3.198 \(\int f^{c (a+b x)^2} \, dx\)

Optimal. Leaf size=41 \[ \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]

[Out]

1/2*erfi((b*x+a)*c^(1/2)*ln(f)^(1/2))*Pi^(1/2)/b/c^(1/2)/ln(f)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2204} \[ \frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(c*(a + b*x)^2),x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b*Sqrt[c]*Sqrt[Log[f]])

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^{c (a+b x)^2} \, dx &=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 41, normalized size = 1.00 \[ \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c*(a + b*x)^2),x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x)*Sqrt[Log[f]]])/(2*b*Sqrt[c]*Sqrt[Log[f]])

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fricas [A]  time = 0.42, size = 45, normalized size = 1.10 \[ -\frac {\sqrt {\pi } \sqrt {-b^{2} c \log \relax (f)} \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \relax (f)} {\left (b x + a\right )}}{b}\right )}{2 \, b^{2} c \log \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*sqrt(-b^2*c*log(f))*erf(sqrt(-b^2*c*log(f))*(b*x + a)/b)/(b^2*c*log(f))

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giac [A]  time = 0.20, size = 33, normalized size = 0.80 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f)} b {\left (x + \frac {a}{b}\right )}\right )}{2 \, \sqrt {-c \log \relax (f)} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(pi)*erf(-sqrt(-c*log(f))*b*(x + a/b))/(sqrt(-c*log(f))*b)

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maple [A]  time = 0.05, size = 41, normalized size = 1.00 \[ -\frac {\sqrt {\pi }\, \erf \left (\frac {a c \ln \relax (f )}{\sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, b x \right )}{2 \sqrt {-c \ln \relax (f )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*Pi^(1/2)/b/(-c*ln(f))^(1/2)*erf(1/(-c*ln(f))^(1/2)*a*c*ln(f)-(-c*ln(f))^(1/2)*b*x)

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maxima [A]  time = 0.59, size = 40, normalized size = 0.98 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c \log \relax (f)} b x - \frac {a c \log \relax (f)}{\sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi)*erf(sqrt(-c*log(f))*b*x - a*c*log(f)/sqrt(-c*log(f)))/(sqrt(-c*log(f))*b)

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mupad [B]  time = 0.04, size = 45, normalized size = 1.10 \[ -\frac {\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,c\,x\,\ln \relax (f)\,b^2+1{}\mathrm {i}\,a\,c\,\ln \relax (f)\,b}{\sqrt {b^2\,c\,\ln \relax (f)}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b^2\,c\,\ln \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(a + b*x)^2),x)

[Out]

-(pi^(1/2)*erf((a*b*c*log(f)*1i + b^2*c*x*log(f)*1i)/(b^2*c*log(f))^(1/2))*1i)/(2*(b^2*c*log(f))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{c \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**2), x)

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