3.5.82 \(\int x^3 (c+d x+e x^2+f x^3) (a+b x^4)^3 \, dx\) [482]

Optimal. Leaf size=156 \[ \frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^3 f x^7+\frac {1}{3} a^2 b d x^9+\frac {3}{10} a^2 b e x^{10}+\frac {3}{11} a^2 b f x^{11}+\frac {3}{13} a b^2 d x^{13}+\frac {3}{14} a b^2 e x^{14}+\frac {1}{5} a b^2 f x^{15}+\frac {1}{17} b^3 d x^{17}+\frac {1}{18} b^3 e x^{18}+\frac {1}{19} b^3 f x^{19}+\frac {c \left (a+b x^4\right )^4}{16 b} \]

[Out]

1/5*a^3*d*x^5+1/6*a^3*e*x^6+1/7*a^3*f*x^7+1/3*a^2*b*d*x^9+3/10*a^2*b*e*x^10+3/11*a^2*b*f*x^11+3/13*a*b^2*d*x^1
3+3/14*a*b^2*e*x^14+1/5*a*b^2*f*x^15+1/17*b^3*d*x^17+1/18*b^3*e*x^18+1/19*b^3*f*x^19+1/16*c*(b*x^4+a)^4/b

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Rubi [A]
time = 0.07, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1596, 1864} \begin {gather*} \frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^3 f x^7+\frac {1}{3} a^2 b d x^9+\frac {3}{10} a^2 b e x^{10}+\frac {3}{11} a^2 b f x^{11}+\frac {3}{13} a b^2 d x^{13}+\frac {3}{14} a b^2 e x^{14}+\frac {1}{5} a b^2 f x^{15}+\frac {c \left (a+b x^4\right )^4}{16 b}+\frac {1}{17} b^3 d x^{17}+\frac {1}{18} b^3 e x^{18}+\frac {1}{19} b^3 f x^{19} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^3*f*x^7)/7 + (a^2*b*d*x^9)/3 + (3*a^2*b*e*x^10)/10 + (3*a^2*b*f*x^11)/11 +
(3*a*b^2*d*x^13)/13 + (3*a*b^2*e*x^14)/14 + (a*b^2*f*x^15)/5 + (b^3*d*x^17)/17 + (b^3*e*x^18)/18 + (b^3*f*x^19
)/19 + (c*(a + b*x^4)^4)/(16*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*} \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^3 \, dx &=\frac {c \left (a+b x^4\right )^4}{16 b}+\int \left (a+b x^4\right )^3 \left (-c x^3+x^3 \left (c+d x+e x^2+f x^3\right )\right ) \, dx\\ &=\frac {c \left (a+b x^4\right )^4}{16 b}+\int \left (a^3 d x^4+a^3 e x^5+a^3 f x^6+3 a^2 b d x^8+3 a^2 b e x^9+3 a^2 b f x^{10}+3 a b^2 d x^{12}+3 a b^2 e x^{13}+3 a b^2 f x^{14}+b^3 d x^{16}+b^3 e x^{17}+b^3 f x^{18}\right ) \, dx\\ &=\frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^3 f x^7+\frac {1}{3} a^2 b d x^9+\frac {3}{10} a^2 b e x^{10}+\frac {3}{11} a^2 b f x^{11}+\frac {3}{13} a b^2 d x^{13}+\frac {3}{14} a b^2 e x^{14}+\frac {1}{5} a b^2 f x^{15}+\frac {1}{17} b^3 d x^{17}+\frac {1}{18} b^3 e x^{18}+\frac {1}{19} b^3 f x^{19}+\frac {c \left (a+b x^4\right )^4}{16 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 185, normalized size = 1.19 \begin {gather*} \frac {1}{4} a^3 c x^4+\frac {1}{5} a^3 d x^5+\frac {1}{6} a^3 e x^6+\frac {1}{7} a^3 f x^7+\frac {3}{8} a^2 b c x^8+\frac {1}{3} a^2 b d x^9+\frac {3}{10} a^2 b e x^{10}+\frac {3}{11} a^2 b f x^{11}+\frac {1}{4} a b^2 c x^{12}+\frac {3}{13} a b^2 d x^{13}+\frac {3}{14} a b^2 e x^{14}+\frac {1}{5} a b^2 f x^{15}+\frac {1}{16} b^3 c x^{16}+\frac {1}{17} b^3 d x^{17}+\frac {1}{18} b^3 e x^{18}+\frac {1}{19} b^3 f x^{19} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (a^3*e*x^6)/6 + (a^3*f*x^7)/7 + (3*a^2*b*c*x^8)/8 + (a^2*b*d*x^9)/3 + (3*a^2*b
*e*x^10)/10 + (3*a^2*b*f*x^11)/11 + (a*b^2*c*x^12)/4 + (3*a*b^2*d*x^13)/13 + (3*a*b^2*e*x^14)/14 + (a*b^2*f*x^
15)/5 + (b^3*c*x^16)/16 + (b^3*d*x^17)/17 + (b^3*e*x^18)/18 + (b^3*f*x^19)/19

