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

Optimal. Leaf size=170 \[ -\frac {\sqrt {\pi } f^2 F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^3 \log ^{\frac {3}{2}}(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^2 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^3 \sqrt {\log (F)}}+\frac {f (d e-c f) F^{a+b (c+d x)^2}}{b d^3 \log (F)}+\frac {f^2 (c+d x) F^{a+b (c+d x)^2}}{2 b d^3 \log (F)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.31, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2226, 2204, 2209, 2212} \[ -\frac {\sqrt {\pi } f^2 F^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^3 \log ^{\frac {3}{2}}(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^2 \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^3 \sqrt {\log (F)}}+\frac {f (d e-c f) F^{a+b (c+d x)^2}}{b d^3 \log (F)}+\frac {f^2 (c+d x) F^{a+b (c+d x)^2}}{2 b d^3 \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2209

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

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])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 105, normalized size = 0.62 \[ \frac {F^a \left (\sqrt {\pi } \left (2 b \log (F) (d e-c f)^2-f^2\right ) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )+2 \sqrt {b} f \sqrt {\log (F)} F^{b (c+d x)^2} (-c f+2 d e+d f x)\right )}{4 b^{3/2} d^3 \log ^{\frac {3}{2}}(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 135, normalized size = 0.79 \[ \frac {\sqrt {\pi } \sqrt {-b d^{2} \log \relax (F)} {\left (f^{2} - 2 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \relax (F)\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) + 2 \, {\left (b d^{2} f^{2} x + 2 \, b d^{2} e f - b c d f^{2}\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} \log \relax (F)}{4 \, b^{2} d^{4} \log \relax (F)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.54, size = 258, normalized size = 1.52 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 2\right )}}{2 \, \sqrt {-b \log \relax (F)} d} + \frac {\frac {\sqrt {\pi } c f \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 1\right )}}{\sqrt {-b \log \relax (F)} d} + \frac {f e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F) + 1\right )}}{b d \log \relax (F)}}{d} - \frac {\frac {\sqrt {\pi } {\left (2 \, b c^{2} f^{2} \log \relax (F) - f^{2}\right )} F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {-b \log \relax (F)} b d \log \relax (F)} - \frac {2 \, {\left (d f^{2} {\left (x + \frac {c}{d}\right )} - 2 \, c f^{2}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{b d \log \relax (F)}}{4 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.08, size = 324, normalized size = 1.91 \[ \frac {f^{2} x \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}-\frac {\sqrt {\pi }\, c^{2} f^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{3}}+\frac {\sqrt {\pi }\, c e f \,F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{\sqrt {-b \ln \relax (F )}\, d^{2}}-\frac {\sqrt {\pi }\, e^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d}-\frac {c \,f^{2} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{3} \ln \relax (F )}+\frac {e f \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{b \,d^{2} \ln \relax (F )}+\frac {\sqrt {\pi }\, f^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{4 \sqrt {-b \ln \relax (F )}\, b \,d^{3} \ln \relax (F )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 2.44, size = 422, normalized size = 2.48 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \relax (F)}{\left (b \log \relax (F)\right )^{\frac {3}{2}} d}\right )} F^{a} e f}{\sqrt {b \log \relax (F)} d} + \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}\right ) - 1\right )} \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{2}} - \frac {{\left (b d^{2} x + b c d\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right ) \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} f^{2}}{2 \, \sqrt {b \log \relax (F)} d} + \frac {\sqrt {\pi } F^{b c^{2} + a} e^{2} \operatorname {erf}\left (\sqrt {-b \log \relax (F)} d x - \frac {b c \log \relax (F)}{\sqrt {-b \log \relax (F)}}\right )}{2 \, \sqrt {-b \log \relax (F)} F^{b c^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(f*x+e)^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*e*f/(sqrt(b*log(F))*d) + 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*l
og(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*f^2/(sqrt(b*lo
g(F))*d) + 1/2*sqrt(pi)*F^(b*c^2 + a)*e^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)

________________________________________________________________________________________

mupad [B]  time = 3.82, size = 194, normalized size = 1.14 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^2\,x}{2\,b\,d^2\,\ln \relax (F)}-\frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,\left (-2\,b\,\ln \relax (F)\,c^2\,f^2+4\,b\,\ln \relax (F)\,c\,d\,e\,f-2\,b\,\ln \relax (F)\,d^2\,e^2+f^2\right )}{4\,b\,d^2\,\ln \relax (F)\,\sqrt {b\,d^2\,\ln \relax (F)}}-F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {c\,f^2}{2\,b\,d^3\,\ln \relax (F)}-\frac {e\,f}{b\,d^2\,\ln \relax (F)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^2*x)/(2*b*d^2*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))*(f^2 - 2*b*c^2*f^2*log(F) - 2*b*d^2*e^2*log(F) + 4*b*c*d*e*f*log(F)))/(4*b*d^2
*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*f^2)/(2*b*d^3*log(F)) - (e*f)/(b
*d^2*log(F)))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________