\(\int (c+d x+e x^2) (a+b x^3)^4 \, dx\) [333]

Optimal result

Integrand size = 20, antiderivative size = 130 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14}+\frac {e \left (a+b x^3\right )^5}{15 b} \]

[Out]

a^4*c*x+1/2*a^4*d*x^2+a^3*b*c*x^4+4/5*a^3*b*d*x^5+6/7*a^2*b^2*c*x^7+3/4*a^2*b^2*d*x^8+2/5*a*b^3*c*x^10+4/11*a*
b^3*d*x^11+1/13*b^4*c*x^13+1/14*b^4*d*x^14+1/15*e*(b*x^3+a)^5/b

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1596, 1864} \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {e \left (a+b x^3\right )^5}{15 b}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14} \]

[In]

Int[(c + d*x + e*x^2)*(a + b*x^3)^4,x]

[Out]

a^4*c*x + (a^4*d*x^2)/2 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (6*a^2*b^2*c*x^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a*b
^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14 + (e*(a + b*x^3)^5)/(15*b)

Rule 1596

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 1]*((a + b*x^n)^(p + 1)/(b*n*(p +
1))), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && I
GtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] &&  !MatchQ[P
x, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Co
eff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a+b x^3\right )^5}{15 b}+\int (c+d x) \left (a+b x^3\right )^4 \, dx \\ & = \frac {e \left (a+b x^3\right )^5}{15 b}+\int \left (a^4 c+a^4 d x+4 a^3 b c x^3+4 a^3 b d x^4+6 a^2 b^2 c x^6+6 a^2 b^2 d x^7+4 a b^3 c x^9+4 a b^3 d x^{10}+b^4 c x^{12}+b^4 d x^{13}\right ) \, dx \\ & = a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14}+\frac {e \left (a+b x^3\right )^5}{15 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.33 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+\frac {1}{3} a^4 e x^3+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {2}{3} a^3 b e x^6+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{3} a^2 b^2 e x^9+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{3} a b^3 e x^{12}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14}+\frac {1}{15} b^4 e x^{15} \]

[In]

Integrate[(c + d*x + e*x^2)*(a + b*x^3)^4,x]

[Out]

a^4*c*x + (a^4*d*x^2)/2 + (a^4*e*x^3)/3 + a^3*b*c*x^4 + (4*a^3*b*d*x^5)/5 + (2*a^3*b*e*x^6)/3 + (6*a^2*b^2*c*x
^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a^2*b^2*e*x^9)/3 + (2*a*b^3*c*x^10)/5 + (4*a*b^3*d*x^11)/11 + (a*b^3*e*x^12)/
3 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14 + (b^4*e*x^15)/15

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.14

method	result	size
gosper	\(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+a^{3} b c \,x^{4}+\frac {4}{5} d \,x^{5} b \,a^{3}+\frac {2}{3} a^{3} b e \,x^{6}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} x^{8} b^{2} d \,a^{2}+\frac {2}{3} a^{2} e \,b^{2} x^{9}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} x^{11} d \,b^{3} a +\frac {1}{3} a \,b^{3} e \,x^{12}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}+\frac {1}{15} e \,b^{4} x^{15}\)	\(148\)
default	\(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+a^{3} b c \,x^{4}+\frac {4}{5} d \,x^{5} b \,a^{3}+\frac {2}{3} a^{3} b e \,x^{6}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} x^{8} b^{2} d \,a^{2}+\frac {2}{3} a^{2} e \,b^{2} x^{9}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} x^{11} d \,b^{3} a +\frac {1}{3} a \,b^{3} e \,x^{12}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}+\frac {1}{15} e \,b^{4} x^{15}\)	\(148\)
norman	\(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+a^{3} b c \,x^{4}+\frac {4}{5} d \,x^{5} b \,a^{3}+\frac {2}{3} a^{3} b e \,x^{6}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} x^{8} b^{2} d \,a^{2}+\frac {2}{3} a^{2} e \,b^{2} x^{9}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} x^{11} d \,b^{3} a +\frac {1}{3} a \,b^{3} e \,x^{12}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}+\frac {1}{15} e \,b^{4} x^{15}\)	\(148\)
risch	\(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+a^{3} b c \,x^{4}+\frac {4}{5} d \,x^{5} b \,a^{3}+\frac {2}{3} a^{3} b e \,x^{6}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} x^{8} b^{2} d \,a^{2}+\frac {2}{3} a^{2} e \,b^{2} x^{9}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} x^{11} d \,b^{3} a +\frac {1}{3} a \,b^{3} e \,x^{12}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}+\frac {1}{15} e \,b^{4} x^{15}\)	\(148\)
parallelrisch	\(a^{4} c x +\frac {1}{2} a^{4} d \,x^{2}+\frac {1}{3} a^{4} e \,x^{3}+a^{3} b c \,x^{4}+\frac {4}{5} d \,x^{5} b \,a^{3}+\frac {2}{3} a^{3} b e \,x^{6}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {3}{4} x^{8} b^{2} d \,a^{2}+\frac {2}{3} a^{2} e \,b^{2} x^{9}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {4}{11} x^{11} d \,b^{3} a +\frac {1}{3} a \,b^{3} e \,x^{12}+\frac {1}{13} b^{4} c \,x^{13}+\frac {1}{14} b^{4} d \,x^{14}+\frac {1}{15} e \,b^{4} x^{15}\)	\(148\)

[In]

int((e*x^2+d*x+c)*(b*x^3+a)^4,x,method=_RETURNVERBOSE)

