3.444 \(\int f^{a+b x+c x^2} (d+e x)^3 \, dx\)

Optimal. Leaf size=266 \[ -\frac{3 \sqrt{\pi } e^2 f^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}+\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} (2 c d-b e)^3 \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}+\frac{e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac{e (d+e x) (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}-\frac{e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)} \]

[Out]

-(e^3*f^(a + b*x + c*x^2))/(2*c^2*Log[f]^2) - (3*e^2*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x
)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (e*(2*c*d - b*e)^2*f^(a + b*x + c*x^2))/(8*c^3*Log[f]
) + (e*(2*c*d - b*e)*f^(a + b*x + c*x^2)*(d + e*x))/(4*c^2*Log[f]) + (e*f^(a + b*x + c*x^2)*(d + e*x)^2)/(2*c*
Log[f]) + ((2*c*d - b*e)^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(16*c^(7/2
)*Sqrt[Log[f]])

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Rubi [A]  time = 0.32375, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2241, 2240, 2234, 2204} \[ -\frac{3 \sqrt{\pi } e^2 f^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}+\frac{\sqrt{\pi } f^{a-\frac{b^2}{4 c}} (2 c d-b e)^3 \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}+\frac{e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac{e (d+e x) (2 c d-b e) f^{a+b x+c x^2}}{4 c^2 \log (f)}-\frac{e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{e (d+e x)^2 f^{a+b x+c x^2}}{2 c \log (f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^3*f^(a + b*x + c*x^2))/(2*c^2*Log[f]^2) - (3*e^2*(2*c*d - b*e)*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x
)*Sqrt[Log[f]])/(2*Sqrt[c])])/(8*c^(5/2)*Log[f]^(3/2)) + (e*(2*c*d - b*e)^2*f^(a + b*x + c*x^2))/(8*c^3*Log[f]
) + (e*(2*c*d - b*e)*f^(a + b*x + c*x^2)*(d + e*x))/(4*c^2*Log[f]) + (e*f^(a + b*x + c*x^2)*(d + e*x)^2)/(2*c*
Log[f]) + ((2*c*d - b*e)^3*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(16*c^(7/2
)*Sqrt[Log[f]])

Rule 2241

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*F^(a + b*x + c*x^2))/(2*c*Log[F]), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[((m - 1)*e^2)/(2*c*Log[F]), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 2240

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] - Dist[(b*e - 2*c*d)/(2*c), Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
NeQ[b*e - 2*c*d, 0]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int f^{a+b x+c x^2} (d+e x)^3 \, dx &=\frac{e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}-\frac{(-2 c d+b e) \int f^{a+b x+c x^2} (d+e x)^2 \, dx}{2 c}-\frac{e^2 \int f^{a+b x+c x^2} (d+e x) \, dx}{c \log (f)}\\ &=-\frac{e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac{(2 c d-b e)^2 \int f^{a+b x+c x^2} (d+e x) \, dx}{4 c^2}-\frac{\left (e^2 (2 c d-b e)\right ) \int f^{a+b x+c x^2} \, dx}{4 c^2 \log (f)}-\frac{\left (e^2 (2 c d-b e)\right ) \int f^{a+b x+c x^2} \, dx}{2 c^2 \log (f)}\\ &=-\frac{e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}+\frac{e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac{(2 c d-b e)^3 \int f^{a+b x+c x^2} \, dx}{8 c^3}-\frac{\left (e^2 (2 c d-b e) f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c^2 \log (f)}-\frac{\left (e^2 (2 c d-b e) f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c^2 \log (f)}\\ &=-\frac{e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{3 e^2 (2 c d-b e) f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}+\frac{e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac{\left ((2 c d-b e)^3 f^{a-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{8 c^3}\\ &=-\frac{e^3 f^{a+b x+c x^2}}{2 c^2 \log ^2(f)}-\frac{3 e^2 (2 c d-b e) f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{8 c^{5/2} \log ^{\frac{3}{2}}(f)}+\frac{e (2 c d-b e)^2 f^{a+b x+c x^2}}{8 c^3 \log (f)}+\frac{e (2 c d-b e) f^{a+b x+c x^2} (d+e x)}{4 c^2 \log (f)}+\frac{e f^{a+b x+c x^2} (d+e x)^2}{2 c \log (f)}+\frac{(2 c d-b e)^3 f^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{16 c^{7/2} \sqrt{\log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.275, size = 169, normalized size = 0.64 \[ \frac{f^{a-\frac{b^2}{4 c}} \left (2 \sqrt{c} e f^{\frac{(b+2 c x)^2}{4 c}} \left (\log (f) \left (b^2 e^2-2 b c e (3 d+e x)+4 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right )-4 c e^2\right )+\sqrt{\pi } \sqrt{\log (f)} (2 c d-b e) \left (\log (f) (b e-2 c d)^2-6 c e^2\right ) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{16 c^{7/2} \log ^2(f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^(a - b^2/(4*c))*((2*c*d - b*e)*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqrt[Log[f]]*(-6*c*e^2
+ (-2*c*d + b*e)^2*Log[f]) + 2*Sqrt[c]*e*f^((b + 2*c*x)^2/(4*c))*(-4*c*e^2 + (b^2*e^2 - 2*b*c*e*(3*d + e*x) +
4*c^2*(3*d^2 + 3*d*e*x + e^2*x^2))*Log[f])))/(16*c^(7/2)*Log[f]^2)

