3.382 \(\int F^{a+b (c+d x)^2} (e+f x)^5 \, dx\)

Optimal. Leaf size=518 \[ \frac {15 \sqrt {\pi } f^4 F^a (d e-c f) \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{8 b^{5/2} d^6 \log ^{\frac {5}{2}}(F)}-\frac {5 \sqrt {\pi } f^2 F^a (d e-c f)^3 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 b^{3/2} d^6 \log ^{\frac {3}{2}}(F)}+\frac {f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}-\frac {15 f^4 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{4 b^2 d^6 \log ^2(F)}-\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac {f^5 (c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^5 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}}+\frac {5 f^4 (c+d x)^3 (d e-c f) F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^3 (c+d x)^2 (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac {5 f^2 (c+d x) (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {f^5 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)} \]

[Out]

f^5*F^(a+b*(d*x+c)^2)/b^3/d^6/ln(F)^3-5*f^3*(-c*f+d*e)^2*F^(a+b*(d*x+c)^2)/b^2/d^6/ln(F)^2-15/4*f^4*(-c*f+d*e)
*F^(a+b*(d*x+c)^2)*(d*x+c)/b^2/d^6/ln(F)^2-f^5*F^(a+b*(d*x+c)^2)*(d*x+c)^2/b^2/d^6/ln(F)^2+5/2*f*(-c*f+d*e)^4*
F^(a+b*(d*x+c)^2)/b/d^6/ln(F)+5*f^2*(-c*f+d*e)^3*F^(a+b*(d*x+c)^2)*(d*x+c)/b/d^6/ln(F)+5*f^3*(-c*f+d*e)^2*F^(a
+b*(d*x+c)^2)*(d*x+c)^2/b/d^6/ln(F)+5/2*f^4*(-c*f+d*e)*F^(a+b*(d*x+c)^2)*(d*x+c)^3/b/d^6/ln(F)+1/2*f^5*F^(a+b*
(d*x+c)^2)*(d*x+c)^4/b/d^6/ln(F)+15/8*f^4*(-c*f+d*e)*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(5/2)/d^
6/ln(F)^(5/2)-5/2*f^2*(-c*f+d*e)^3*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(3/2)/d^6/ln(F)^(3/2)+1/2*
(-c*f+d*e)^5*F^a*erfi((d*x+c)*b^(1/2)*ln(F)^(1/2))*Pi^(1/2)/d^6/b^(1/2)/ln(F)^(1/2)

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Rubi [A]  time = 0.94, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2226, 2204, 2209, 2212} \[ -\frac {5 \sqrt {\pi } f^2 F^a (d e-c f)^3 \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 b^{3/2} d^6 \log ^{\frac {3}{2}}(F)}+\frac {15 \sqrt {\pi } f^4 F^a (d e-c f) \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{8 b^{5/2} d^6 \log ^{\frac {5}{2}}(F)}-\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac {15 f^4 (c+d x) (d e-c f) F^{a+b (c+d x)^2}}{4 b^2 d^6 \log ^2(F)}-\frac {f^5 (c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}+\frac {f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}+\frac {\sqrt {\pi } F^a (d e-c f)^5 \text {Erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}}+\frac {5 f^4 (c+d x)^3 (d e-c f) F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^3 (c+d x)^2 (d e-c f)^2 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac {5 f^2 (c+d x) (d e-c f)^3 F^{a+b (c+d x)^2}}{b d^6 \log (F)}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {f^5 (c+d x)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)*(e + f*x)^5,x]

[Out]

