3.211 \(\int (a+c x^2) (1+(d+a x+\frac {c x^3}{3})^5) \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{6} \left (a x+\frac {c x^3}{3}+d\right )^6+a x+\frac {c x^3}{3} \]

[Out]

a*x+1/3*c*x^3+1/6*(d+a*x+1/3*c*x^3)^6

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Rubi [A]  time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1591} \[ \frac {1}{6} \left (a x+\frac {c x^3}{3}+d\right )^6+a x+\frac {c x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*x + (c*x^3)/3 + (d + a*x + (c*x^3)/3)^6/6

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

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Mathematica [B]  time = 0.05, size = 140, normalized size = 4.52 \[ \frac {x \left (3 a+c x^2\right ) \left (243 a^5 x^5+405 a^4 c x^7+270 a^3 c^2 x^9+90 a^2 c^3 x^{11}+15 a c^4 x^{13}+1215 d^4 \left (3 a x+c x^3\right )+540 d^3 \left (3 a x+c x^3\right )^2+135 d^2 \left (3 a x+c x^3\right )^3+18 d \left (3 a x+c x^3\right )^4+c^5 x^{15}+1458 d^5+1458\right )}{4374} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(3*a + c*x^2)*(1458 + 1458*d^5 + 243*a^5*x^5 + 405*a^4*c*x^7 + 270*a^3*c^2*x^9 + 90*a^2*c^3*x^11 + 15*a*c^4
*x^13 + c^5*x^15 + 1215*d^4*(3*a*x + c*x^3) + 540*d^3*(3*a*x + c*x^3)^2 + 135*d^2*(3*a*x + c*x^3)^3 + 18*d*(3*
a*x + c*x^3)^4))/4374

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fricas [B]  time = 0.99, size = 291, normalized size = 9.39 \[ \frac {1}{4374} x^{18} c^{6} + \frac {1}{243} x^{16} c^{5} a + \frac {1}{243} x^{15} d c^{5} + \frac {5}{162} x^{14} c^{4} a^{2} + \frac {5}{81} x^{13} d c^{4} a + \frac {5}{162} x^{12} d^{2} c^{4} + \frac {10}{81} x^{12} c^{3} a^{3} + \frac {10}{27} x^{11} d c^{3} a^{2} + \frac {10}{27} x^{10} d^{2} c^{3} a + \frac {5}{18} x^{10} c^{2} a^{4} + \frac {10}{81} x^{9} d^{3} c^{3} + \frac {10}{9} x^{9} d c^{2} a^{3} + \frac {5}{3} x^{8} d^{2} c^{2} a^{2} + \frac {1}{3} x^{8} c a^{5} + \frac {10}{9} x^{7} d^{3} c^{2} a + \frac {5}{3} x^{7} d c a^{4} + \frac {5}{18} x^{6} d^{4} c^{2} + \frac {10}{3} x^{6} d^{2} c a^{3} + \frac {1}{6} x^{6} a^{6} + \frac {10}{3} x^{5} d^{3} c a^{2} + x^{5} d a^{5} + \frac {5}{3} x^{4} d^{4} c a + \frac {5}{2} x^{4} d^{2} a^{4} + \frac {1}{3} x^{3} d^{5} c + \frac {10}{3} x^{3} d^{3} a^{3} + \frac {5}{2} x^{2} d^{4} a^{2} + x d^{5} a + \frac {1}{3} x^{3} c + x a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*x^18*c^6 + 1/243*x^16*c^5*a + 1/243*x^15*d*c^5 + 5/162*x^14*c^4*a^2 + 5/81*x^13*d*c^4*a + 5/162*x^12*d^
2*c^4 + 10/81*x^12*c^3*a^3 + 10/27*x^11*d*c^3*a^2 + 10/27*x^10*d^2*c^3*a + 5/18*x^10*c^2*a^4 + 10/81*x^9*d^3*c
^3 + 10/9*x^9*d*c^2*a^3 + 5/3*x^8*d^2*c^2*a^2 + 1/3*x^8*c*a^5 + 10/9*x^7*d^3*c^2*a + 5/3*x^7*d*c*a^4 + 5/18*x^
6*d^4*c^2 + 10/3*x^6*d^2*c*a^3 + 1/6*x^6*a^6 + 10/3*x^5*d^3*c*a^2 + x^5*d*a^5 + 5/3*x^4*d^4*c*a + 5/2*x^4*d^2*
a^4 + 1/3*x^3*d^5*c + 10/3*x^3*d^3*a^3 + 5/2*x^2*d^4*a^2 + x*d^5*a + 1/3*x^3*c + x*a

