3.272 \(\int F^{a+b (c+d x)^2} (c+d x)^2 \, dx\)

Optimal. Leaf size=77 \[ \frac {(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)} \]

[Out]

1/2*F^(a+b*(d*x+c)^2)*(d*x+c)/b/d/ln(F)-1/4*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(3/2)/d/ln(F)^(3/
2)

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Rubi [A]  time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2212, 2204} \[ \frac {(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^2,x]

[Out]

-(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(4*b^(3/2)*d*Log[F]^(3/2)) + (F^(a + b*(c + d*x)^2)*(c +
d*x))/(2*b*d*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]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 77, normalized size = 1.00 \[ \frac {(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^2,x]

[Out]

-1/4*(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(b^(3/2)*d*Log[F]^(3/2)) + (F^(a + b*(c + d*x)^2)*(c
+ d*x))/(2*b*d*Log[F])

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fricas [A]  time = 0.66, size = 88, normalized size = 1.14 \[ \frac {\sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) + 2 \, {\left (b d^{2} x + b c d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} \log \relax (F)}{4 \, b^{2} d^{2} \log \relax (F)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(pi)*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) + 2*(b*d^2*x + b*c*d)*F^(b*d^2*x^2
+ 2*b*c*d*x + b*c^2 + a)*log(F))/(b^2*d^2*log(F)^2)

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giac [A]  time = 0.27, size = 91, normalized size = 1.18 \[ \frac {{\left (x + \frac {c}{d}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{2 \, b \log \relax (F)} + \frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{4 \, \sqrt {-b \log \relax (F)} b d \log \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x + c/d)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b*log(F)) + 1/4*sqrt(pi)*F^a*
erf(-sqrt(-b*log(F))*d*(x + c/d))/(sqrt(-b*log(F))*b*d*log(F))

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maple [B]  time = 0.07, size = 131, normalized size = 1.70 \[ \frac {x \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \ln \relax (F )}+\frac {c \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b d \ln \relax (F )}+\frac {\sqrt {\pi }\, F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{4 \sqrt {-b \ln \relax (F )}\, b d \ln \relax (F )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+1/2/d*c/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*
F^a+1/4/d/ln(F)/b*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf(1/(-b*ln(F))^(1/2)*b*c*ln(F)-(-b*ln(F))^(1/2)*d*x)

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maxima [B]  time = 3.67, size = 413, normalized size = 5.36 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \relax (F)}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d}\right )} F^{a} c}{\sqrt {b \log \relax (F)}} + \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{2}} - \frac {{\left (b d^{2} x + b c d\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right ) \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} d}{2 \, \sqrt {b \log \relax (F)}} + \frac {\sqrt {\pi } F^{b c^{2} + a} c^{2} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} d x - \frac {b c \log \relax (F)}{\sqrt {-b \log \relax (F)}}\right )}{2 \, \sqrt {-b \log \relax (F)} F^{b c^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^2/((b*log(F))^(3/
2)*d^2*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*log(F)/((b*log(F))^(3/2)
*d))*F^a*c/sqrt(b*log(F)) + 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d
^2))) - 1)*log(F)^3/((b*log(F))^(5/2)*d^3*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^((b*d^2*x + b*c*d)^
2/(b*d^2))*b^2*c*log(F)^2/((b*log(F))^(5/2)*d^2) - (b*d^2*x + b*c*d)^3*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/
(b*d^2))*log(F)^3/((b*log(F))^(5/2)*d^5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*d/sqrt(b*log(F)) + 1
/2*sqrt(pi)*F^(b*c^2 + a)*c^2*erf(sqrt(-b*log(F))*d*x - b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)
*d)

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mupad [B]  time = 3.59, size = 130, normalized size = 1.69 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x}{2\,b\,\ln \relax (F)}-\frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )}{4\,b\,\ln \relax (F)\,\sqrt {b\,d^2\,\ln \relax (F)}}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,c}{2\,b\,d\,\ln \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x)/(2*b*log(F)) - (F^a*pi^(1/2)*erfi((b*c*d*log(F) + b*d^2*x*log(F)
)/(b*d^2*log(F))^(1/2)))/(4*b*log(F)*(b*d^2*log(F))^(1/2)) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*c)/(2*
b*d*log(F))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**2,x)

[Out]

Integral(F**(a + b*(c + d*x)**2)*(c + d*x)**2, x)

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