3.3.11 \(\int (a+c x^2) (1+(d+a x+\frac {c x^3}{3})^5) \, dx\) [211]

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

[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 = 0.03, 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 = {1605} \begin {gather*} \frac {1}{6} \left (a x+\frac {c x^3}{3}+d\right )^6+a x+\frac {c x^3}{3} \end {gather*}

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 1605

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 &=\text {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] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(31)=62\).
time = 0.04, size = 140, normalized size = 4.52 \begin {gather*} \frac {x \left (3 a+c x^2\right ) \left (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 \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\right )}{4374} \end {gather*}

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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Maple [A]
time = 0.10, size = 27, normalized size = 0.87

method	result	size
default	\(\frac {\left (d +a x +\frac {1}{3} c \,x^{3}\right )^{6}}{6}+d +a x +\frac {c \,x^{3}}{3}\)	\(27\)
norman	\(\left (\frac {10}{81} a^{3} c^{3}+\frac {5}{162} c^{4} d^{2}\right ) x^{12}+\left (\frac {5}{18} a^{4} c^{2}+\frac {10}{27} a \,c^{3} d^{2}\right ) x^{10}+\left (\frac {5}{2} a^{4} d^{2}+\frac {5}{3} a c \,d^{4}\right ) x^{4}+\left (\frac {1}{3} a^{5} c +\frac {5}{3} a^{2} c^{2} d^{2}\right ) x^{8}+\left (a^{5} d +\frac {10}{3} a^{2} c \,d^{3}\right ) x^{5}+\left (\frac {10}{9} a^{3} c^{2} d +\frac {10}{81} c^{3} d^{3}\right ) x^{9}+\left (\frac {5}{3} a^{4} c d +\frac {10}{9} a \,c^{2} d^{3}\right ) x^{7}+\left (\frac {1}{6} a^{6}+\frac {10}{3} a^{3} c \,d^{2}+\frac {5}{18} c^{2} d^{4}\right ) x^{6}+\left (\frac {10}{3} a^{3} d^{3}+\frac {1}{3} d^{5} c +\frac {1}{3} c \right ) x^{3}+\left (d^{5} a +a \right ) x +\frac {c^{6} x^{18}}{4374}+\frac {a \,c^{5} x^{16}}{243}+\frac {5 a^{2} c^{4} x^{14}}{162}+\frac {5 a^{2} d^{4} x^{2}}{2}+\frac {c^{5} d \,x^{15}}{243}+\frac {5 a \,c^{4} d \,x^{13}}{81}+\frac {10 a^{2} c^{3} d \,x^{11}}{27}\)	\(277\)
risch	\(\frac {1}{3} c \,x^{3}+\frac {1}{6} a^{6} x^{6}+\frac {5}{162} x^{12} c^{4} d^{2}+\frac {10}{81} x^{9} c^{3} d^{3}+a x +\frac {5}{81} a \,c^{4} d \,x^{13}+\frac {5}{2} a^{2} d^{4} x^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {1}{3} a^{5} c \,x^{8}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {10}{81} a^{3} c^{3} x^{12}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {1}{243} a \,c^{5} x^{16}+\frac {5}{3} a^{4} c d \,x^{7}+\frac {10}{9} a^{3} c^{2} d \,x^{9}+\frac {10}{27} a^{2} c^{3} d \,x^{11}+a \,d^{5} x +\frac {1}{4374} c^{6} x^{18}+\frac {10}{27} x^{10} a \,c^{3} d^{2}+\frac {5}{3} x^{8} a^{2} c^{2} d^{2}+\frac {10}{9} x^{7} a \,c^{2} d^{3}+\frac {10}{3} x^{6} a^{3} c \,d^{2}+\frac {10}{3} x^{5} a^{2} c \,d^{3}+\frac {5}{3} x^{4} a c \,d^{4}+\frac {5}{18} x^{6} c^{2} d^{4}+x^{5} a^{5} d +\frac {5}{2} x^{4} a^{4} d^{2}+\frac {10}{3} x^{3} a^{3} d^{3}+\frac {1}{3} x^{3} d^{5} c\)	\(292\)
gosper	\(\frac {x \left (c^{6} x^{17}+18 a \,c^{5} x^{15}+18 c^{5} d \,x^{14}+135 a^{2} c^{4} x^{13}+270 a \,c^{4} d \,x^{12}+540 a^{3} c^{3} x^{11}+135 x^{11} c^{4} d^{2}+1620 a^{2} c^{3} d \,x^{10}+1215 a^{4} c^{2} x^{9}+1620 x^{9} a \,c^{3} d^{2}+4860 a^{3} c^{2} d \,x^{8}+540 c^{3} d^{3} x^{8}+1458 a^{5} c \,x^{7}+7290 a^{2} c^{2} d^{2} x^{7}+7290 a^{4} c d \,x^{6}+4860 a \,c^{2} d^{3} x^{6}+729 a^{6} x^{5}+14580 x^{5} a^{3} c \,d^{2}+1215 x^{5} c^{2} d^{4}+4374 a^{5} d \,x^{4}+14580 a^{2} c \,d^{3} x^{4}+10935 x^{3} a^{4} d^{2}+7290 x^{3} a c \,d^{4}+14580 a^{3} d^{3} x^{2}+1458 c \,d^{5} x^{2}+10935 a^{2} d^{4} x +4374 d^{5} a +1458 c \,x^{2}+4374 a \right )}{4374}\)	\(293\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (27) = 54\).
time = 0.27, size = 280, normalized size = 9.03 \begin {gather*} \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 \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (27) = 54\).
time = 0.40, size = 280, normalized size = 9.03 \begin {gather*} \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 \end {gather*}

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*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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (24) = 48\).
time = 0.05, size = 314, normalized size = 10.13 \begin {gather*} \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} \cdot \left (\frac {10 a^{3} c^{3}}{81} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{10} \cdot \left (\frac {5 a^{4} c^{2}}{18} + \frac {10 a c^{3} d^{2}}{27}\right ) + x^{9} \cdot \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} \cdot \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} \cdot \left (\frac {5 a^{4} d^{2}}{2} + \frac {5 a c d^{4}}{3}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3}}{3} + \frac {c d^{5}}{3} + \frac {c}{3}\right ) + x \left (a d^{5} + a\right ) \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (27) = 54\).
time = 3.79, size = 105, normalized size = 3.39 \begin {gather*} \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 \end {gather*}

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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Mupad [B]
time = 2.27, size = 266, normalized size = 8.58 \begin {gather*} 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} \end {gather*}

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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