\(\int (a+b x^3)^2 (a c+a d x+b c x^3+b d x^4) \, dx\) [54]

Optimal result

Integrand size = 30, antiderivative size = 88 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^3 c x+\frac {1}{2} a^3 d x^2+\frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1864} \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^3 c x+\frac {1}{2} a^3 d x^2+\frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11} \]

[In]

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

[Out]

a^3*c*x + (a^3*d*x^2)/2 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (b^3
*c*x^10)/10 + (b^3*d*x^11)/11

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^3 c x+\frac {1}{2} a^3 d x^2+\frac {3}{4} a^2 b c x^4+\frac {3}{5} a^2 b d x^5+\frac {3}{7} a b^2 c x^7+\frac {3}{8} a b^2 d x^8+\frac {1}{10} b^3 c x^{10}+\frac {1}{11} b^3 d x^{11} \]

[In]

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

[Out]

a^3*c*x + (a^3*d*x^2)/2 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (b^3
*c*x^10)/10 + (b^3*d*x^11)/11

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85

method	result	size
default	\(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {3}{4} a^{2} b c \,x^{4}+\frac {3}{5} x^{5} b d \,a^{2}+\frac {3}{7} a \,b^{2} c \,x^{7}+\frac {3}{8} x^{8} b^{2} d a +\frac {1}{10} b^{3} c \,x^{10}+\frac {1}{11} b^{3} d \,x^{11}\)	\(75\)
norman	\(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {3}{4} a^{2} b c \,x^{4}+\frac {3}{5} x^{5} b d \,a^{2}+\frac {3}{7} a \,b^{2} c \,x^{7}+\frac {3}{8} x^{8} b^{2} d a +\frac {1}{10} b^{3} c \,x^{10}+\frac {1}{11} b^{3} d \,x^{11}\)	\(75\)
risch	\(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {3}{4} a^{2} b c \,x^{4}+\frac {3}{5} x^{5} b d \,a^{2}+\frac {3}{7} a \,b^{2} c \,x^{7}+\frac {3}{8} x^{8} b^{2} d a +\frac {1}{10} b^{3} c \,x^{10}+\frac {1}{11} b^{3} d \,x^{11}\)	\(75\)
parallelrisch	\(a^{3} c x +\frac {1}{2} a^{3} d \,x^{2}+\frac {3}{4} a^{2} b c \,x^{4}+\frac {3}{5} x^{5} b d \,a^{2}+\frac {3}{7} a \,b^{2} c \,x^{7}+\frac {3}{8} x^{8} b^{2} d a +\frac {1}{10} b^{3} c \,x^{10}+\frac {1}{11} b^{3} d \,x^{11}\)	\(75\)
gosper	\(\frac {x \left (280 b^{3} d \,x^{10}+308 b^{3} c \,x^{9}+1155 a \,b^{2} d \,x^{7}+1320 a \,b^{2} c \,x^{6}+1848 a^{2} b d \,x^{4}+2310 a^{2} x^{3} b c +1540 a^{3} d x +3080 c \,a^{3}\right )}{3080}\)	\(76\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {1}{11} \, b^{3} d x^{11} + \frac {1}{10} \, b^{3} c x^{10} + \frac {3}{8} \, a b^{2} d x^{8} + \frac {3}{7} \, a b^{2} c x^{7} + \frac {3}{5} \, a^{2} b d x^{5} + \frac {3}{4} \, a^{2} b c x^{4} + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=a^{3} c x + \frac {a^{3} d x^{2}}{2} + \frac {3 a^{2} b c x^{4}}{4} + \frac {3 a^{2} b d x^{5}}{5} + \frac {3 a b^{2} c x^{7}}{7} + \frac {3 a b^{2} d x^{8}}{8} + \frac {b^{3} c x^{10}}{10} + \frac {b^{3} d x^{11}}{11} \]

[In]

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

[Out]

a**3*c*x + a**3*d*x**2/2 + 3*a**2*b*c*x**4/4 + 3*a**2*b*d*x**5/5 + 3*a*b**2*c*x**7/7 + 3*a*b**2*d*x**8/8 + b**
3*c*x**10/10 + b**3*d*x**11/11

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {1}{11} \, b^{3} d x^{11} + \frac {1}{10} \, b^{3} c x^{10} + \frac {3}{8} \, a b^{2} d x^{8} + \frac {3}{7} \, a b^{2} c x^{7} + \frac {3}{5} \, a^{2} b d x^{5} + \frac {3}{4} \, a^{2} b c x^{4} + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {1}{11} \, b^{3} d x^{11} + \frac {1}{10} \, b^{3} c x^{10} + \frac {3}{8} \, a b^{2} d x^{8} + \frac {3}{7} \, a b^{2} c x^{7} + \frac {3}{5} \, a^{2} b d x^{5} + \frac {3}{4} \, a^{2} b c x^{4} + \frac {1}{2} \, a^{3} d x^{2} + a^{3} c x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {d\,a^3\,x^2}{2}+c\,a^3\,x+\frac {3\,d\,a^2\,b\,x^5}{5}+\frac {3\,c\,a^2\,b\,x^4}{4}+\frac {3\,d\,a\,b^2\,x^8}{8}+\frac {3\,c\,a\,b^2\,x^7}{7}+\frac {d\,b^3\,x^{11}}{11}+\frac {c\,b^3\,x^{10}}{10} \]

[In]

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

[Out]

(a^3*d*x^2)/2 + (b^3*c*x^10)/10 + (b^3*d*x^11)/11 + a^3*c*x + (3*a^2*b*c*x^4)/4 + (3*a*b^2*c*x^7)/7 + (3*a^2*b
*d*x^5)/5 + (3*a*b^2*d*x^8)/8