(f^5*F^(a + b*(c + d*x)^2))/(b^3*d^6*Log[F]^3) + (15*f^4*(d*e - c*f)*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[
Log[F]]])/(8*b^(5/2)*d^6*Log[F]^(5/2)) - (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2))/(b^2*d^6*Log[F]^2) - (15*
f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x))/(4*b^2*d^6*Log[F]^2) - (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)^2
)/(b^2*d^6*Log[F]^2) - (5*f^2*(d*e - c*f)^3*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*b^(3/2)*d^6*
Log[F]^(3/2)) + (5*f*(d*e - c*f)^4*F^(a + b*(c + d*x)^2))/(2*b*d^6*Log[F]) + (5*f^2*(d*e - c*f)^3*F^(a + b*(c
+ d*x)^2)*(c + d*x))/(b*d^6*Log[F]) + (5*f^3*(d*e - c*f)^2*F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b*d^6*Log[F]) +
(5*f^4*(d*e - c*f)*F^(a + b*(c + d*x)^2)*(c + d*x)^3)/(2*b*d^6*Log[F]) + (f^5*F^(a + b*(c + d*x)^2)*(c + d*x)
^4)/(2*b*d^6*Log[F]) + ((d*e - c*f)^5*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]])/(2*Sqrt[b]*d^6*Sqrt[L
og[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)^5 \, dx &=\int \left (\frac {(d e-c f)^5 F^{a+b (c+d x)^2}}{d^5}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2} (c+d x)}{d^5}+\frac {10 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)^2}{d^5}+\frac {10 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^3}{d^5}+\frac {5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^4}{d^5}+\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^5}{d^5}\right ) \, dx\\ &=\frac {f^5 \int F^{a+b (c+d x)^2} (c+d x)^5 \, dx}{d^5}+\frac {\left (5 f^4 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x)^4 \, dx}{d^5}+\frac {\left (10 f^3 (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{d^5}+\frac {\left (10 f^2 (d e-c f)^3\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{d^5}+\frac {\left (5 f (d e-c f)^4\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{d^5}+\frac {(d e-c f)^5 \int F^{a+b (c+d x)^2} \, dx}{d^5}\\ &=\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac {5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac {(d e-c f)^5 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}}-\frac {\left (2 f^5\right ) \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{b d^5 \log (F)}-\frac {\left (15 f^4 (d e-c f)\right ) \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx}{2 b d^5 \log (F)}-\frac {\left (10 f^3 (d e-c f)^2\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b d^5 \log (F)}-\frac {\left (5 f^2 (d e-c f)^3\right ) \int F^{a+b (c+d x)^2} \, dx}{b d^5 \log (F)}\\ &=-\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac {15 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^6 \log ^2(F)}-\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d^6 \log ^2(F)}-\frac {5 f^2 (d e-c f)^3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 b^{3/2} d^6 \log ^{\frac {3}{2}}(F)}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac {5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac {(d e-c f)^5 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}}+\frac {\left (2 f^5\right ) \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b^2 d^5 \log ^2(F)}+\frac {\left (15 f^4 (d e-c f)\right ) \int F^{a+b (c+d x)^2} \, dx}{4 b^2 d^5 \log ^2(F)}\\ &=\frac {f^5 F^{a+b (c+d x)^2}}{b^3 d^6 \log ^3(F)}+\frac {15 f^4 (d e-c f) F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{8 b^{5/2} d^6 \log ^{\frac {5}{2}}(F)}-\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2}}{b^2 d^6 \log ^2(F)}-\frac {15 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)}{4 b^2 d^6 \log ^2(F)}-\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d^6 \log ^2(F)}-\frac {5 f^2 (d e-c f)^3 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 b^{3/2} d^6 \log ^{\frac {3}{2}}(F)}+\frac {5 f (d e-c f)^4 F^{a+b (c+d x)^2}}{2 b d^6 \log (F)}+\frac {5 f^2 (d e-c f)^3 F^{a+b (c+d x)^2} (c+d x)}{b d^6 \log (F)}+\frac {5 f^3 (d e-c f)^2 F^{a+b (c+d x)^2} (c+d x)^2}{b d^6 \log (F)}+\frac {5 f^4 (d e-c f) F^{a+b (c+d x)^2} (c+d x)^3}{2 b d^6 \log (F)}+\frac {f^5 F^{a+b (c+d x)^2} (c+d x)^4}{2 b d^6 \log (F)}+\frac {(d e-c f)^5 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d^6 \sqrt {\log (F)}}\\ \end {align*}

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Mathematica [A]  time = 0.62, size = 412, normalized size = 0.80 \[ \frac {F^a \left (4 \sqrt {\pi } b^{3/2} \log ^{\frac {3}{2}}(F) (d e-c f)^5 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )+\frac {15 f^4 (c f-d e) \left (2 \sqrt {b} \sqrt {\log (F)} (c+d x) F^{b (c+d x)^2}-\sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )\right )}{\sqrt {b} \sqrt {\log (F)}}+20 \sqrt {\pi } \sqrt {b} f^2 \sqrt {\log (F)} (c f-d e)^3 \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )+20 b f^4 \log (F) (c+d x)^3 (d e-c f) F^{b (c+d x)^2}-40 f^3 (d e-c f)^2 F^{b (c+d x)^2}+40 b f^3 \log (F) (c+d x)^2 (d e-c f)^2 F^{b (c+d x)^2}+40 b f^2 \log (F) (c+d x) (d e-c f)^3 F^{b (c+d x)^2}+20 b f \log (F) (d e-c f)^4 F^{b (c+d x)^2}+4 b f^5 \log (F) (c+d x)^4 F^{b (c+d x)^2}+\frac {8 f^5 F^{b (c+d x)^2} \left (1-b \log (F) (c+d x)^2\right )}{b \log (F)}\right )}{8 b^2 d^6 \log ^2(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)*(e + f*x)^5,x]