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giac [B]  time = 0.34, size = 105, normalized size = 3.39 \[ \frac {1}{4374} \, {\left (c x^{3} + 3 \, a x\right )}^{6} + \frac {1}{243} \, {\left (c x^{3} + 3 \, a x\right )}^{5} d + \frac {5}{162} \, {\left (c x^{3} + 3 \, a x\right )}^{4} d^{2} + \frac {10}{81} \, {\left (c x^{3} + 3 \, a x\right )}^{3} d^{3} + \frac {5}{18} \, {\left (c x^{3} + 3 \, a x\right )}^{2} d^{4} + \frac {1}{3} \, {\left (c x^{3} + 3 \, a x\right )} d^{5} + \frac {1}{3} \, c x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*(c*x^3 + 3*a*x)^6 + 1/243*(c*x^3 + 3*a*x)^5*d + 5/162*(c*x^3 + 3*a*x)^4*d^2 + 10/81*(c*x^3 + 3*a*x)^3*d
^3 + 5/18*(c*x^3 + 3*a*x)^2*d^4 + 1/3*(c*x^3 + 3*a*x)*d^5 + 1/3*c*x^3 + a*x

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maple [B]  time = 0.00, size = 618, normalized size = 19.94 \[ \frac {c^{6} x^{18}}{4374}+\frac {a \,c^{5} x^{16}}{243}+\frac {c^{5} d \,x^{15}}{243}+\frac {5 a^{2} c^{4} x^{14}}{162}+\frac {5 a \,c^{4} d \,x^{13}}{81}+\frac {10 a^{2} c^{3} d \,x^{11}}{27}+\frac {\left (\frac {10 a^{3} c^{3}}{27}+\left (\frac {2 a^{3} c^{2}}{3}+\frac {4 c^{3} d^{2}}{27}+\frac {\left (\frac {4}{3} a^{3} c +\frac {2}{3} c^{2} d^{2}\right ) c}{3}\right ) c \right ) x^{12}}{12}+\frac {\left (\left (\frac {2 a^{3} c^{2}}{3}+\frac {4 c^{3} d^{2}}{27}+\frac {\left (\frac {4}{3} a^{3} c +\frac {2}{3} c^{2} d^{2}\right ) c}{3}\right ) a +\left (\frac {4 a \,c^{2} d^{2}}{3}+\left (\frac {4}{3} a^{3} c +\frac {2}{3} c^{2} d^{2}\right ) a +\frac {\left (a^{4}+4 a c \,d^{2}\right ) c}{3}\right ) c \right ) x^{10}}{10}+\frac {\left (\frac {10 a^{3} c^{2} d}{3}+\left (4 a^{3} c d +\frac {\left (4 a^{3} d +\frac {4}{3} c \,d^{3}\right ) c}{3}+\left (\frac {4}{3} a^{3} c +\frac {2}{3} c^{2} d^{2}\right ) d \right ) c \right ) x^{9}}{9}+\frac {5 a^{2} d^{4} x^{2}}{2}+\frac {\left (\left (\frac {4 a \,c^{2} d^{2}}{3}+\left (\frac {4}{3} a^{3} c +\frac {2}{3} c^{2} d^{2}\right ) a +\frac {\left (a^{4}+4 a c \,d^{2}\right ) c}{3}\right ) a +\left (6 a^{2} c \,d^{2}+\left (a^{4}+4 a c \,d^{2}\right ) a \right ) c \right ) x^{8}}{8}+\frac {\left (\left (4 a^{3} c d +\frac {\left (4 a^{3} d +\frac {4}{3} c \,d^{3}\right ) c}{3}+\left (\frac {4}{3} a^{3} c +\frac {2}{3} c^{2} d^{2}\right ) d \right ) a +\left (\frac {4 a c \,d^{3}}{3}+\left (4 a^{3} d +\frac {4}{3} c \,d^{3}\right ) a +\left (a^{4}+4 a c \,d^{2}\right ) d \right ) c \right ) x^{7}}{7}+\frac {\left (\left (6 a^{2} c \,d^{2}+\left (a^{4}+4 a c \,d^{2}\right ) a \right ) a +\left (6 a^{3} d^{2}+\frac {c \,d^{4}}{3}+\left (4 a^{3} d +\frac {4}{3} c \,d^{3}\right ) d \right ) c \right ) x^{6}}{6}+\frac {\left (10 a^{2} c \,d^{3}+\left (\frac {4 a c \,d^{3}}{3}+\left (4 a^{3} d +\frac {4}{3} c \,d^{3}\right ) a +\left (a^{4}+4 a c \,d^{2}\right ) d \right ) a \right ) x^{5}}{5}+\frac {\left (5 a c \,d^{4}+\left (6 a^{3} d^{2}+\frac {c \,d^{4}}{3}+\left (4 a^{3} d +\frac {4}{3} c \,d^{3}\right ) d \right ) a \right ) x^{4}}{4}+\frac {\left (10 a^{3} d^{3}+\left (d^{5}+1\right ) c \right ) x^{3}}{3}+\left (d^{5}+1\right ) a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*c^6*x^18+1/243*a*c^5*x^16+1/243*c^5*d*x^15+5/162*a^2*c^4*x^14+5/81*a*c^4*d*x^13+1/12*(10/27*a^3*c^3+c*(
4/27*d^2*c^3+2/3*a^3*c^2+1/3*c*(2/3*d^2*c^2+4/3*a^3*c)))*x^12+10/27*a^2*c^3*d*x^11+1/10*(a*(4/27*d^2*c^3+2/3*a
^3*c^2+1/3*c*(2/3*d^2*c^2+4/3*a^3*c))+c*(4/3*d^2*a*c^2+a*(2/3*d^2*c^2+4/3*a^3*c)+1/3*c*(a^4+4*a*c*d^2)))*x^10+
1/9*(10/3*a^3*d*c^2+c*(d*(2/3*d^2*c^2+4/3*a^3*c)+4*a^3*d*c+1/3*c*(4/3*d^3*c+4*a^3*d)))*x^9+1/8*(a*(4/3*d^2*a*c
^2+a*(2/3*d^2*c^2+4/3*a^3*c)+1/3*c*(a^4+4*a*c*d^2))+c*(6*d^2*a^2*c+a*(a^4+4*a*c*d^2)))*x^8+1/7*(a*(d*(2/3*d^2*
c^2+4/3*a^3*c)+4*a^3*d*c+1/3*c*(4/3*d^3*c+4*a^3*d))+c*(d*(a^4+4*a*c*d^2)+a*(4/3*d^3*c+4*a^3*d)+4/3*c*d^3*a))*x
^7+1/6*(a*(6*d^2*a^2*c+a*(a^4+4*a*c*d^2))+c*(d*(4/3*d^3*c+4*a^3*d)+6*a^3*d^2+1/3*c*d^4))*x^6+1/5*(a*(d*(a^4+4*
a*c*d^2)+a*(4/3*d^3*c+4*a^3*d)+4/3*c*d^3*a)+10*a^2*c*d^3)*x^5+1/4*(a*(d*(4/3*d^3*c+4*a^3*d)+6*a^3*d^2+1/3*c*d^
4)+5*a*c*d^4)*x^4+1/3*(10*a^3*d^3+c*(d^5+1))*x^3+5/2*a^2*d^4*x^2+a*(d^5+1)*x

