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

Optimal. Leaf size=258 \[ -\frac {3 \sqrt {\pi } f^2 F^a (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^3 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}+\frac {3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.44, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2226, 2204, 2209, 2212} \[ -\frac {3 \sqrt {\pi } f^2 F^a (d e-c f) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d^4 \log ^{\frac {3}{2}}(F)}-\frac {f^3 F^{a+b (c+d x)^2}}{2 b^2 d^4 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^3 \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^4 \sqrt {\log (F)}}+\frac {3 f^2 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {3 f (d e-c f)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)}+\frac {f^3 (c+d x)^2 F^{a+b (c+d x)^2}}{2 b d^4 \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.26, size = 148, normalized size = 0.57 \[ \frac {F^a \left (2 f F^{b (c+d x)^2} \left (b \log (F) \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-f^2\right )+\sqrt {\pi } \sqrt {b} \sqrt {\log (F)} (d e-c f) \left (2 b \log (F) (d e-c f)^2-3 f^2\right ) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )\right )}{4 b^2 d^4 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 208, normalized size = 0.81 \[ \frac {\sqrt {\pi } {\left (3 \, d e f^{2} - 3 \, c f^{3} - 2 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \log \relax (F)\right )} \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (d f^{3} - {\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} + {\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \log \relax (F)\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{4 \, b^{2} d^{5} \log \relax (F)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(pi)*(3*d*e*f^2 - 3*c*f^3 - 2*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(F))*sqr
t(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) - 2*(d*f^3 - (b*d^3*f^3*x^2 + 3*b*d^3*e^2*f - 3*b*c*
d^2*e*f^2 + b*c^2*d*f^3 + (3*b*d^3*e*f^2 - b*c*d^2*f^3)*x)*log(F))*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(b^2
*d^5*log(F)^2)

________________________________________________________________________________________

giac [A]  time = 0.48, size = 426, normalized size = 1.65 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 3\right )}}{2 \, \sqrt {-b \log \relax (F)} d} + \frac {3 \, {\left (\frac {\sqrt {\pi } c f \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 2\right )}}{\sqrt {-b \log \relax (F)} d} + \frac {f e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F) + 2\right )}}{b d \log \relax (F)}\right )}}{2 \, d} - \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, b c^{2} f^{2} \log \relax (F) - f^{2}\right )} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right ) e^{\left (a \log \relax (F) + 1\right )}}{\sqrt {-b \log \relax (F)} b d \log \relax (F)} - \frac {2 \, {\left (d f^{2} {\left (x + \frac {c}{d}\right )} - 2 \, c f^{2}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F) + 1\right )}}{b d \log \relax (F)}\right )}}{4 \, d^{2}} + \frac {\frac {\sqrt {\pi } {\left (2 \, b c^{3} f^{3} \log \relax (F) - 3 \, c f^{3}\right )} F^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (F)} d {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {-b \log \relax (F)} b d \log \relax (F)} + \frac {2 \, {\left (b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} \log \relax (F) - 3 \, b c d f^{3} {\left (x + \frac {c}{d}\right )} \log \relax (F) + 3 \, b c^{2} f^{3} \log \relax (F) - f^{3}\right )} e^{\left (b d^{2} x^{2} \log \relax (F) + 2 \, b c d x \log \relax (F) + b c^{2} \log \relax (F) + a \log \relax (F)\right )}}{b^{2} d \log \relax (F)^{2}}}{4 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.09, size = 617, normalized size = 2.39 \[ \frac {f^{3} x^{2} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}-\frac {c \,f^{3} x \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{3} \ln \relax (F )}+\frac {3 e \,f^{2} x \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}+\frac {\sqrt {\pi }\, c^{3} f^{3} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{4}}-\frac {3 \sqrt {\pi }\, c^{2} e \,f^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{3}}+\frac {3 \sqrt {\pi }\, c \,e^{2} f \,F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d^{2}}-\frac {\sqrt {\pi }\, e^{3} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{2 \sqrt {-b \ln \relax (F )}\, d}+\frac {c^{2} f^{3} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{4} \ln \relax (F )}-\frac {3 c e \,f^{2} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{3} \ln \relax (F )}+\frac {3 e^{2} f \,F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b \,d^{2} \ln \relax (F )}-\frac {3 \sqrt {\pi }\, c \,f^{3} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{4 \sqrt {-b \ln \relax (F )}\, b \,d^{4} \ln \relax (F )}+\frac {3 \sqrt {\pi }\, e \,f^{2} F^{a} \erf \left (\frac {b c \ln \relax (F )}{\sqrt {-b \ln \relax (F )}}-\sqrt {-b \ln \relax (F )}\, d x \right )}{4 \sqrt {-b \ln \relax (F )}\, b \,d^{3} \ln \relax (F )}-\frac {f^{3} F^{a} F^{b \,c^{2}} F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 b^{2} d^{4} \ln \relax (F )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3*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^3/ln(F)/
b/d^2*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-1/2*f^3*c/d^3/ln(F)/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c
^2)*F^a+1/2*f^3*c^2/d^4/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+1/2*f^3*c^3/d^4*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)-3/4*f^3*c/d^4/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)-1/2*f^3/ln(F)^2/b^2/d^4*F^(b*d^2*x^2)*F^(2*b*c*d
*x)*F^(b*c^2)*F^a+3/2*e*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-3/2*e*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-3/2*e*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)+3/4*e*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)+3/2*e^2*f/ln(F)/b/d^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+3/2*e^2*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 = 3.28, size = 695, normalized size = 2.69 \[ -\frac {3 \, {\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^{2} f}{2 \, \sqrt {b \log \relax (F)} d} + \frac {3 \, {\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} e f^{2}}{2 \, \sqrt {b \log \relax (F)} d} - \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{3} c^{3} {\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)^{4}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{4} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}}} - \frac {3 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{3} c^{2} \log \relax (F)^{3}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{3}} - \frac {3 \, {\left (b d^{2} x + b c d\right )}^{3} b c \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)^{4}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{6} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right )^{\frac {3}{2}}} + \frac {b^{2} \Gamma \left (2, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \relax (F)}{b d^{2}}\right ) \log \relax (F)^{2}}{\left (b \log \relax (F)\right )^{\frac {7}{2}} d^{3}}\right )} F^{a} f^{3}}{2 \, \sqrt {b \log \relax (F)} d} + \frac {\sqrt {\pi } F^{b c^{2} + a} e^{3} \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)^3,x, algorithm="maxima")

