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]

-1/2*e^3*f^(c*x^2+b*x+a)/c^2/ln(f)^2+1/8*e*(-b*e+2*c*d)^2*f^(c*x^2+b*x+a)/c^3/ln(f)+1/4*e*(-b*e+2*c*d)*f^(c*x^
2+b*x+a)*(e*x+d)/c^2/ln(f)+1/2*e*f^(c*x^2+b*x+a)*(e*x+d)^2/c/ln(f)-3/8*e^2*(-b*e+2*c*d)*f^(a-1/4/c*b^2)*erfi(1
/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(5/2)/ln(f)^(3/2)+1/16*(-b*e+2*c*d)^3*f^(a-1/4/c*b^2)*erfi(1/2*(2
*c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(7/2)/ln(f)^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.29, 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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fricas [A]  time = 0.42, size = 204, normalized size = 0.77 \[ -\frac {2 \, {\left (4 \, c^{2} e^{3} - {\left (4 \, c^{3} e^{3} x^{2} + 12 \, c^{3} d^{2} e - 6 \, b c^{2} d e^{2} + b^{2} c e^{3} + 2 \, {\left (6 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x\right )} \log \relax (f)\right )} f^{c x^{2} + b x + a} - \frac {\sqrt {\pi } {\left (12 \, c^{2} d e^{2} - 6 \, b c e^{3} - {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \relax (f)\right )} \sqrt {-c \log \relax (f)} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \relax (f)}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{16 \, c^{4} \log \relax (f)^{2}} \]

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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giac [A]  time = 0.54, size = 401, normalized size = 1.51 \[ -\frac {\sqrt {\pi } d^{3} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f)}{4 \, c}\right )}}{2 \, \sqrt {-c \log \relax (f)}} + \frac {3 \, {\left (\frac {\sqrt {\pi } b d^{2} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 4 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)}} + \frac {2 \, d^{2} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 1\right )}}{\log \relax (f)}\right )}}{4 \, c} - \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (b^{2} d \log \relax (f) - 2 \, c d\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 8 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)} \log \relax (f)} - \frac {2 \, {\left (c d {\left (2 \, x + \frac {b}{c}\right )} - 2 \, b d\right )} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 2\right )}}{\log \relax (f)}\right )}}{8 \, c^{2}} + \frac {\frac {\sqrt {\pi } {\left (b^{3} \log \relax (f) - 6 \, b c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \relax (f)} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \relax (f) - 4 \, a c \log \relax (f) - 12 \, c}{4 \, c}\right )}}{\sqrt {-c \log \relax (f)} \log \relax (f)} + \frac {2 \, {\left (c^{2} {\left (2 \, x + \frac {b}{c}\right )}^{2} \log \relax (f) - 3 \, b c {\left (2 \, x + \frac {b}{c}\right )} \log \relax (f) + 3 \, b^{2} \log \relax (f) - 4 \, c\right )} e^{\left (c x^{2} \log \relax (f) + b x \log \relax (f) + a \log \relax (f) + 3\right )}}{\log \relax (f)^{2}}}{16 \, c^{3}} \]

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

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maple [B]  time = 0.08, size = 550, normalized size = 2.07 \[ \frac {\sqrt {\pi }\, b^{3} e^{3} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{16 \sqrt {-c \ln \relax (f )}\, c^{3}}-\frac {3 \sqrt {\pi }\, b^{2} d \,e^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{8 \sqrt {-c \ln \relax (f )}\, c^{2}}+\frac {3 \sqrt {\pi }\, b \,d^{2} e \,f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}\, c}+\frac {e^{3} x^{2} f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}-\frac {\sqrt {\pi }\, d^{3} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{2 \sqrt {-c \ln \relax (f )}}-\frac {b \,e^{3} x \,f^{a} f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \relax (f )}+\frac {3 d \,e^{2} x \,f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}+\frac {b^{2} e^{3} f^{a} f^{b x} f^{c \,x^{2}}}{8 c^{3} \ln \relax (f )}-\frac {3 b d \,e^{2} f^{a} f^{b x} f^{c \,x^{2}}}{4 c^{2} \ln \relax (f )}-\frac {3 \sqrt {\pi }\, b \,e^{3} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{8 \sqrt {-c \ln \relax (f )}\, c^{2} \ln \relax (f )}+\frac {3 d^{2} e \,f^{a} f^{b x} f^{c \,x^{2}}}{2 c \ln \relax (f )}+\frac {3 \sqrt {\pi }\, d \,e^{2} f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (\frac {b \ln \relax (f )}{2 \sqrt {-c \ln \relax (f )}}-\sqrt {-c \ln \relax (f )}\, x \right )}{4 \sqrt {-c \ln \relax (f )}\, c \ln \relax (f )}-\frac {e^{3} f^{a} f^{b x} f^{c \,x^{2}}}{2 c^{2} \ln \relax (f )^{2}} \]