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maxima [B]  time = 0.46, size = 280, normalized size = 9.03 \[ \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{243} \, a c^{5} x^{16} + \frac {1}{243} \, c^{5} d x^{15} + \frac {5}{162} \, a^{2} c^{4} x^{14} + \frac {5}{81} \, a c^{4} d x^{13} + \frac {10}{27} \, a^{2} c^{3} d x^{11} + \frac {5}{162} \, {\left (4 \, a^{3} c^{3} + c^{4} d^{2}\right )} x^{12} + \frac {5}{54} \, {\left (3 \, a^{4} c^{2} + 4 \, a c^{3} d^{2}\right )} x^{10} + \frac {10}{81} \, {\left (9 \, a^{3} c^{2} d + c^{3} d^{3}\right )} x^{9} + \frac {1}{3} \, {\left (a^{5} c + 5 \, a^{2} c^{2} d^{2}\right )} x^{8} + \frac {5}{2} \, a^{2} d^{4} x^{2} + \frac {5}{9} \, {\left (3 \, a^{4} c d + 2 \, a c^{2} d^{3}\right )} x^{7} + \frac {1}{18} \, {\left (3 \, a^{6} + 60 \, a^{3} c d^{2} + 5 \, c^{2} d^{4}\right )} x^{6} + \frac {1}{3} \, {\left (3 \, a^{5} d + 10 \, a^{2} c d^{3}\right )} x^{5} + \frac {5}{6} \, {\left (3 \, a^{4} d^{2} + 2 \, a c d^{4}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} d^{3} + c d^{5} + c\right )} x^{3} + {\left (a d^{5} + a\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*c^6*x^18 + 1/243*a*c^5*x^16 + 1/243*c^5*d*x^15 + 5/162*a^2*c^4*x^14 + 5/81*a*c^4*d*x^13 + 10/27*a^2*c^3
*d*x^11 + 5/162*(4*a^3*c^3 + c^4*d^2)*x^12 + 5/54*(3*a^4*c^2 + 4*a*c^3*d^2)*x^10 + 10/81*(9*a^3*c^2*d + c^3*d^
3)*x^9 + 1/3*(a^5*c + 5*a^2*c^2*d^2)*x^8 + 5/2*a^2*d^4*x^2 + 5/9*(3*a^4*c*d + 2*a*c^2*d^3)*x^7 + 1/18*(3*a^6 +
60*a^3*c*d^2 + 5*c^2*d^4)*x^6 + 1/3*(3*a^5*d + 10*a^2*c*d^3)*x^5 + 5/6*(3*a^4*d^2 + 2*a*c*d^4)*x^4 + 1/3*(10*
a^3*d^3 + c*d^5 + c)*x^3 + (a*d^5 + a)*x

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mupad [B]  time = 2.27, size = 266, normalized size = 8.58 \[ x^5\,\left (a^5\,d+\frac {10\,c\,a^2\,d^3}{3}\right )+x^4\,\left (\frac {5\,a^4\,d^2}{2}+\frac {5\,c\,a\,d^4}{3}\right )+x^3\,\left (\frac {10\,a^3\,d^3}{3}+\frac {c\,d^5}{3}+\frac {c}{3}\right )+x^6\,\left (\frac {a^6}{6}+\frac {10\,a^3\,c\,d^2}{3}+\frac {5\,c^2\,d^4}{18}\right )+\frac {c^6\,x^{18}}{4374}+\frac {a\,c^5\,x^{16}}{243}+a\,x\,\left (d^5+1\right )+\frac {c^5\,d\,x^{15}}{243}+\frac {5\,a^2\,c^4\,x^{14}}{162}+\frac {5\,a^2\,d^4\,x^2}{2}+\frac {5\,c^3\,x^{12}\,\left (4\,a^3+c\,d^2\right )}{162}+\frac {a^2\,c\,x^8\,\left (a^3+5\,c\,d^2\right )}{3}+\frac {10\,a^2\,c^3\,d\,x^{11}}{27}+\frac {5\,a\,c^2\,x^{10}\,\left (3\,a^3+4\,c\,d^2\right )}{54}+\frac {10\,c^2\,d\,x^9\,\left (9\,a^3+c\,d^2\right )}{81}+\frac {5\,a\,c^4\,d\,x^{13}}{81}+\frac {5\,a\,c\,d\,x^7\,\left (3\,a^3+2\,c\,d^2\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*(a^5*d + (10*a^2*c*d^3)/3) + x^4*((5*a^4*d^2)/2 + (5*a*c*d^4)/3) + x^3*(c/3 + (c*d^5)/3 + (10*a^3*d^3)/3)
+ x^6*(a^6/6 + (5*c^2*d^4)/18 + (10*a^3*c*d^2)/3) + (c^6*x^18)/4374 + (a*c^5*x^16)/243 + a*x*(d^5 + 1) + (c^5*
d*x^15)/243 + (5*a^2*c^4*x^14)/162 + (5*a^2*d^4*x^2)/2 + (5*c^3*x^12*(c*d^2 + 4*a^3))/162 + (a^2*c*x^8*(5*c*d^
2 + a^3))/3 + (10*a^2*c^3*d*x^11)/27 + (5*a*c^2*x^10*(4*c*d^2 + 3*a^3))/54 + (10*c^2*d*x^9*(c*d^2 + 9*a^3))/81
+ (5*a*c^4*d*x^13)/81 + (5*a*c*d*x^7*(2*c*d^2 + 3*a^3))/9