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Maple [A]
time = 0.38, size = 154, normalized size = 0.99

method	result	size
gosper	\(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} a^{3} f \,x^{7}+\frac {3}{8} c \,a^{2} b \,x^{8}+\frac {1}{3} a^{2} b d \,x^{9}+\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {1}{4} a c \,b^{2} x^{12}+\frac {3}{13} a \,b^{2} d \,x^{13}+\frac {3}{14} a \,b^{2} e \,x^{14}+\frac {1}{5} a \,b^{2} f \,x^{15}+\frac {1}{16} b^{3} c \,x^{16}+\frac {1}{17} b^{3} d \,x^{17}+\frac {1}{18} b^{3} e \,x^{18}+\frac {1}{19} b^{3} f \,x^{19}\)	\(154\)
default	\(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} a^{3} f \,x^{7}+\frac {3}{8} c \,a^{2} b \,x^{8}+\frac {1}{3} a^{2} b d \,x^{9}+\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {1}{4} a c \,b^{2} x^{12}+\frac {3}{13} a \,b^{2} d \,x^{13}+\frac {3}{14} a \,b^{2} e \,x^{14}+\frac {1}{5} a \,b^{2} f \,x^{15}+\frac {1}{16} b^{3} c \,x^{16}+\frac {1}{17} b^{3} d \,x^{17}+\frac {1}{18} b^{3} e \,x^{18}+\frac {1}{19} b^{3} f \,x^{19}\)	\(154\)
norman	\(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} a^{3} f \,x^{7}+\frac {3}{8} c \,a^{2} b \,x^{8}+\frac {1}{3} a^{2} b d \,x^{9}+\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {1}{4} a c \,b^{2} x^{12}+\frac {3}{13} a \,b^{2} d \,x^{13}+\frac {3}{14} a \,b^{2} e \,x^{14}+\frac {1}{5} a \,b^{2} f \,x^{15}+\frac {1}{16} b^{3} c \,x^{16}+\frac {1}{17} b^{3} d \,x^{17}+\frac {1}{18} b^{3} e \,x^{18}+\frac {1}{19} b^{3} f \,x^{19}\)	\(154\)
risch	\(\frac {1}{4} a^{3} c \,x^{4}+\frac {1}{5} a^{3} d \,x^{5}+\frac {1}{6} a^{3} e \,x^{6}+\frac {1}{7} a^{3} f \,x^{7}+\frac {3}{8} c \,a^{2} b \,x^{8}+\frac {1}{3} a^{2} b d \,x^{9}+\frac {3}{10} a^{2} b e \,x^{10}+\frac {3}{11} a^{2} b f \,x^{11}+\frac {1}{4} a c \,b^{2} x^{12}+\frac {3}{13} a \,b^{2} d \,x^{13}+\frac {3}{14} a \,b^{2} e \,x^{14}+\frac {1}{5} a \,b^{2} f \,x^{15}+\frac {1}{16} b^{3} c \,x^{16}+\frac {1}{17} b^{3} d \,x^{17}+\frac {1}{18} b^{3} e \,x^{18}+\frac {1}{19} b^{3} f \,x^{19}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*c*x^4+1/5*a^3*d*x^5+1/6*a^3*e*x^6+1/7*a^3*f*x^7+3/8*c*a^2*b*x^8+1/3*a^2*b*d*x^9+3/10*a^2*b*e*x^10+3/11
*a^2*b*f*x^11+1/4*a*c*b^2*x^12+3/13*a*b^2*d*x^13+3/14*a*b^2*e*x^14+1/5*a*b^2*f*x^15+1/16*b^3*c*x^16+1/17*b^3*d
*x^17+1/18*b^3*e*x^18+1/19*b^3*f*x^19