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

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maxima [B]  time = 2.71, size = 539, normalized size = 2.03 \[ -\frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)}{\left (c \log \relax (f)\right )^{\frac {3}{2}}}\right )} d^{2} e f^{a - \frac {b^{2}}{4 \, c}}}{4 \, \sqrt {c \log \relax (f)}} + \frac {3 \, {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {5}{2}}}\right )} d e^{2} f^{a - \frac {b^{2}}{4 \, c}}}{8 \, \sqrt {c \log \relax (f)}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{3} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}}\right ) - 1\right )} \log \relax (f)^{4}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}} \left (c \log \relax (f)\right )^{\frac {7}{2}}} - \frac {12 \, {\left (2 \, c x + b\right )}^{3} b \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{4}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{c}\right )^{\frac {3}{2}} \left (c \log \relax (f)\right )^{\frac {7}{2}}} - \frac {6 \, b^{2} c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \relax (f)^{3}}{\left (c \log \relax (f)\right )^{\frac {7}{2}}} + \frac {8 \, c^{2} \Gamma \left (2, -\frac {{\left (2 \, c x + b\right )}^{2} \log \relax (f)}{4 \, c}\right ) \log \relax (f)^{2}}{\left (c \log \relax (f)\right )^{\frac {7}{2}}}\right )} e^{3} f^{a - \frac {b^{2}}{4 \, c}}}{16 \, \sqrt {c \log \relax (f)}} + \frac {\sqrt {\pi } d^{3} f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f)} x - \frac {b \log \relax (f)}{2 \, \sqrt {-c \log \relax (f)}}\right )}{2 \, \sqrt {-c \log \relax (f)} f^{\frac {b^{2}}{4 \, c}}} \]

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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mupad [B]  time = 3.89, size = 251, normalized size = 0.94 \[ \frac {e^3\,f^a\,f^{c\,x^2}\,f^{b\,x}\,x^2}{2\,c\,\ln \relax (f)}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \relax (f)}{2}+c\,x\,\ln \relax (f)}{\sqrt {c\,\ln \relax (f)}}\right )\,\left (\frac {\ln \relax (f)\,b^3\,e^3}{16}-\frac {3\,\ln \relax (f)\,b^2\,c\,d\,e^2}{8}+\frac {3\,\ln \relax (f)\,b\,c^2\,d^2\,e}{4}-\frac {3\,b\,c\,e^3}{8}-\frac {\ln \relax (f)\,c^3\,d^3}{2}+\frac {3\,c^2\,d\,e^2}{4}\right )}{c^3\,\ln \relax (f)\,\sqrt {c\,\ln \relax (f)}}-\frac {f^a\,f^{c\,x^2}\,f^{b\,x}\,x\,\left (b\,e^3-6\,c\,d\,e^2\right )}{4\,c^2\,\ln \relax (f)}-f^a\,f^{c\,x^2}\,f^{b\,x}\,\left (\frac {e^3}{2\,c^2\,{\ln \relax (f)}^2}-\frac {3\,d^2\,e}{2\,c\,\ln \relax (f)}-\frac {b^2\,e^3}{8\,c^3\,\ln \relax (f)}+\frac {3\,b\,d\,e^2}{4\,c^2\,\ln \relax (f)}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^3*f^a*f^(c*x^2)*f^(b*x)*x^2)/(2*c*log(f)) - (f^(a - b^2/(4*c))*pi^(1/2)*erfi(((b*log(f))/2 + c*x*log(f))/(c
*log(f))^(1/2))*((3*c^2*d*e^2)/4 + (b^3*e^3*log(f))/16 - (c^3*d^3*log(f))/2 - (3*b*c*e^3)/8 + (3*b*c^2*d^2*e*l
og(f))/4 - (3*b^2*c*d*e^2*log(f))/8))/(c^3*log(f)*(c*log(f))^(1/2)) - (f^a*f^(c*x^2)*f^(b*x)*x*(b*e^3 - 6*c*d*
e^2))/(4*c^2*log(f)) - f^a*f^(c*x^2)*f^(b*x)*(e^3/(2*c^2*log(f)^2) - (3*d^2*e)/(2*c*log(f)) - (b^2*e^3)/(8*c^3
*log(f)) + (3*b*d*e^2)/(4*c^2*log(f)))

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

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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