[Out]

(F^a*(-40*f^3*(d*e - c*f)^2*F^(b*(c + d*x)^2) + (15*f^4*(-(d*e) + c*f)*(-(Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt
[Log[F]]]) + 2*Sqrt[b]*F^(b*(c + d*x)^2)*(c + d*x)*Sqrt[Log[F]]))/(Sqrt[b]*Sqrt[Log[F]]) + 20*Sqrt[b]*f^2*(-(d
*e) + c*f)^3*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]] + 20*b*f*(d*e - c*f)^4*F^(b*(c + d*x)^
2)*Log[F] + 40*b*f^2*(d*e - c*f)^3*F^(b*(c + d*x)^2)*(c + d*x)*Log[F] + 40*b*f^3*(d*e - c*f)^2*F^(b*(c + d*x)^
2)*(c + d*x)^2*Log[F] + 20*b*f^4*(d*e - c*f)*F^(b*(c + d*x)^2)*(c + d*x)^3*Log[F] + 4*b*f^5*F^(b*(c + d*x)^2)*
(c + d*x)^4*Log[F] + 4*b^(3/2)*(d*e - c*f)^5*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(3/2) + (8*f
^5*F^(b*(c + d*x)^2)*(1 - b*(c + d*x)^2*Log[F]))/(b*Log[F])))/(8*b^2*d^6*Log[F]^2)

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fricas [A]  time = 0.47, size = 531, normalized size = 1.03 \[ -\frac {\sqrt {\pi } {\left (15 \, d e f^{4} - 15 \, c f^{5} + 4 \, {\left (b^{2} d^{5} e^{5} - 5 \, b^{2} c d^{4} e^{4} f + 10 \, b^{2} c^{2} d^{3} e^{3} f^{2} - 10 \, b^{2} c^{3} d^{2} e^{2} f^{3} + 5 \, b^{2} c^{4} d e f^{4} - b^{2} c^{5} f^{5}\right )} \log \relax (F)^{2} - 20 \, {\left (b d^{3} e^{3} f^{2} - 3 \, b c d^{2} e^{2} f^{3} + 3 \, b c^{2} d e f^{4} - b c^{3} f^{5}\right )} \log \relax (F)\right )} \sqrt {-b d^{2} \log \relax (F)} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \relax (F)} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (4 \, d f^{5} + 2 \, {\left (b^{2} d^{5} f^{5} x^{4} + 5 \, b^{2} d^{5} e^{4} f - 10 \, b^{2} c d^{4} e^{3} f^{2} + 10 \, b^{2} c^{2} d^{3} e^{2} f^{3} - 5 \, b^{2} c^{3} d^{2} e f^{4} + b^{2} c^{4} d f^{5} + {\left (5 \, b^{2} d^{5} e f^{4} - b^{2} c d^{4} f^{5}\right )} x^{3} + {\left (10 \, b^{2} d^{5} e^{2} f^{3} - 5 \, b^{2} c d^{4} e f^{4} + b^{2} c^{2} d^{3} f^{5}\right )} x^{2} + {\left (10 \, b^{2} d^{5} e^{3} f^{2} - 10 \, b^{2} c d^{4} e^{2} f^{3} + 5 \, b^{2} c^{2} d^{3} e f^{4} - b^{2} c^{3} d^{2} f^{5}\right )} x\right )} \log \relax (F)^{2} - {\left (4 \, b d^{3} f^{5} x^{2} + 20 \, b d^{3} e^{2} f^{3} - 25 \, b c d^{2} e f^{4} + 9 \, b c^{2} d f^{5} + {\left (15 \, b d^{3} e f^{4} - 7 \, b c d^{2} f^{5}\right )} x\right )} \log \relax (F)\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{8 \, b^{3} d^{7} \log \relax (F)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(f*x+e)^5,x, algorithm="fricas")

[Out]

