3.212 \(\int (b x+c x^2) (1+(\frac {b x^2}{2}+\frac {c x^3}{3})^5) \, dx\)

Optimal. Leaf size=34 \[ \frac {x^{12} (3 b+2 c x)^6}{279936}+\frac {b x^2}{2}+\frac {c x^3}{3} \]

[Out]

1/2*b*x^2+1/3*c*x^3+1/279936*x^12*(2*c*x+3*b)^6

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1591} \[ \frac {x^{12} (3 b+2 c x)^6}{279936}+\frac {b x^2}{2}+\frac {c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)*(1 + ((b*x^2)/2 + (c*x^3)/3)^5),x]

[Out]

(b*x^2)/2 + (c*x^3)/3 + (x^12*(3*b + 2*c*x)^6)/279936

Rule 1591

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,
r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

Rubi steps

\begin {align*} \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx &=\operatorname {Subst}\left (\int \left (1+x^5\right ) \, dx,x,\frac {b x^2}{2}+\frac {c x^3}{3}\right )\\ &=\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {x^{12} (3 b+2 c x)^6}{279936}\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 98, normalized size = 2.88 \[ \frac {b^6 x^{12}}{384}+\frac {1}{96} b^5 c x^{13}+\frac {5}{288} b^4 c^2 x^{14}+\frac {5}{324} b^3 c^3 x^{15}+\frac {5}{648} b^2 c^4 x^{16}+\frac {1}{486} b c^5 x^{17}+\frac {b x^2}{2}+\frac {c^6 x^{18}}{4374}+\frac {c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)*(1 + ((b*x^2)/2 + (c*x^3)/3)^5),x]

[Out]

(b*x^2)/2 + (c*x^3)/3 + (b^6*x^12)/384 + (b^5*c*x^13)/96 + (5*b^4*c^2*x^14)/288 + (5*b^3*c^3*x^15)/324 + (5*b^
2*c^4*x^16)/648 + (b*c^5*x^17)/486 + (c^6*x^18)/4374

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fricas [B]  time = 0.68, size = 80, normalized size = 2.35 \[ \frac {1}{4374} x^{18} c^{6} + \frac {1}{486} x^{17} c^{5} b + \frac {5}{648} x^{16} c^{4} b^{2} + \frac {5}{324} x^{15} c^{3} b^{3} + \frac {5}{288} x^{14} c^{2} b^{4} + \frac {1}{96} x^{13} c b^{5} + \frac {1}{384} x^{12} b^{6} + \frac {1}{3} x^{3} c + \frac {1}{2} x^{2} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)*(1+(1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="fricas")

[Out]

1/4374*x^18*c^6 + 1/486*x^17*c^5*b + 5/648*x^16*c^4*b^2 + 5/324*x^15*c^3*b^3 + 5/288*x^14*c^2*b^4 + 1/96*x^13*
c*b^5 + 1/384*x^12*b^6 + 1/3*x^3*c + 1/2*x^2*b

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giac [A]  time = 0.29, size = 30, normalized size = 0.88 \[ \frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{6} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)*(1+(1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="giac")

[Out]

1/279936*(2*c*x^3 + 3*b*x^2)^6 + 1/3*c*x^3 + 1/2*b*x^2

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maple [B]  time = 0.00, size = 81, normalized size = 2.38 \[ \frac {1}{4374} c^{6} x^{18}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {1}{96} b^{5} c \,x^{13}+\frac {1}{384} b^{6} x^{12}+\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)*(1+(1/2*b*x^2+1/3*c*x^3)^5),x)

[Out]

1/4374*c^6*x^18+1/486*b*c^5*x^17+5/648*b^2*c^4*x^16+5/324*b^3*c^3*x^15+5/288*b^4*c^2*x^14+1/96*b^5*c*x^13+1/38
4*b^6*x^12+1/3*c*x^3+1/2*b*x^2

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maxima [B]  time = 0.44, size = 80, normalized size = 2.35 \[ \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {5}{324} \, b^{3} c^{3} x^{15} + \frac {5}{288} \, b^{4} c^{2} x^{14} + \frac {1}{96} \, b^{5} c x^{13} + \frac {1}{384} \, b^{6} x^{12} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)*(1+(1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="maxima")

[Out]

1/4374*c^6*x^18 + 1/486*b*c^5*x^17 + 5/648*b^2*c^4*x^16 + 5/324*b^3*c^3*x^15 + 5/288*b^4*c^2*x^14 + 1/96*b^5*c
*x^13 + 1/384*b^6*x^12 + 1/3*c*x^3 + 1/2*b*x^2

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mupad [B]  time = 0.07, size = 80, normalized size = 2.35 \[ \frac {b^6\,x^{12}}{384}+\frac {b^5\,c\,x^{13}}{96}+\frac {5\,b^4\,c^2\,x^{14}}{288}+\frac {5\,b^3\,c^3\,x^{15}}{324}+\frac {5\,b^2\,c^4\,x^{16}}{648}+\frac {b\,c^5\,x^{17}}{486}+\frac {b\,x^2}{2}+\frac {c^6\,x^{18}}{4374}+\frac {c\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(((b*x^2)/2 + (c*x^3)/3)^5 + 1),x)

[Out]

(b*x^2)/2 + (c*x^3)/3 + (b^6*x^12)/384 + (c^6*x^18)/4374 + (b^5*c*x^13)/96 + (b*c^5*x^17)/486 + (5*b^4*c^2*x^1
4)/288 + (5*b^3*c^3*x^15)/324 + (5*b^2*c^4*x^16)/648

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sympy [B]  time = 0.10, size = 90, normalized size = 2.65 \[ \frac {b^{6} x^{12}}{384} + \frac {b^{5} c x^{13}}{96} + \frac {5 b^{4} c^{2} x^{14}}{288} + \frac {5 b^{3} c^{3} x^{15}}{324} + \frac {5 b^{2} c^{4} x^{16}}{648} + \frac {b c^{5} x^{17}}{486} + \frac {b x^{2}}{2} + \frac {c^{6} x^{18}}{4374} + \frac {c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)*(1+(1/2*b*x**2+1/3*c*x**3)**5),x)

[Out]

b**6*x**12/384 + b**5*c*x**13/96 + 5*b**4*c**2*x**14/288 + 5*b**3*c**3*x**15/324 + 5*b**2*c**4*x**16/648 + b*c
**5*x**17/486 + b*x**2/2 + c**6*x**18/4374 + c*x**3/3

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