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Maxima [A]
time = 0.27, size = 157, normalized size = 1.01 \begin {gather*} \frac {1}{19} \, b^{3} f x^{19} + \frac {1}{18} \, b^{3} x^{18} e + \frac {1}{17} \, b^{3} d x^{17} + \frac {1}{16} \, b^{3} c x^{16} + \frac {1}{5} \, a b^{2} f x^{15} + \frac {3}{14} \, a b^{2} x^{14} e + \frac {3}{13} \, a b^{2} d x^{13} + \frac {1}{4} \, a b^{2} c x^{12} + \frac {3}{11} \, a^{2} b f x^{11} + \frac {3}{10} \, a^{2} b x^{10} e + \frac {1}{3} \, a^{2} b d x^{9} + \frac {3}{8} \, a^{2} b c x^{8} + \frac {1}{7} \, a^{3} f x^{7} + \frac {1}{6} \, a^{3} x^{6} e + \frac {1}{5} \, a^{3} d x^{5} + \frac {1}{4} \, a^{3} c x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^3*f*x^19 + 1/18*b^3*x^18*e + 1/17*b^3*d*x^17 + 1/16*b^3*c*x^16 + 1/5*a*b^2*f*x^15 + 3/14*a*b^2*x^14*e +
3/13*a*b^2*d*x^13 + 1/4*a*b^2*c*x^12 + 3/11*a^2*b*f*x^11 + 3/10*a^2*b*x^10*e + 1/3*a^2*b*d*x^9 + 3/8*a^2*b*c*
x^8 + 1/7*a^3*f*x^7 + 1/6*a^3*x^6*e + 1/5*a^3*d*x^5 + 1/4*a^3*c*x^4

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Fricas [A]
time = 0.37, size = 153, normalized size = 0.98 \begin {gather*} \frac {1}{19} \, b^{3} f x^{19} + \frac {1}{18} \, b^{3} e x^{18} + \frac {1}{17} \, b^{3} d x^{17} + \frac {1}{16} \, b^{3} c x^{16} + \frac {1}{5} \, a b^{2} f x^{15} + \frac {3}{14} \, a b^{2} e x^{14} + \frac {3}{13} \, a b^{2} d x^{13} + \frac {1}{4} \, a b^{2} c x^{12} + \frac {3}{11} \, a^{2} b f x^{11} + \frac {3}{10} \, a^{2} b e x^{10} + \frac {1}{3} \, a^{2} b d x^{9} + \frac {3}{8} \, a^{2} b c x^{8} + \frac {1}{7} \, a^{3} f x^{7} + \frac {1}{6} \, a^{3} e x^{6} + \frac {1}{5} \, a^{3} d x^{5} + \frac {1}{4} \, a^{3} c x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^3*f*x^19 + 1/18*b^3*e*x^18 + 1/17*b^3*d*x^17 + 1/16*b^3*c*x^16 + 1/5*a*b^2*f*x^15 + 3/14*a*b^2*e*x^14 +
3/13*a*b^2*d*x^13 + 1/4*a*b^2*c*x^12 + 3/11*a^2*b*f*x^11 + 3/10*a^2*b*e*x^10 + 1/3*a^2*b*d*x^9 + 3/8*a^2*b*c*
x^8 + 1/7*a^3*f*x^7 + 1/6*a^3*e*x^6 + 1/5*a^3*d*x^5 + 1/4*a^3*c*x^4