-1/8*(sqrt(pi)*(15*d*e*f^4 - 15*c*f^5 + 4*(b^2*d^5*e^5 - 5*b^2*c*d^4*e^4*f + 10*b^2*c^2*d^3*e^3*f^2 - 10*b^2*c
^3*d^2*e^2*f^3 + 5*b^2*c^4*d*e*f^4 - b^2*c^5*f^5)*log(F)^2 - 20*(b*d^3*e^3*f^2 - 3*b*c*d^2*e^2*f^3 + 3*b*c^2*d
*e*f^4 - b*c^3*f^5)*log(F))*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) - 2*(4*d*f^5 + 2*(b^2
*d^5*f^5*x^4 + 5*b^2*d^5*e^4*f - 10*b^2*c*d^4*e^3*f^2 + 10*b^2*c^2*d^3*e^2*f^3 - 5*b^2*c^3*d^2*e*f^4 + b^2*c^4
*d*f^5 + (5*b^2*d^5*e*f^4 - b^2*c*d^4*f^5)*x^3 + (10*b^2*d^5*e^2*f^3 - 5*b^2*c*d^4*e*f^4 + b^2*c^2*d^3*f^5)*x^
2 + (10*b^2*d^5*e^3*f^2 - 10*b^2*c*d^4*e^2*f^3 + 5*b^2*c^2*d^3*e*f^4 - b^2*c^3*d^2*f^5)*x)*log(F)^2 - (4*b*d^3
*f^5*x^2 + 20*b*d^3*e^2*f^3 - 25*b*c*d^2*e*f^4 + 9*b*c^2*d*f^5 + (15*b*d^3*e*f^4 - 7*b*c*d^2*f^5)*x)*log(F))*F
^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(b^3*d^7*log(F)^3)

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giac [A]  time = 0.50, size = 942, normalized size = 1.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(f*x+e)^5,x, algorithm="giac")

[Out]

-1/2*sqrt(pi)*erf(-sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 5)/(sqrt(-b*log(F))*d) + 5/2*(sqrt(pi)*c*f*erf(-
sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 4)/(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) + 4)/(b*d*log(F)))/d - 5/2*(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) + 3)/(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) + 3)/(b*d*log(F)))/d^2 + 5/2*(sqrt(pi)*(2*b*c^3*f^3*log(F) - 3*c
*f^3)*erf(-sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 2)/(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) + 2)/(b^2*d*log(F)^2))/d^3 - 5/8*(sqrt(pi)*(4*b^2*c^4*f^4*log(F)^2 - 12*b*c^2*f^4*log(
F) + 3*f^4)*erf(-sqrt(-b*log(F))*d*(x + c/d))*e^(a*log(F) + 1)/(sqrt(-b*log(F))*b^2*d*log(F)^2) - 2*(2*b*d^3*f
^4*(x + c/d)^3*log(F) - 8*b*c*d^2*f^4*(x + c/d)^2*log(F) + 12*b*c^2*d*f^4*(x + c/d)*log(F) - 8*b*c^3*f^4*log(F
) - 3*d*f^4*(x + c/d) + 8*c*f^4)*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^2*d*
log(F)^2))/d^4 + 1/8*(sqrt(pi)*(4*b^2*c^5*f^5*log(F)^2 - 20*b*c^3*f^5*log(F) + 15*c*f^5)*F^a*erf(-sqrt(-b*log(
F))*d*(x + c/d))/(sqrt(-b*log(F))*b^2*d*log(F)^2) + 2*(2*b^2*d^4*f^5*(x + c/d)^4*log(F)^2 - 10*b^2*c*d^3*f^5*(
x + c/d)^3*log(F)^2 + 20*b^2*c^2*d^2*f^5*(x + c/d)^2*log(F)^2 - 20*b^2*c^3*d*f^5*(x + c/d)*log(F)^2 + 10*b^2*c
^4*f^5*log(F)^2 - 4*b*d^2*f^5*(x + c/d)^2*log(F) + 15*b*c*d*f^5*(x + c/d)*log(F) - 20*b*c^2*f^5*log(F) + 4*f^5
)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b^3*d*log(F)^3))/d^5

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maple [B]  time = 0.12, size = 1657, normalized size = 3.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+(d*x+c)^2*b)*(f*x+e)^5,x)

[Out]

