3.3.59 \(\int F^{a+b (c+d x)^2} (c+d x)^3 \, dx\) [259]

Optimal. Leaf size=62 \[ -\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^2}{2 b d \log (F)} \]

[Out]

-1/2*F^(a+b*(d*x+c)^2)/b^2/d/ln(F)^2+1/2*F^(a+b*(d*x+c)^2)*(d*x+c)^2/b/d/ln(F)

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Rubi [A]
time = 0.07, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2240} \begin {gather*} \frac {(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*F^(a + b*(c + d*x)^2)/(b^2*d*Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(2*b*d*Log[F])

Rule 2240

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 2243

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])

Rubi steps

\begin {align*} \int F^{a+b (c+d x)^2} (c+d x)^3 \, dx &=\frac {F^{a+b (c+d x)^2} (c+d x)^2}{2 b d \log (F)}-\frac {\int F^{a+b (c+d x)^2} (c+d x) \, dx}{b \log (F)}\\ &=-\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^2}{2 b d \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 40, normalized size = 0.65 \begin {gather*} \frac {F^{a+b (c+d x)^2} \left (-1+b (c+d x)^2 \log (F)\right )}{2 b^2 d \log ^2(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(a + b*(c + d*x)^2)*(-1 + b*(c + d*x)^2*Log[F]))/(2*b^2*d*Log[F]^2)

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Maple [A]
time = 0.08, size = 63, normalized size = 1.02

method	result	size
gosper	\(\frac {\left (\ln \left (F \right ) b \,d^{2} x^{2}+2 \ln \left (F \right ) b c d x +\ln \left (F \right ) b \,c^{2}-1\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 \ln \left (F \right )^{2} b^{2} d}\)	\(63\)
risch	\(\frac {\left (\ln \left (F \right ) b \,d^{2} x^{2}+2 \ln \left (F \right ) b c d x +\ln \left (F \right ) b \,c^{2}-1\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 \ln \left (F \right )^{2} b^{2} d}\)	\(63\)
norman	\(\frac {c x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {\left (\ln \left (F \right ) b \,c^{2}-1\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} b^{2} d}+\frac {d \,x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^2)*(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(ln(F)*b*d^2*x^2+2*ln(F)*b*c*d*x+ln(F)*b*c^2-1)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)/ln(F)^2/b^2/d

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.52, size = 683, normalized size = 11.02 \begin {gather*} -\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 \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \left (F\right )}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d}\right )} F^{a} c^{2}}{2 \, \sqrt {b \log \left (F\right )}} + \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 \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \left (F\right )^{2}}{\left (b \log \left (F\right )\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 \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} c d}{2 \, \sqrt {b \log \left (F\right )}} - \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 \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{4} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{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 \left (F\right )^{3}}{\left (b \log \left (F\right )\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 \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{4}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{6} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{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 \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {7}{2}} d^{3}}\right )} F^{a} d^{2}}{2 \, \sqrt {b \log \left (F\right )}} + \frac {\sqrt {\pi } F^{b c^{2} + a} c^{3} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} d x - \frac {b c \log \left (F\right )}{\sqrt {-b \log \left (F\right )}}\right )}{2 \, \sqrt {-b \log \left (F\right )} F^{b c^{2}} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^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*c^2/sqrt(b*log(F)) + 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*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*c*d/sqrt(b*log
(F)) - 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*lo
g(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*lo
g(F)/(b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*d^2/sqrt(b*log(F)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^3*erf(sqr
t(-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 = 0.36, size = 60, normalized size = 0.97 \begin {gather*} \frac {{\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 1\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{2} d \log \left (F\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) - 1)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)/(b^2*d*log(F)^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (49) = 98\).
time = 0.07, size = 99, normalized size = 1.60 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b c^{2} \log {\left (F \right )} + 2 b c d x \log {\left (F \right )} + b d^{2} x^{2} \log {\left (F \right )} - 1\right )}{2 b^{2} d \log {\left (F \right )}^{2}} & \text {for}\: b^{2} d \log {\left (F \right )}^{2} \neq 0 \\c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((F**(a + b*(c + d*x)**2)*(b*c**2*log(F) + 2*b*c*d*x*log(F) + b*d**2*x**2*log(F) - 1)/(2*b**2*d*log(F
)**2), Ne(b**2*d*log(F)**2, 0)), (c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4, True))

