3.53 \(\int (a+b x^3)^3 (a c+a d x+b c x^3+b d x^4) \, dx\)

Optimal. Leaf size=113 \[ a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14} \]

[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

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Rubi [A]  time = 0.10, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1850} \[ \frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+a^4 c x+\frac {1}{2} a^4 d x^2+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3*(a*c + a*d*x + b*c*x^3 + b*d*x^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

Rule 1850

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 \left (a+b x^3\right )^3 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx &=\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}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 113, normalized size = 1.00 \[ a^4 c x+\frac {1}{2} a^4 d x^2+a^3 b c x^4+\frac {4}{5} a^3 b d x^5+\frac {6}{7} a^2 b^2 c x^7+\frac {3}{4} a^2 b^2 d x^8+\frac {2}{5} a b^3 c x^{10}+\frac {4}{11} a b^3 d x^{11}+\frac {1}{13} b^4 c x^{13}+\frac {1}{14} b^4 d x^{14} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3*(a*c + a*d*x + b*c*x^3 + b*d*x^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

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fricas [A]  time = 0.74, size = 97, normalized size = 0.86 \[ \frac {1}{14} x^{14} d b^{4} + \frac {1}{13} x^{13} c b^{4} + \frac {4}{11} x^{11} d b^{3} a + \frac {2}{5} x^{10} c b^{3} a + \frac {3}{4} x^{8} d b^{2} a^{2} + \frac {6}{7} x^{7} c b^{2} a^{2} + \frac {4}{5} x^{5} d b a^{3} + x^{4} c b a^{3} + \frac {1}{2} x^{2} d a^{4} + x c a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 97, normalized size = 0.86 \[ \frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 98, normalized size = 0.87 \[ \frac {1}{14} b^{4} d \,x^{14}+\frac {1}{13} b^{4} c \,x^{13}+\frac {4}{11} a \,b^{3} d \,x^{11}+\frac {2}{5} a \,b^{3} c \,x^{10}+\frac {3}{4} a^{2} b^{2} d \,x^{8}+\frac {6}{7} a^{2} b^{2} c \,x^{7}+\frac {4}{5} a^{3} b d \,x^{5}+a^{3} b c \,x^{4}+\frac {1}{2} a^{4} d \,x^{2}+a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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maxima [A]  time = 1.39, size = 97, normalized size = 0.86 \[ \frac {1}{14} \, b^{4} d x^{14} + \frac {1}{13} \, b^{4} c x^{13} + \frac {4}{11} \, a b^{3} d x^{11} + \frac {2}{5} \, a b^{3} c x^{10} + \frac {3}{4} \, a^{2} b^{2} d x^{8} + \frac {6}{7} \, a^{2} b^{2} c x^{7} + \frac {4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 97, normalized size = 0.86 \[ \frac {d\,a^4\,x^2}{2}+c\,a^4\,x+\frac {4\,d\,a^3\,b\,x^5}{5}+c\,a^3\,b\,x^4+\frac {3\,d\,a^2\,b^2\,x^8}{4}+\frac {6\,c\,a^2\,b^2\,x^7}{7}+\frac {4\,d\,a\,b^3\,x^{11}}{11}+\frac {2\,c\,a\,b^3\,x^{10}}{5}+\frac {d\,b^4\,x^{14}}{14}+\frac {c\,b^4\,x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^4*d*x^2)/2 + (b^4*c*x^13)/13 + (b^4*d*x^14)/14 + a^4*c*x + (6*a^2*b^2*c*x^7)/7 + (3*a^2*b^2*d*x^8)/4 + 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

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sympy [A]  time = 0.73, size = 117, normalized size = 1.04 \[ a^{4} c x + \frac {a^{4} d x^{2}}{2} + a^{3} b c x^{4} + \frac {4 a^{3} b d x^{5}}{5} + \frac {6 a^{2} b^{2} c x^{7}}{7} + \frac {3 a^{2} b^{2} d x^{8}}{4} + \frac {2 a b^{3} c x^{10}}{5} + \frac {4 a b^{3} d x^{11}}{11} + \frac {b^{4} c x^{13}}{13} + \frac {b^{4} d x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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