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Sympy [A]
time = 0.02, size = 184, normalized size = 1.18 \begin {gather*} \frac {a^{3} c x^{4}}{4} + \frac {a^{3} d x^{5}}{5} + \frac {a^{3} e x^{6}}{6} + \frac {a^{3} f x^{7}}{7} + \frac {3 a^{2} b c x^{8}}{8} + \frac {a^{2} b d x^{9}}{3} + \frac {3 a^{2} b e x^{10}}{10} + \frac {3 a^{2} b f x^{11}}{11} + \frac {a b^{2} c x^{12}}{4} + \frac {3 a b^{2} d x^{13}}{13} + \frac {3 a b^{2} e x^{14}}{14} + \frac {a b^{2} f x^{15}}{5} + \frac {b^{3} c x^{16}}{16} + \frac {b^{3} d x^{17}}{17} + \frac {b^{3} e x^{18}}{18} + \frac {b^{3} f x^{19}}{19} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c*x**4/4 + a**3*d*x**5/5 + a**3*e*x**6/6 + a**3*f*x**7/7 + 3*a**2*b*c*x**8/8 + a**2*b*d*x**9/3 + 3*a**2*b
*e*x**10/10 + 3*a**2*b*f*x**11/11 + a*b**2*c*x**12/4 + 3*a*b**2*d*x**13/13 + 3*a*b**2*e*x**14/14 + a*b**2*f*x*
*15/5 + b**3*c*x**16/16 + b**3*d*x**17/17 + b**3*e*x**18/18 + b**3*f*x**19/19

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Giac [A]
time = 0.63, size = 157, normalized size = 1.01 \begin {gather*} \frac {1}{19} \, b^{3} f x^{19} + \frac {1}{18} \, b^{3} x^{18} e + \frac {1}{17} \, b^{3} d x^{17} + \frac {1}{16} \, b^{3} c x^{16} + \frac {1}{5} \, a b^{2} f x^{15} + \frac {3}{14} \, a b^{2} x^{14} e + \frac {3}{13} \, a b^{2} d x^{13} + \frac {1}{4} \, a b^{2} c x^{12} + \frac {3}{11} \, a^{2} b f x^{11} + \frac {3}{10} \, a^{2} b x^{10} e + \frac {1}{3} \, a^{2} b d x^{9} + \frac {3}{8} \, a^{2} b c x^{8} + \frac {1}{7} \, a^{3} f x^{7} + \frac {1}{6} \, a^{3} x^{6} e + \frac {1}{5} \, a^{3} d x^{5} + \frac {1}{4} \, a^{3} c x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b^3*f*x^19 + 1/18*b^3*x^18*e + 1/17*b^3*d*x^17 + 1/16*b^3*c*x^16 + 1/5*a*b^2*f*x^15 + 3/14*a*b^2*x^14*e +
3/13*a*b^2*d*x^13 + 1/4*a*b^2*c*x^12 + 3/11*a^2*b*f*x^11 + 3/10*a^2*b*x^10*e + 1/3*a^2*b*d*x^9 + 3/8*a^2*b*c*
x^8 + 1/7*a^3*f*x^7 + 1/6*a^3*x^6*e + 1/5*a^3*d*x^5 + 1/4*a^3*c*x^4

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Mupad [B]
time = 0.16, size = 153, normalized size = 0.98 \begin {gather*} \frac {f\,a^3\,x^7}{7}+\frac {e\,a^3\,x^6}{6}+\frac {d\,a^3\,x^5}{5}+\frac {c\,a^3\,x^4}{4}+\frac {3\,f\,a^2\,b\,x^{11}}{11}+\frac {3\,e\,a^2\,b\,x^{10}}{10}+\frac {d\,a^2\,b\,x^9}{3}+\frac {3\,c\,a^2\,b\,x^8}{8}+\frac {f\,a\,b^2\,x^{15}}{5}+\frac {3\,e\,a\,b^2\,x^{14}}{14}+\frac {3\,d\,a\,b^2\,x^{13}}{13}+\frac {c\,a\,b^2\,x^{12}}{4}+\frac {f\,b^3\,x^{19}}{19}+\frac {e\,b^3\,x^{18}}{18}+\frac {d\,b^3\,x^{17}}{17}+\frac {c\,b^3\,x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*c*x^4)/4 + (a^3*d*x^5)/5 + (b^3*c*x^16)/16 + (a^3*e*x^6)/6 + (b^3*d*x^17)/17 + (a^3*f*x^7)/7 + (b^3*e*x^1
8)/18 + (b^3*f*x^19)/19 + (3*a^2*b*c*x^8)/8 + (a*b^2*c*x^12)/4 + (a^2*b*d*x^9)/3 + (3*a*b^2*d*x^13)/13 + (3*a^
2*b*e*x^10)/10 + (3*a*b^2*e*x^14)/14 + (3*a^2*b*f*x^11)/11 + (a*b^2*f*x^15)/5

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