3.584 \(\int x^m (a+b x^3)^5 \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0370078, 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, Rules used = {270} \[ \frac{10 a^3 b^2 x^{m+7}}{m+7}+\frac{10 a^2 b^3 x^{m+10}}{m+10}+\frac{5 a^4 b x^{m+4}}{m+4}+\frac{a^5 x^{m+1}}{m+1}+\frac{5 a b^4 x^{m+13}}{m+13}+\frac{b^5 x^{m+16}}{m+16} \]

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)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^m \left (a+b x^3\right )^5 \, dx &=\int \left (a^5 x^m+5 a^4 b x^{3+m}+10 a^3 b^2 x^{6+m}+10 a^2 b^3 x^{9+m}+5 a b^4 x^{12+m}+b^5 x^{15+m}\right ) \, dx\\ &=\frac{a^5 x^{1+m}}{1+m}+\frac{5 a^4 b x^{4+m}}{4+m}+\frac{10 a^3 b^2 x^{7+m}}{7+m}+\frac{10 a^2 b^3 x^{10+m}}{10+m}+\frac{5 a b^4 x^{13+m}}{13+m}+\frac{b^5 x^{16+m}}{16+m}\\ \end{align*}

Mathematica [A]  time = 0.043415, size = 88, normalized size = 0.91 \[ x^{m+1} \left (\frac{10 a^2 b^3 x^9}{m+10}+\frac{10 a^3 b^2 x^6}{m+7}+\frac{5 a^4 b x^3}{m+4}+\frac{a^5}{m+1}+\frac{5 a b^4 x^{12}}{m+13}+\frac{b^5 x^{15}}{m+16}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^3)^5,x]

[Out]

x^(1 + m)*(a^5/(1 + m) + (5*a^4*b*x^3)/(4 + m) + (10*a^3*b^2*x^6)/(7 + m) + (10*a^2*b^3*x^9)/(10 + m) + (5*a*b
^4*x^12)/(13 + m) + (b^5*x^15)/(16 + m))

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 432, normalized size = 4.5 \begin{align*}{\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 ) }} \end{align*}

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+5
714*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+3
4840*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+1
00630*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)

________________________________________________________________________________________

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(x^m*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.55624, size = 880, normalized size = 9.07 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^5,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^16 + 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)

________________________________________________________________________________________

Sympy [A]  time = 12.4051, size = 2006, normalized size = 20.68 \begin{align*} \text{result too large to display} \end{align*}

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**12) - 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*l
og(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 + 95064*m + 58240) + 8650*a**5*m**2*x*x**m/(m**6 + 51*m**5 + 100
5*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 + 58
240) + 235*a**4*b*m**4*x**4*x**m/(m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240) + 408
5*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 + 95064*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 + 58240) + 47260*a**3*b**2*m**2*x**7*x**m/(m**6 + 51*m**5 + 1005*m**4 + 960
5*m**3 + 45474*m**2 + 95064*m + 58240) + 123920*a**3*b**2*m*x**7*x**m/(m**6 + 51*m**5 + 1005*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 + 9605*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 + 582
40) + 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 + 95064*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**13*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 + 45474*m**2 + 95064*m + 58240) + 3640*b**5*x**16*x**m/(
m**6 + 51*m**5 + 1005*m**4 + 9605*m**3 + 45474*m**2 + 95064*m + 58240), True))