[Out]

a^4*c*x+1/2*a^4*d*x^2+1/3*a^4*e*x^3+a^3*b*c*x^4+4/5*d*x^5*b*a^3+2/3*a^3*b*e*x^6+6/7*a^2*b^2*c*x^7+3/4*x^8*b^2*
d*a^2+2/3*a^2*e*b^2*x^9+2/5*a*b^3*c*x^10+4/11*x^11*d*b^3*a+1/3*a*b^3*e*x^12+1/13*b^4*c*x^13+1/14*b^4*d*x^14+1/
15*e*b^4*x^15

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.13 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{15} \, b^{4} e x^{15} + \frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {1}{3} \, a b^{3} e x^{12} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {2}{3} \, a^{2} b^{2} e x^{9} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {2}{3} \, a^{3} b e x^{6} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^4,x, algorithm="fricas")

[Out]

1/15*b^4*e*x^15 + 1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 1/3*a*b^3*e*x^12 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10
+ 2/3*a^2*b^2*e*x^9 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 2/3*a^3*b*e*x^6 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4
+ 1/3*a^4*e*x^3 + 1/2*a^4*d*x^2 + a^4*c*x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.37 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=a^{4} c x + \frac {a^{4} d x^{2}}{2} + \frac {a^{4} e x^{3}}{3} + a^{3} b c x^{4} + \frac {4 a^{3} b d x^{5}}{5} + \frac {2 a^{3} b e x^{6}}{3} + \frac {6 a^{2} b^{2} c x^{7}}{7} + \frac {3 a^{2} b^{2} d x^{8}}{4} + \frac {2 a^{2} b^{2} e x^{9}}{3} + \frac {2 a b^{3} c x^{10}}{5} + \frac {4 a b^{3} d x^{11}}{11} + \frac {a b^{3} e x^{12}}{3} + \frac {b^{4} c x^{13}}{13} + \frac {b^{4} d x^{14}}{14} + \frac {b^{4} e x^{15}}{15} \]

[In]

integrate((e*x**2+d*x+c)*(b*x**3+a)**4,x)

[Out]

a**4*c*x + a**4*d*x**2/2 + a**4*e*x**3/3 + a**3*b*c*x**4 + 4*a**3*b*d*x**5/5 + 2*a**3*b*e*x**6/3 + 6*a**2*b**2
*c*x**7/7 + 3*a**2*b**2*d*x**8/4 + 2*a**2*b**2*e*x**9/3 + 2*a*b**3*c*x**10/5 + 4*a*b**3*d*x**11/11 + a*b**3*e*
x**12/3 + b**4*c*x**13/13 + b**4*d*x**14/14 + b**4*e*x**15/15

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.13 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{15} \, b^{4} e x^{15} + \frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {1}{3} \, a b^{3} e x^{12} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {2}{3} \, a^{2} b^{2} e x^{9} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {2}{3} \, a^{3} b e x^{6} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^4,x, algorithm="maxima")

[Out]

1/15*b^4*e*x^15 + 1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 1/3*a*b^3*e*x^12 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10
+ 2/3*a^2*b^2*e*x^9 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 2/3*a^3*b*e*x^6 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4
+ 1/3*a^4*e*x^3 + 1/2*a^4*d*x^2 + a^4*c*x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.13 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {1}{15} \, b^{4} e x^{15} + \frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {1}{3} \, a b^{3} e x^{12} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {2}{3} \, a^{2} b^{2} e x^{9} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {2}{3} \, a^{3} b e x^{6} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{3} \, a^{4} e x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^4,x, algorithm="giac")

[Out]

1/15*b^4*e*x^15 + 1/14*b^4*d*x^14 + 1/13*b^4*c*x^13 + 1/3*a*b^3*e*x^12 + 4/11*a*b^3*d*x^11 + 2/5*a*b^3*c*x^10
+ 2/3*a^2*b^2*e*x^9 + 3/4*a^2*b^2*d*x^8 + 6/7*a^2*b^2*c*x^7 + 2/3*a^3*b*e*x^6 + 4/5*a^3*b*d*x^5 + a^3*b*c*x^4
+ 1/3*a^4*e*x^3 + 1/2*a^4*d*x^2 + a^4*c*x

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.13 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^4 \, dx=\frac {e\,a^4\,x^3}{3}+\frac {d\,a^4\,x^2}{2}+c\,a^4\,x+\frac {2\,e\,a^3\,b\,x^6}{3}+\frac {4\,d\,a^3\,b\,x^5}{5}+c\,a^3\,b\,x^4+\frac {2\,e\,a^2\,b^2\,x^9}{3}+\frac {3\,d\,a^2\,b^2\,x^8}{4}+\frac {6\,c\,a^2\,b^2\,x^7}{7}+\frac {e\,a\,b^3\,x^{12}}{3}+\frac {4\,d\,a\,b^3\,x^{11}}{11}+\frac {2\,c\,a\,b^3\,x^{10}}{5}+\frac {e\,b^4\,x^{15}}{15}+\frac {d\,b^4\,x^{14}}{14}+\frac {c\,b^4\,x^{13}}{13} \]

[In]

int((a + b*x^3)^4*(c + d*x + e*x^2),x)

[Out]

(a^4*d*x^2)/2 + (b^4*c*x^13)/13 + (a^4*e*x^3)/3 + (b^4*d*x^14)/14 + (b^4*e*x^15)/15 + a^4*c*x + (6*a^2*b^2*c*x
^7)/7 + (3*a^2*b^2*d*x^8)/4 + (2*a^2*b^2*e*x^9)/3 + a^3*b*c*x^4 + (2*a*b^3*c*x^10)/5 + (4*a^3*b*d*x^5)/5 + (4*
a*b^3*d*x^11)/11 + (2*a^3*b*e*x^6)/3 + (a*b^3*e*x^12)/3