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sympy [B]  time = 0.14, size = 314, normalized size = 10.13 \[ \frac {5 a^{2} c^{4} x^{14}}{162} + \frac {10 a^{2} c^{3} d x^{11}}{27} + \frac {5 a^{2} d^{4} x^{2}}{2} + \frac {a c^{5} x^{16}}{243} + \frac {5 a c^{4} d x^{13}}{81} + \frac {c^{6} x^{18}}{4374} + \frac {c^{5} d x^{15}}{243} + x^{12} \left (\frac {10 a^{3} c^{3}}{81} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{10} \left (\frac {5 a^{4} c^{2}}{18} + \frac {10 a c^{3} d^{2}}{27}\right ) + x^{9} \left (\frac {10 a^{3} c^{2} d}{9} + \frac {10 c^{3} d^{3}}{81}\right ) + x^{8} \left (\frac {a^{5} c}{3} + \frac {5 a^{2} c^{2} d^{2}}{3}\right ) + x^{7} \left (\frac {5 a^{4} c d}{3} + \frac {10 a c^{2} d^{3}}{9}\right ) + x^{6} \left (\frac {a^{6}}{6} + \frac {10 a^{3} c d^{2}}{3} + \frac {5 c^{2} d^{4}}{18}\right ) + x^{5} \left (a^{5} d + \frac {10 a^{2} c d^{3}}{3}\right ) + x^{4} \left (\frac {5 a^{4} d^{2}}{2} + \frac {5 a c d^{4}}{3}\right ) + x^{3} \left (\frac {10 a^{3} d^{3}}{3} + \frac {c d^{5}}{3} + \frac {c}{3}\right ) + x \left (a d^{5} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a**2*c**4*x**14/162 + 10*a**2*c**3*d*x**11/27 + 5*a**2*d**4*x**2/2 + a*c**5*x**16/243 + 5*a*c**4*d*x**13/81
+ c**6*x**18/4374 + c**5*d*x**15/243 + x**12*(10*a**3*c**3/81 + 5*c**4*d**2/162) + x**10*(5*a**4*c**2/18 + 10*
a*c**3*d**2/27) + x**9*(10*a**3*c**2*d/9 + 10*c**3*d**3/81) + x**8*(a**5*c/3 + 5*a**2*c**2*d**2/3) + x**7*(5*a
**4*c*d/3 + 10*a*c**2*d**3/9) + x**6*(a**6/6 + 10*a**3*c*d**2/3 + 5*c**2*d**4/18) + x**5*(a**5*d + 10*a**2*c*d
**3/3) + x**4*(5*a**4*d**2/2 + 5*a*c*d**4/3) + x**3*(10*a**3*d**3/3 + c*d**5/3 + c/3) + x*(a*d**5 + a)

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