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Maple [B]  time = 0.052, size = 550, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d^3*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+1/
2*e^3/c/ln(f)*x^2*f^(c*x^2)*f^(b*x)*f^a-1/4*e^3/c^2*b/ln(f)*x*f^(c*x^2)*f^(b*x)*f^a+1/8*e^3/c^3*b^2/ln(f)*f^(c
*x^2)*f^(b*x)*f^a+1/16*e^3/c^3*b^3*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*
ln(f)/(-c*ln(f))^(1/2))-3/8*e^3/c^2*b/ln(f)*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)
*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))-1/2*e^3/c^2/ln(f)^2*f^(c*x^2)*f^(b*x)*f^a+3/2*d*e^2/c/ln(f)*x*f^(c*x^2)*f^(b*
x)*f^a-3/4*d*e^2/c^2*b/ln(f)*f^(c*x^2)*f^(b*x)*f^a-3/8*d*e^2/c^2*b^2*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1
/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+3/4*d*e^2/c/ln(f)*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln
(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))+3/2*e*d^2/c/ln(f)*f^(c*x^2)*f^(b*x)*f^a+3/4*e
*d^2*b/c*Pi^(1/2)*f^a*f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))

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Maxima [B]  time = 1.35878, size = 728, normalized size = 2.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^2/(sqrt(-(2*c*x + b)^2*log(f)
/c)*(c*log(f))^(3/2)) - 2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)/(c*log(f))^(3/2))*d^2*e*f^(a - 1/4*b^2/c)/sqrt(c*lo
g(f)) + 3/8*(sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f)^3/(sqrt(-(2*c*x + b)
^2*log(f)/c)*(c*log(f))^(5/2)) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^3/((-(2*c*x +
b)^2*log(f)/c)^(3/2)*(c*log(f))^(5/2)) - 4*b*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^2/(c*log(f))^(5/2))*d*e^2*f^(a -
1/4*b^2/c)/sqrt(c*log(f)) - 1/16*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2*log(f)/c)) - 1)*log(f
)^4/(sqrt(-(2*c*x + b)^2*log(f)/c)*(c*log(f))^(7/2)) - 12*(2*c*x + b)^3*b*gamma(3/2, -1/4*(2*c*x + b)^2*log(f)
/c)*log(f)^4/((-(2*c*x + b)^2*log(f)/c)^(3/2)*(c*log(f))^(7/2)) - 6*b^2*c*f^(1/4*(2*c*x + b)^2/c)*log(f)^3/(c*
log(f))^(7/2) + 8*c^2*gamma(2, -1/4*(2*c*x + b)^2*log(f)/c)*log(f)^2/(c*log(f))^(7/2))*e^3*f^(a - 1/4*b^2/c)/s
qrt(c*log(f)) + 1/2*sqrt(pi)*d^3*f^a*erf(sqrt(-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*log(f))*f^
(1/4*b^2/c))

