3.258 \(\int F^{a+b (c+d x)^2} (c+d x)^5 \, dx\)
Optimal. Leaf size=91 \[ -\frac{(c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2}}{b^3 d \log ^3(F)}+\frac{(c+d x)^4 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]
[Out]
F^(a + b*(c + d*x)^2)/(b^3*d*Log[F]^3) - (F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b^2*d*Log[F]^2) + (F^(a + b*(c +
d*x)^2)*(c + d*x)^4)/(2*b*d*Log[F])
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Rubi [A] time = 0.175959, antiderivative size = 91, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used =
{2212, 2209} \[ -\frac{(c+d x)^2 F^{a+b (c+d x)^2}}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2}}{b^3 d \log ^3(F)}+\frac{(c+d x)^4 F^{a+b (c+d x)^2}}{2 b d \log (F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b*(c + d*x)^2)*(c + d*x)^5,x]
[Out]
F^(a + b*(c + d*x)^2)/(b^3*d*Log[F]^3) - (F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(b^2*d*Log[F]^2) + (F^(a + b*(c +
d*x)^2)*(c + d*x)^4)/(2*b*d*Log[F])
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 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]
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^2} (c+d x)^5 \, dx &=\frac{F^{a+b (c+d x)^2} (c+d x)^4}{2 b d \log (F)}-\frac{2 \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx}{b \log (F)}\\ &=-\frac{F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2} (c+d x)^4}{2 b d \log (F)}+\frac{2 \int F^{a+b (c+d x)^2} (c+d x) \, dx}{b^2 \log ^2(F)}\\ &=\frac{F^{a+b (c+d x)^2}}{b^3 d \log ^3(F)}-\frac{F^{a+b (c+d x)^2} (c+d x)^2}{b^2 d \log ^2(F)}+\frac{F^{a+b (c+d x)^2} (c+d x)^4}{2 b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0328125, size = 56, normalized size = 0.62 \[ \frac{F^{a+b (c+d x)^2} \left (b^2 \log ^2(F) (c+d x)^4-2 b \log (F) (c+d x)^2+2\right )}{2 b^3 d \log ^3(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^5,x]
[Out]
(F^(a + b*(c + d*x)^2)*(2 - 2*b*(c + d*x)^2*Log[F] + b^2*(c + d*x)^4*Log[F]^2))/(2*b^3*d*Log[F]^3)
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Maple [A] time = 0.005, size = 138, normalized size = 1.5 \begin{align*}{\frac{ \left ({d}^{4}{x}^{4}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+4\,{d}^{3}c{x}^{3}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+6\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{d}^{2}{x}^{2}+4\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{3}dx+ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{4}-2\,\ln \left ( F \right ) b{d}^{2}{x}^{2}-4\,\ln \left ( F \right ) bcdx-2\,\ln \left ( F \right ) b{c}^{2}+2 \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+{c}^{2}b+a}}{2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a+b*(d*x+c)^2)*(d*x+c)^5,x)
[Out]
1/2*(d^4*x^4*b^2*ln(F)^2+4*d^3*c*x^3*b^2*ln(F)^2+6*ln(F)^2*b^2*c^2*d^2*x^2+4*ln(F)^2*b^2*c^3*d*x+ln(F)^2*b^2*c
^4-2*ln(F)*b*d^2*x^2-4*ln(F)*b*c*d*x-2*ln(F)*b*c^2+2)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)/ln(F)^3/b^3/d
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Maxima [C] time = 2.15168, size = 2030, normalized size = 22.31 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^5,x, algorithm="maxima")
[Out]
-5/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*d*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^2/((b*d^2*l
og(F))^(3/2)*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*d^2*log(F)/(b*d^2*
log(F))^(3/2))*F^a*c^4*d/sqrt(b*d^2*log(F)) + 5*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*d^2*(erf(sqrt(-(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*d^2*log(F))^(5/2)*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*d^3*log(F)^2/(b*d^2*log(F))^(5/2) - (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*d^2*log(F))^(5/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^
a*c^3*d^2/sqrt(b*d^2*log(F)) - 5*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*d^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)
/(b*d^2))) - 1)*log(F)^4/((b*d^2*log(F))^(7/2)*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*d^4*log(F)^3/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x + b*c*d)^3*b*c*d*gamma(3/2, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*d^2*log(F))^(7/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*d^