[Out]

-3/2*(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^2*f/(sqrt(b*log(F))*d) + 3/2*(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*e*f^2/(s
qrt(b*log(F))*d) - 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1
)*log(F)^4/((b*log(F))^(7/2)*d^4*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*d)^2/(b*d^2)
)*b^3*c^2*log(F)^3/((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(
b*d^2))*log(F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*f^3/(sqrt(b*log(F))*d) + 1/2*sqrt(pi)*F^(b*c^2
+ a)*e^3*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 = 3.76, size = 313, normalized size = 1.21 \[ \frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,\left (-2\,b\,\ln \relax (F)\,c^3\,f^3+6\,b\,\ln \relax (F)\,c^2\,d\,e\,f^2-6\,b\,\ln \relax (F)\,c\,d^2\,e^2\,f+3\,c\,f^3+2\,b\,\ln \relax (F)\,d^3\,e^3-3\,d\,e\,f^2\right )}{4\,b\,d^3\,\ln \relax (F)\,\sqrt {b\,d^2\,\ln \relax (F)}}-F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (\frac {f^3}{2\,b^2\,d^4\,{\ln \relax (F)}^2}-\frac {3\,e^2\,f}{2\,b\,d^2\,\ln \relax (F)}-\frac {c^2\,f^3}{2\,b\,d^4\,\ln \relax (F)}+\frac {3\,c\,e\,f^2}{2\,b\,d^3\,\ln \relax (F)}\right )-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (c\,f^3-3\,d\,e\,f^2\right )}{2\,b\,d^3\,\ln \relax (F)}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^3\,x^2}{2\,b\,d^2\,\ln \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^a*pi^(1/2)*erfi((b*c*d*log(F) + b*d^2*x*log(F))/(b*d^2*log(F))^(1/2))*(3*c*f^3 - 3*d*e*f^2 - 2*b*c^3*f^3*lo
g(F) + 2*b*d^3*e^3*log(F) - 6*b*c*d^2*e^2*f*log(F) + 6*b*c^2*d*e*f^2*log(F)))/(4*b*d^3*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)*(f^3/(2*b^2*d^4*log(F)^2) - (3*e^2*f)/(2*b*d^2*log(F)) - (c^
2*f^3)/(2*b*d^4*log(F)) + (3*c*e*f^2)/(2*b*d^3*log(F))) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x*(c*f^3
- 3*d*e*f^2))/(2*b*d^3*log(F)) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^3*x^2)/(2*b*d^2*log(F))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________