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Giac [C] Result contains complex when optimal does not.
time = 3.19, size = 1227, normalized size = 19.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*((pi*b^2*d^2*log(abs(F))*sgn(F) - pi*b^2*d^2*log(abs(F)))*(pi*b*c^2*d*sgn(F) + pi*(d*x^2 + 2*c*x)*b*d^2
*sgn(F) - pi*b*c^2*d - pi*(d*x^2 + 2*c*x)*b*d^2)/((pi^2*b^2*d^2*sgn(F) - pi^2*b^2*d^2 + 2*b^2*d^2*log(abs(F))^
2)^2 + 4*(pi*b^2*d^2*log(abs(F))*sgn(F) - pi*b^2*d^2*log(abs(F)))^2) + (pi^2*b^2*d^2*sgn(F) - pi^2*b^2*d^2 + 2
*b^2*d^2*log(abs(F))^2)*(b*c^2*d*log(abs(F)) + (d*x^2 + 2*c*x)*b*d^2*log(abs(F)) - d)/((pi^2*b^2*d^2*sgn(F) -
pi^2*b^2*d^2 + 2*b^2*d^2*log(abs(F))^2)^2 + 4*(pi*b^2*d^2*log(abs(F))*sgn(F) - pi*b^2*d^2*log(abs(F)))^2))*cos
(-1/2*pi*b*d^2*x^2*sgn(F) + 1/2*pi*b*d^2*x^2 - pi*b*c*d*x*sgn(F) + pi*b*c*d*x - 1/2*pi*b*c^2*sgn(F) + 1/2*pi*b
*c^2 - 1/2*pi*a*sgn(F) + 1/2*pi*a) + ((pi^2*b^2*d^2*sgn(F) - pi^2*b^2*d^2 + 2*b^2*d^2*log(abs(F))^2)*(pi*b*c^2
*d*sgn(F) + pi*(d*x^2 + 2*c*x)*b*d^2*sgn(F) - pi*b*c^2*d - pi*(d*x^2 + 2*c*x)*b*d^2)/((pi^2*b^2*d^2*sgn(F) - p
i^2*b^2*d^2 + 2*b^2*d^2*log(abs(F))^2)^2 + 4*(pi*b^2*d^2*log(abs(F))*sgn(F) - pi*b^2*d^2*log(abs(F)))^2) - 4*(
pi*b^2*d^2*log(abs(F))*sgn(F) - pi*b^2*d^2*log(abs(F)))*(b*c^2*d*log(abs(F)) + (d*x^2 + 2*c*x)*b*d^2*log(abs(F
)) - d)/((pi^2*b^2*d^2*sgn(F) - pi^2*b^2*d^2 + 2*b^2*d^2*log(abs(F))^2)^2 + 4*(pi*b^2*d^2*log(abs(F))*sgn(F) -
pi*b^2*d^2*log(abs(F)))^2))*sin(-1/2*pi*b*d^2*x^2*sgn(F) + 1/2*pi*b*d^2*x^2 - pi*b*c*d*x*sgn(F) + pi*b*c*d*x
- 1/2*pi*b*c^2*sgn(F) + 1/2*pi*b*c^2 - 1/2*pi*a*sgn(F) + 1/2*pi*a))*e^(b*c^2*log(abs(F)) + (d*x^2 + 2*c*x)*b*d
*log(abs(F)) + a*log(abs(F))) - 1/4*I*((pi*b*c^2*d*sgn(F) + pi*(d*x^2 + 2*c*x)*b*d^2*sgn(F) - pi*b*c^2*d - pi*
(d*x^2 + 2*c*x)*b*d^2 - 2*I*b*c^2*d*log(abs(F)) + 2*(-I*d*x^2 - 2*I*c*x)*b*d^2*log(abs(F)) + 2*I*d)*e^(1/2*I*p
i*b*d^2*x^2*sgn(F) - 1/2*I*pi*b*d^2*x^2 + I*pi*b*c*d*x*sgn(F) - I*pi*b*c*d*x + 1/2*I*pi*b*c^2*sgn(F) - 1/2*I*p
i*b*c^2 + 1/2*I*pi*a*sgn(F) - 1/2*I*pi*a)/(pi^2*b^2*d^2*sgn(F) + 2*I*pi*b^2*d^2*log(abs(F))*sgn(F) - pi^2*b^2*
d^2 - 2*I*pi*b^2*d^2*log(abs(F)) + 2*b^2*d^2*log(abs(F))^2) + (pi*b*c^2*d*sgn(F) + pi*(d*x^2 + 2*c*x)*b*d^2*sg
n(F) - pi*b*c^2*d - pi*(d*x^2 + 2*c*x)*b*d^2 + 2*I*b*c^2*d*log(abs(F)) - 2*(-I*d*x^2 - 2*I*c*x)*b*d^2*log(abs(
F)) - 2*I*d)*e^(-1/2*I*pi*b*d^2*x^2*sgn(F) + 1/2*I*pi*b*d^2*x^2 - I*pi*b*c*d*x*sgn(F) + I*pi*b*c*d*x - 1/2*I*p
i*b*c^2*sgn(F) + 1/2*I*pi*b*c^2 - 1/2*I*pi*a*sgn(F) + 1/2*I*pi*a)/(pi^2*b^2*d^2*sgn(F) - 2*I*pi*b^2*d^2*log(ab
s(F))*sgn(F) - pi^2*b^2*d^2 + 2*I*pi*b^2*d^2*log(abs(F)) + 2*b^2*d^2*log(abs(F))^2))*e^(b*c^2*log(abs(F)) + (d
*x^2 + 2*c*x)*b*d*log(abs(F)) + a*log(abs(F)))

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Mupad [B]
time = 3.55, size = 67, normalized size = 1.08 \begin {gather*} \frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (b\,\ln \left (F\right )\,c^2+2\,b\,\ln \left (F\right )\,c\,d\,x+b\,\ln \left (F\right )\,d^2\,x^2-1\right )}{2\,b^2\,d\,{\ln \left (F\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(b*c^2*log(F) + b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) - 1))/(2*b^2*d*
log(F)^2)

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