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Fricas [A]  time = 1.56033, size = 458, normalized size = 1.72 \begin{align*} -\frac{2 \,{\left (4 \, c^{2} e^{3} -{\left (4 \, c^{3} e^{3} x^{2} + 12 \, c^{3} d^{2} e - 6 \, b c^{2} d e^{2} + b^{2} c e^{3} + 2 \,{\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x\right )} \log \left (f\right )\right )} f^{c x^{2} + b x + a} - \frac{\sqrt{\pi }{\left (12 \, c^{2} d e^{2} - 6 \, b c e^{3} -{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (f\right )\right )} \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2} - 4 \, a c}{4 \, c}}}}{16 \, c^{4} \log \left (f\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*(4*c^2*e^3 - (4*c^3*e^3*x^2 + 12*c^3*d^2*e - 6*b*c^2*d*e^2 + b^2*c*e^3 + 2*(6*c^3*d*e^2 - b*c^2*e^3)*
x)*log(f))*f^(c*x^2 + b*x + a) - sqrt(pi)*(12*c^2*d*e^2 - 6*b*c*e^3 - (8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*
e^2 - b^3*e^3)*log(f))*sqrt(-c*log(f))*erf(1/2*(2*c*x + b)*sqrt(-c*log(f))/c)/f^(1/4*(b^2 - 4*a*c)/c))/(c^4*lo
g(f)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x + c x^{2}} \left (d + e x\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x + c*x**2)*(d + e*x)**3, x)

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Giac [A]  time = 1.36433, size = 541, normalized size = 2.03 \begin{align*} -\frac{\sqrt{\pi } d^{3} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{2 \, \sqrt{-c \log \left (f\right )}} + \frac{3 \,{\left (\frac{\sqrt{\pi } b d^{2} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right ) - 4 \, c}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )}} + \frac{2 \, d^{2} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right ) + 1\right )}}{\log \left (f\right )}\right )}}{4 \, c} - \frac{3 \,{\left (\frac{\sqrt{\pi }{\left (b^{2} d \log \left (f\right ) - 2 \, c d\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right ) - 8 \, c}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )} \log \left (f\right )} - \frac{2 \,{\left (c d{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b d\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right ) + 2\right )}}{\log \left (f\right )}\right )}}{8 \, c^{2}} + \frac{\frac{\sqrt{\pi }{\left (b^{3} \log \left (f\right ) - 6 \, b c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right ) - 12 \, c}{4 \, c}\right )}}{\sqrt{-c \log \left (f\right )} \log \left (f\right )} + \frac{2 \,{\left (c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} \log \left (f\right ) - 3 \, b c{\left (2 \, x + \frac{b}{c}\right )} \log \left (f\right ) + 3 \, b^{2} \log \left (f\right ) - 4 \, c\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right ) + 3\right )}}{\log \left (f\right )^{2}}}{16 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*d^3*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f))/c)/sqrt(-c*log(f))
+ 3/4*(sqrt(pi)*b*d^2*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f) - 4*c)/c)/sqrt
(-c*log(f)) + 2*d^2*e^(c*x^2*log(f) + b*x*log(f) + a*log(f) + 1)/log(f))/c - 3/8*(sqrt(pi)*(b^2*d*log(f) - 2*c
*d)*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f) - 8*c)/c)/(sqrt(-c*log(f))*log(f)
) - 2*(c*d*(2*x + b/c) - 2*b*d)*e^(c*x^2*log(f) + b*x*log(f) + a*log(f) + 2)/log(f))/c^2 + 1/16*(sqrt(pi)*(b^3
*log(f) - 6*b*c)*erf(-1/2*sqrt(-c*log(f))*(2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f) - 12*c)/c)/(sqrt(-c*
log(f))*log(f)) + 2*(c^2*(2*x + b/c)^2*log(f) - 3*b*c*(2*x + b/c)*log(f) + 3*b^2*log(f) - 4*c)*e^(c*x^2*log(f)
+ b*x*log(f) + a*log(f) + 3)/log(f)^2)/c^3