4*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^2/(b*d^2*log(F))^(7/2))*F^a*c^2*d^3/sqrt(b*d^2*log(F))
+ 5/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^4*c^4*d^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^5/((
b*d^2*log(F))^(9/2)*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*d^5
*log(F)^4/(b*d^2*log(F))^(9/2) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*d^2*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d
^2))*log(F)^5/((b*d^2*log(F))^(9/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 4*b^3*c*d^5*gamma(2, -(b*d^
2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/(b*d^2*log(F))^(9/2) - (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*d^2*log(F))^(9/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F^a*c*d^4/sq
rt(b*d^2*log(F)) - 1/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^5*c^5*d^5*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2)))
- 1)*log(F)^6/((b*d^2*log(F))^(11/2)*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*d^6*log(F)^5/(b*d^2*log(F))^(11/2) - 10*(b*d^2*x + b*c*d)^3*b^3*c^3*d^3*gamma(3/2, -(b*d^2*x +
b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*d^2*log(F))^(11/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)) + 10*b^4
*c^2*d^6*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/(b*d^2*log(F))^(11/2) - b^3*d^6*gamma(3, -(b*d
^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/(b*d^2*log(F))^(11/2) - 5*(b*d^2*x + b*c*d)^5*b*c*d*gamma(5/2, -(b*d^
2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^6/((b*d^2*log(F))^(11/2)*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(5/2)))*F
^a*d^5/sqrt(b*d^2*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^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)
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Fricas [A] time = 1.54344, size = 265, normalized size = 2.91 \begin{align*} \frac{{\left ({\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} - 2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 2\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{3} d \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^5,x, algorithm="fricas")
[Out]
1/2*((b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(F)^2 - 2*(b*d^2*x^2 + 2
*b*c*d*x + b*c^2)*log(F) + 2)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)/(b^3*d*log(F)^3)
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Sympy [A] time = 0.212551, size = 214, normalized size = 2.35 \begin{align*} \begin{cases} \frac{F^{a + b \left (c + d x\right )^{2}} \left (b^{2} c^{4} \log{\left (F \right )}^{2} + 4 b^{2} c^{3} d x \log{\left (F \right )}^{2} + 6 b^{2} c^{2} d^{2} x^{2} \log{\left (F \right )}^{2} + 4 b^{2} c d^{3} x^{3} \log{\left (F \right )}^{2} + b^{2} d^{4} x^{4} \log{\left (F \right )}^{2} - 2 b c^{2} \log{\left (F \right )} - 4 b c d x \log{\left (F \right )} - 2 b d^{2} x^{2} \log{\left (F \right )} + 2\right )}{2 b^{3} d \log{\left (F \right )}^{3}} & \text{for}\: 2 b^{3} d \log{\left (F \right )}^{3} \neq 0 \\c^{5} x + \frac{5 c^{4} d x^{2}}{2} + \frac{10 c^{3} d^{2} x^{3}}{3} + \frac{5 c^{2} d^{3} x^{4}}{2} + c d^{4} x^{5} + \frac{d^{5} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**5,x)
[Out]
Piecewise((F**(a + b*(c + d*x)**2)*(b**2*c**4*log(F)**2 + 4*b**2*c**3*d*x*log(F)**2 + 6*b**2*c**2*d**2*x**2*lo
g(F)**2 + 4*b**2*c*d**3*x**3*log(F)**2 + b**2*d**4*x**4*log(F)**2 - 2*b*c**2*log(F) - 4*b*c*d*x*log(F) - 2*b*d
**2*x**2*log(F) + 2)/(2*b**3*d*log(F)**3), Ne(2*b**3*d*log(F)**3, 0)), (c**5*x + 5*c**4*d*x**2/2 + 10*c**3*d**
2*x**3/3 + 5*c**2*d**3*x**4/2 + c*d**4*x**5 + d**5*x**6/6, True))
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Giac [A] time = 1.27021, size = 111, normalized size = 1.22 \begin{align*} \frac{{\left (b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{4} \log \left (F\right )^{2} - 2 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{2} \log \left (F\right ) + 2\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{3} d \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^5,x, algorithm="giac")
[Out]
1/2*(b^2*d^4*(x + c/d)^4*log(F)^2 - 2*b*d^2*(x + c/d)^2*log(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^3*d*log(F)^3)