5*x^m,x, algorithm="giac")

[Out]

(b^5*m^5*x^16*e^(m*ln(x)) + 35*b^5*m^4*x^16*e^(m*ln(x)) + 445*b^5*m^3*x^16*e^(m*
ln(x)) + 5*a*b^4*m^5*x^13*e^(m*ln(x)) + 2485*b^5*m^2*x^16*e^(m*ln(x)) + 190*a*b^
4*m^4*x^13*e^(m*ln(x)) + 5714*b^5*m*x^16*e^(m*ln(x)) + 2555*a*b^4*m^3*x^13*e^(m*
ln(x)) + 3640*b^5*x^16*e^(m*ln(x)) + 10*a^2*b^3*m^5*x^10*e^(m*ln(x)) + 14810*a*b
^4*m^2*x^13*e^(m*ln(x)) + 410*a^2*b^3*m^4*x^10*e^(m*ln(x)) + 34840*a*b^4*m*x^13*
e^(m*ln(x)) + 5950*a^2*b^3*m^3*x^10*e^(m*ln(x)) + 22400*a*b^4*x^13*e^(m*ln(x)) +
10*a^3*b^2*m^5*x^7*e^(m*ln(x)) + 36550*a^2*b^3*m^2*x^10*e^(m*ln(x)) + 440*a^3*b
^2*m^4*x^7*e^(m*ln(x)) + 89240*a^2*b^3*m*x^10*e^(m*ln(x)) + 6970*a^3*b^2*m^3*x^7
*e^(m*ln(x)) + 58240*a^2*b^3*x^10*e^(m*ln(x)) + 5*a^4*b*m^5*x^4*e^(m*ln(x)) + 47
260*a^3*b^2*m^2*x^7*e^(m*ln(x)) + 235*a^4*b*m^4*x^4*e^(m*ln(x)) + 123920*a^3*b^2
*m*x^7*e^(m*ln(x)) + 4085*a^4*b*m^3*x^4*e^(m*ln(x)) + 83200*a^3*b^2*x^7*e^(m*ln(
x)) + a^5*m^5*x*e^(m*ln(x)) + 31685*a^4*b*m^2*x^4*e^(m*ln(x)) + 50*a^5*m^4*x*e^(
m*ln(x)) + 100630*a^4*b*m*x^4*e^(m*ln(x)) + 955*a^5*m^3*x*e^(m*ln(x)) + 72800*a^
4*b*x^4*e^(m*ln(x)) + 8650*a^5*m^2*x*e^(m*ln(x)) + 36824*a^5*m*x*e^(m*ln(x)) + 5
8240*a^5*x*e^(m*ln(x)))/(m^6 + 51*m^5 + 1005*m^4 + 9605*m^3 + 45474*m^2 + 95064*
m + 58240)