3.5.80 \(\int x^3 (c+d x+e x^2+f x^3) (a+b x^4)^2 \, dx\) [480]

Optimal. Leaf size=114 \[ \frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{7} a^2 f x^7+\frac {2}{9} a b d x^9+\frac {1}{5} a b e x^{10}+\frac {2}{11} a b f x^{11}+\frac {1}{13} b^2 d x^{13}+\frac {1}{14} b^2 e x^{14}+\frac {1}{15} b^2 f x^{15}+\frac {c \left (a+b x^4\right )^3}{12 b} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 114, 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^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{7} a^2 f x^7+\frac {c \left (a+b x^4\right )^3}{12 b}+\frac {2}{9} a b d x^9+\frac {1}{5} a b e x^{10}+\frac {2}{11} a b f x^{11}+\frac {1}{13} b^2 d x^{13}+\frac {1}{14} b^2 e x^{14}+\frac {1}{15} b^2 f x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.00, size = 129, normalized size = 1.13 \begin {gather*} \frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{7} a^2 f x^7+\frac {1}{4} a b c x^8+\frac {2}{9} a b d x^9+\frac {1}{5} a b e x^{10}+\frac {2}{11} a b f x^{11}+\frac {1}{12} b^2 c x^{12}+\frac {1}{13} b^2 d x^{13}+\frac {1}{14} b^2 e x^{14}+\frac {1}{15} b^2 f x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.36, size = 106, normalized size = 0.93

method	result	size
gosper	\(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} x^{5} a^{2} d +\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {1}{4} a b c \,x^{8}+\frac {2}{9} x^{9} a b d +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{12} b^{2} c \,x^{12}+\frac {1}{13} b^{2} d \,x^{13}+\frac {1}{14} b^{2} e \,x^{14}+\frac {1}{15} b^{2} f \,x^{15}\)	\(106\)
default	\(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} x^{5} a^{2} d +\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {1}{4} a b c \,x^{8}+\frac {2}{9} x^{9} a b d +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{12} b^{2} c \,x^{12}+\frac {1}{13} b^{2} d \,x^{13}+\frac {1}{14} b^{2} e \,x^{14}+\frac {1}{15} b^{2} f \,x^{15}\)	\(106\)
norman	\(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} x^{5} a^{2} d +\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {1}{4} a b c \,x^{8}+\frac {2}{9} x^{9} a b d +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{12} b^{2} c \,x^{12}+\frac {1}{13} b^{2} d \,x^{13}+\frac {1}{14} b^{2} e \,x^{14}+\frac {1}{15} b^{2} f \,x^{15}\)	\(106\)
risch	\(\frac {1}{4} a^{2} c \,x^{4}+\frac {1}{5} x^{5} a^{2} d +\frac {1}{6} a^{2} e \,x^{6}+\frac {1}{7} a^{2} f \,x^{7}+\frac {1}{4} a b c \,x^{8}+\frac {2}{9} x^{9} a b d +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{12} b^{2} c \,x^{12}+\frac {1}{13} b^{2} d \,x^{13}+\frac {1}{14} b^{2} e \,x^{14}+\frac {1}{15} b^{2} f \,x^{15}\)	\(106\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 108, normalized size = 0.95 \begin {gather*} \frac {1}{15} \, b^{2} f x^{15} + \frac {1}{14} \, b^{2} x^{14} e + \frac {1}{13} \, b^{2} d x^{13} + \frac {1}{12} \, b^{2} c x^{12} + \frac {2}{11} \, a b f x^{11} + \frac {1}{5} \, a b x^{10} e + \frac {2}{9} \, a b d x^{9} + \frac {1}{4} \, a b c x^{8} + \frac {1}{7} \, a^{2} f x^{7} + \frac {1}{6} \, a^{2} x^{6} e + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{4} \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 105, normalized size = 0.92 \begin {gather*} \frac {1}{15} \, b^{2} f x^{15} + \frac {1}{14} \, b^{2} e x^{14} + \frac {1}{13} \, b^{2} d x^{13} + \frac {1}{12} \, b^{2} c x^{12} + \frac {2}{11} \, a b f x^{11} + \frac {1}{5} \, a b e x^{10} + \frac {2}{9} \, a b d x^{9} + \frac {1}{4} \, a b c x^{8} + \frac {1}{7} \, a^{2} f x^{7} + \frac {1}{6} \, a^{2} e x^{6} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{4} \, a^{2} 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.01, size = 124, normalized size = 1.09 \begin {gather*} \frac {a^{2} c x^{4}}{4} + \frac {a^{2} d x^{5}}{5} + \frac {a^{2} e x^{6}}{6} + \frac {a^{2} f x^{7}}{7} + \frac {a b c x^{8}}{4} + \frac {2 a b d x^{9}}{9} + \frac {a b e x^{10}}{5} + \frac {2 a b f x^{11}}{11} + \frac {b^{2} c x^{12}}{12} + \frac {b^{2} d x^{13}}{13} + \frac {b^{2} e x^{14}}{14} + \frac {b^{2} f x^{15}}{15} \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)**2,x)

[Out]

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

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Giac [A]
time = 0.53, size = 108, normalized size = 0.95 \begin {gather*} \frac {1}{15} \, b^{2} f x^{15} + \frac {1}{14} \, b^{2} x^{14} e + \frac {1}{13} \, b^{2} d x^{13} + \frac {1}{12} \, b^{2} c x^{12} + \frac {2}{11} \, a b f x^{11} + \frac {1}{5} \, a b x^{10} e + \frac {2}{9} \, a b d x^{9} + \frac {1}{4} \, a b c x^{8} + \frac {1}{7} \, a^{2} f x^{7} + \frac {1}{6} \, a^{2} x^{6} e + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{4} \, a^{2} 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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.07, size = 105, normalized size = 0.92 \begin {gather*} \frac {f\,a^2\,x^7}{7}+\frac {e\,a^2\,x^6}{6}+\frac {d\,a^2\,x^5}{5}+\frac {c\,a^2\,x^4}{4}+\frac {2\,f\,a\,b\,x^{11}}{11}+\frac {e\,a\,b\,x^{10}}{5}+\frac {2\,d\,a\,b\,x^9}{9}+\frac {c\,a\,b\,x^8}{4}+\frac {f\,b^2\,x^{15}}{15}+\frac {e\,b^2\,x^{14}}{14}+\frac {d\,b^2\,x^{13}}{13}+\frac {c\,b^2\,x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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