-5/2*e*f^4*c/d^3/ln(F)/b*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+5/2*e*f^4*c^2/d^4/ln(F)/b*x*F^(b*d^2*x^
2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-5*e^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-5*e^2*f^3/l
n(F)^2/b^2/d^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+5/2*e^3*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)+5/2*e^4*f/ln(F)/b/d^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b
*c^2)*F^a+1/2*f^5/ln(F)/b/d^2*x^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+1/2*f^5*c^4/d^6/ln(F)/b*F^(b*d^2*x
^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-5/2*f^5*c^3/d^6/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)-9/4*f^5*c^2/d^6/ln(F)^2/b^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+15/8*f^5*
c/d^6/ln(F)^2/b^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)-f^5/ln(
F)^2/b^2/d^4*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-15/8*e*f^4/ln(F)^2/b^2/d^5*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^5*c/d^3/ln(F)/b*x^3*F^(b*d^2*x^2)*F^(2*b*c*
d*x)*F^(b*c^2)*F^a+1/2*f^5*c^2/d^4/ln(F)/b*x^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-1/2*f^5*c^3/d^5/ln(F)
/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+7/4*f^5*c/d^5/ln(F)^2/b^2*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^
2)*F^a+5/2*e*f^4/ln(F)/b/d^2*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-5/2*e*f^4*c^3/d^5/ln(F)/b*F^(b*d^2*
x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+15/2*e*f^4*c^2/d^5/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)+25/4*e*f^4*c/d^5/ln(F)^2/b^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a-15/4
*e*f^4/ln(F)^2/b^2/d^4*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+5*e^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+5*e^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-15/2*e^2*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)-5*e^3*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+5*e^3*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*e^5*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)+f^5/ln(F
)^3/b^3/d^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)*F^(b*c^2)*F^a+1/2*f^5*c^5/d^6*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)-5*e^3*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)+5/2*e^4*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)-5/2*e*f^4*c^4/d^5*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)+5*e^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)

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maxima [B]  time = 5.38, size = 1456, normalized size = 2.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(f*x+e)^5,x, algorithm="maxima")

[Out]

-5/2*(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^4*f/(sqrt(b*log(F))*d) + 5*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf(sqrt(-(b*d^2*x + b*c*d)^2*l
og(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^3*f^2/(s
qrt(b*log(F))*d) - 5*(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*e^2*f^3/(sqrt(b*log(F))*d) + 5/2*(sqrt(pi)*(b*d^2
*x + b*c*d)*b^4*c^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^5/((b*log(F))^(9/2)*d^5*sqrt(-
(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 4*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^4*c^3*log(F)^4/((b*log(F))^(9/2)*d^
4) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*log(F))^(9/2)*
d^7*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 4*b^3*c*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F
)^3/((b*log(F))^(9/2)*d^4) - (b*d^2*x + b*c*d)^5*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^5/((b*
log(F))^(9/2)*d^9*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*e*f^4/(sqrt(b*log(F))*d) - 1/2*(sqrt(pi)*(
b*d^2*x + b*c*d)*b^5*c^5*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^6/((b*log(F))^(11/2)*d^6*
sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 5*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^5*c^4*log(F)^5/((b*log(F))^(1
1/2)*d^5) - 10*(b*d^2*x + b*c*d)^3*b^3*c^3*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F)
)^(11/2)*d^8*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 10*b^4*c^2*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b
*d^2))*log(F)^4/((b*log(F))^(11/2)*d^5) - b^3*gamma(3, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F
))^(11/2)*d^5) - 5*(b*d^2*x + b*c*d)^5*b*c*gamma(5/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*log(F)
)^(11/2)*d^10*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*f^5/(sqrt(b*log(F))*d) + 1/2*sqrt(pi)*F^(b*c^2
+ a)*e^5*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 = 4.11, size = 716, normalized size = 1.38 \[ \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (f^5+\frac {{\ln \relax (F)}^2\,\left (2\,F^a\,b^2\,c^4\,f^5+10\,F^a\,b^2\,d^4\,e^4\,f+20\,F^a\,b^2\,c^2\,d^2\,e^2\,f^3-10\,F^a\,b^2\,c^3\,d\,e\,f^4-20\,F^a\,b^2\,c\,d^3\,e^3\,f^2\right )}{4\,F^a}-\frac {\ln \relax (F)\,\left (9\,F^a\,b\,c^2\,f^5+20\,F^a\,b\,d^2\,e^2\,f^3-25\,F^a\,b\,c\,d\,e\,f^4\right )}{4\,F^a}\right )}{b^3\,d^6\,{\ln \relax (F)}^3}-\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (F)\,d^2+b\,c\,\ln \relax (F)\,d}{\sqrt {b\,d^2\,\ln \relax (F)}}\right )\,\left (\frac {\frac {F^a\,\sqrt {\pi }\,\left (15\,c\,f^5-15\,d\,e\,f^4\right )}{8\,\sqrt {b\,d^2\,\ln \relax (F)}}-\frac {F^a\,\sqrt {\pi }\,\ln \relax (F)\,\left (20\,b\,c^3\,f^5-60\,b\,c^2\,d\,e\,f^4+60\,b\,c\,d^2\,e^2\,f^3-20\,b\,d^3\,e^3\,f^2\right )}{8\,\sqrt {b\,d^2\,\ln \relax (F)}}}{b^2\,d^5\,{\ln \relax (F)}^2}+\frac {F^a\,\sqrt {\pi }\,\left (4\,b^2\,c^5\,f^5-20\,b^2\,c^4\,d\,e\,f^4+40\,b^2\,c^3\,d^2\,e^2\,f^3-40\,b^2\,c^2\,d^3\,e^3\,f^2+20\,b^2\,c\,d^4\,e^4\,f-4\,b^2\,d^5\,e^5\right )}{8\,b^2\,d^5\,\sqrt {b\,d^2\,\ln \relax (F)}}\right )-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x\,\left (\ln \relax (F)\,\left (\frac {b\,c^3\,f^5}{2}-\frac {5\,b\,c^2\,d\,e\,f^4}{2}+5\,b\,c\,d^2\,e^2\,f^3-5\,b\,d^3\,e^3\,f^2\right )-\frac {7\,c\,f^5}{4}+\frac {15\,d\,e\,f^4}{4}\right )}{b^2\,d^5\,{\ln \relax (F)}^2}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^5\,x^4}{2\,b\,d^2\,\ln \relax (F)}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x^2\,\left (f^5-\frac {b\,f^3\,\left (\ln \relax (F)\,c^2\,f^2-5\,\ln \relax (F)\,c\,d\,e\,f+10\,\ln \relax (F)\,d^2\,e^2\right )}{2}\right )}{b^2\,d^4\,{\ln \relax (F)}^2}-\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,f^4\,x^3\,\left (c\,f-5\,d\,e\right )}{2\,b\,d^3\,\ln \relax (F)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^2)*(e + f*x)^5,x)