________________________________________________________________________________________

Giac [B]  time = 1.67369, size = 729, normalized size = 7.52 \begin{align*} \frac{b^{5} m^{5} x^{16} x^{m} + 35 \, b^{5} m^{4} x^{16} x^{m} + 445 \, b^{5} m^{3} x^{16} x^{m} + 5 \, a b^{4} m^{5} x^{13} x^{m} + 2485 \, b^{5} m^{2} x^{16} x^{m} + 190 \, a b^{4} m^{4} x^{13} x^{m} + 5714 \, b^{5} m x^{16} x^{m} + 2555 \, a b^{4} m^{3} x^{13} x^{m} + 3640 \, b^{5} x^{16} x^{m} + 10 \, a^{2} b^{3} m^{5} x^{10} x^{m} + 14810 \, a b^{4} m^{2} x^{13} x^{m} + 410 \, a^{2} b^{3} m^{4} x^{10} x^{m} + 34840 \, a b^{4} m x^{13} x^{m} + 5950 \, a^{2} b^{3} m^{3} x^{10} x^{m} + 22400 \, a b^{4} x^{13} x^{m} + 10 \, a^{3} b^{2} m^{5} x^{7} x^{m} + 36550 \, a^{2} b^{3} m^{2} x^{10} x^{m} + 440 \, a^{3} b^{2} m^{4} x^{7} x^{m} + 89240 \, a^{2} b^{3} m x^{10} x^{m} + 6970 \, a^{3} b^{2} m^{3} x^{7} x^{m} + 58240 \, a^{2} b^{3} x^{10} x^{m} + 5 \, a^{4} b m^{5} x^{4} x^{m} + 47260 \, a^{3} b^{2} m^{2} x^{7} x^{m} + 235 \, a^{4} b m^{4} x^{4} x^{m} + 123920 \, a^{3} b^{2} m x^{7} x^{m} + 4085 \, a^{4} b m^{3} x^{4} x^{m} + 83200 \, a^{3} b^{2} x^{7} x^{m} + a^{5} m^{5} x x^{m} + 31685 \, a^{4} b m^{2} x^{4} x^{m} + 50 \, a^{5} m^{4} x x^{m} + 100630 \, a^{4} b m x^{4} x^{m} + 955 \, a^{5} m^{3} x x^{m} + 72800 \, a^{4} b x^{4} x^{m} + 8650 \, a^{5} m^{2} x x^{m} + 36824 \, a^{5} m x x^{m} + 58240 \, a^{5} x x^{m}}{m^{6} + 51 \, m^{5} + 1005 \, m^{4} + 9605 \, m^{3} + 45474 \, m^{2} + 95064 \, m + 58240} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*m^5*x^16*x^m + 35*b^5*m^4*x^16*x^m + 445*b^5*m^3*x^16*x^m + 5*a*b^4*m^5*x^13*x^m + 2485*b^5*m^2*x^16*x^m
+ 190*a*b^4*m^4*x^13*x^m + 5714*b^5*m*x^16*x^m + 2555*a*b^4*m^3*x^13*x^m + 3640*b^5*x^16*x^m + 10*a^2*b^3*m^5*
x^10*x^m + 14810*a*b^4*m^2*x^13*x^m + 410*a^2*b^3*m^4*x^10*x^m + 34840*a*b^4*m*x^13*x^m + 5950*a^2*b^3*m^3*x^1
0*x^m + 22400*a*b^4*x^13*x^m + 10*a^3*b^2*m^5*x^7*x^m + 36550*a^2*b^3*m^2*x^10*x^m + 440*a^3*b^2*m^4*x^7*x^m +
89240*a^2*b^3*m*x^10*x^m + 6970*a^3*b^2*m^3*x^7*x^m + 58240*a^2*b^3*x^10*x^m + 5*a^4*b*m^5*x^4*x^m + 47260*a^
3*b^2*m^2*x^7*x^m + 235*a^4*b*m^4*x^4*x^m + 123920*a^3*b^2*m*x^7*x^m + 4085*a^4*b*m^3*x^4*x^m + 83200*a^3*b^2*
x^7*x^m + a^5*m^5*x*x^m + 31685*a^4*b*m^2*x^4*x^m + 50*a^5*m^4*x*x^m + 100630*a^4*b*m*x^4*x^m + 955*a^5*m^3*x*
x^m + 72800*a^4*b*x^4*x^m + 8650*a^5*m^2*x*x^m + 36824*a^5*m*x*x^m + 58240*a^5*x*x^m)/(m^6 + 51*m^5 + 1005*m^4
+ 9605*m^3 + 45474*m^2 + 95064*m + 58240)