[Out]

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(f^5 + (log(F)^2*(2*F^a*b^2*c^4*f^5 + 10*F^a*b^2*d^4*e^4*f + 20*F^a
*b^2*c^2*d^2*e^2*f^3 - 10*F^a*b^2*c^3*d*e*f^4 - 20*F^a*b^2*c*d^3*e^3*f^2))/(4*F^a) - (log(F)*(9*F^a*b*c^2*f^5
+ 20*F^a*b*d^2*e^2*f^3 - 25*F^a*b*c*d*e*f^4))/(4*F^a)))/(b^3*d^6*log(F)^3) - erfi((b*c*d*log(F) + b*d^2*x*log(
F))/(b*d^2*log(F))^(1/2))*(((F^a*pi^(1/2)*(15*c*f^5 - 15*d*e*f^4))/(8*(b*d^2*log(F))^(1/2)) - (F^a*pi^(1/2)*lo
g(F)*(20*b*c^3*f^5 - 20*b*d^3*e^3*f^2 - 60*b*c^2*d*e*f^4 + 60*b*c*d^2*e^2*f^3))/(8*(b*d^2*log(F))^(1/2)))/(b^2
*d^5*log(F)^2) + (F^a*pi^(1/2)*(4*b^2*c^5*f^5 - 4*b^2*d^5*e^5 - 40*b^2*c^2*d^3*e^3*f^2 + 40*b^2*c^3*d^2*e^2*f^
3 + 20*b^2*c*d^4*e^4*f - 20*b^2*c^4*d*e*f^4))/(8*b^2*d^5*(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)*x*(log(F)*((b*c^3*f^5)/2 - 5*b*d^3*e^3*f^2 - (5*b*c^2*d*e*f^4)/2 + 5*b*c*d^2*e^2*f^3) - (7*c*f^
5)/4 + (15*d*e*f^4)/4))/(b^2*d^5*log(F)^2) + (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^5*x^4)/(2*b*d^2*log(
F)) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*x^2*(f^5 - (b*f^3*(c^2*f^2*log(F) + 10*d^2*e^2*log(F) - 5*c*d
*e*f*log(F)))/2))/(b^2*d^4*log(F)^2) - (F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*f^4*x^3*(c*f - 5*d*e))/(2*b*
d^3*log(F))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + b \left (c + d x\right )^{2}} \left (e + f x\right )^{5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**2)*(f*x+e)**5,x)

[Out]

Integral(F**(a + b*(c + d*x)**2)*(e + f*x)**5, x)

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