3.323 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^7} \, dx\)

Optimal. Leaf size=91 \[ \frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4} \]

[Out]

-(F^(a + b/(c + d*x)^2)/(b^3*d*Log[F]^3)) + F^(a + b/(c + d*x)^2)/(b^2*d*(c + d*x)^2*Log[F]^2) - F^(a + b/(c +
d*x)^2)/(2*b*d*(c + d*x)^4*Log[F])

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Rubi [A]  time = 0.137092, 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} \[ \frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2)/(c + d*x)^7,x]

[Out]

-(F^(a + b/(c + d*x)^2)/(b^3*d*Log[F]^3)) + F^(a + b/(c + d*x)^2)/(b^2*d*(c + d*x)^2*Log[F]^2) - F^(a + b/(c +
d*x)^2)/(2*b*d*(c + d*x)^4*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 \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^7} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}-\frac{2 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^5} \, dx}{b \log (F)}\\ &=\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}+\frac{2 \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^3} \, dx}{b^2 \log ^2(F)}\\ &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^3 d \log ^3(F)}+\frac{F^{a+\frac{b}{(c+d x)^2}}}{b^2 d (c+d x)^2 \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d (c+d x)^4 \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0325184, size = 64, normalized size = 0.7 \[ -\frac{F^{a+\frac{b}{(c+d x)^2}} \left (b^2 \log ^2(F)-2 b \log (F) (c+d x)^2+2 (c+d x)^4\right )}{2 b^3 d \log ^3(F) (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^7,x]

[Out]

-(F^(a + b/(c + d*x)^2)*(2*(c + d*x)^4 - 2*b*(c + d*x)^2*Log[F] + b^2*Log[F]^2))/(2*b^3*d*(c + d*x)^4*Log[F]^3
)

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Maple [B]  time = 0.061, size = 301, normalized size = 3.3 \begin{align*}{\frac{1}{ \left ( dx+c \right ) ^{6}} \left ({\frac{{d}^{3} \left ( b\ln \left ( F \right ) -15\,{c}^{2} \right ){x}^{4}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{{d}^{5}{x}^{6}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{c \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-4\,\ln \left ( F \right ) b{c}^{2}+6\,{c}^{4} \right ) x}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{d \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-12\,\ln \left ( F \right ) b{c}^{2}+30\,{c}^{4} \right ){x}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-6\,{\frac{c{d}^{4}{x}^{5}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-2\,\ln \left ( F \right ) b{c}^{2}+2\,{c}^{4} \right ){c}^{2}}{2\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}d}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}}+4\,{\frac{c{d}^{2} \left ( b\ln \left ( F \right ) -5\,{c}^{2} \right ){x}^{3}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{2}}} \right ) \ln \left ( F \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b/(d*x+c)^2)/(d*x+c)^7,x)

[Out]

(d^3*(b*ln(F)-15*c^2)/ln(F)^3/b^3*x^4*exp((a+b/(d*x+c)^2)*ln(F))-d^5/ln(F)^3/b^3*x^6*exp((a+b/(d*x+c)^2)*ln(F)
)-c*(ln(F)^2*b^2-4*ln(F)*b*c^2+6*c^4)/b^3/ln(F)^3*x*exp((a+b/(d*x+c)^2)*ln(F))-1/2*d*(ln(F)^2*b^2-12*ln(F)*b*c
^2+30*c^4)/ln(F)^3/b^3*x^2*exp((a+b/(d*x+c)^2)*ln(F))-6*d^4*c/ln(F)^3/b^3*x^5*exp((a+b/(d*x+c)^2)*ln(F))-1/2*(
ln(F)^2*b^2-2*ln(F)*b*c^2+2*c^4)*c^2/b^3/ln(F)^3/d*exp((a+b/(d*x+c)^2)*ln(F))+4*c*d^2*(b*ln(F)-5*c^2)/ln(F)^3/
b^3*x^3*exp((a+b/(d*x+c)^2)*ln(F)))/(d*x+c)^6

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Maxima [B]  time = 1.02802, size = 281, normalized size = 3.09 \begin{align*} -\frac{{\left (2 \, F^{a} d^{4} x^{4} + 8 \, F^{a} c d^{3} x^{3} + 2 \, F^{a} c^{4} - 2 \, F^{a} b c^{2} \log \left (F\right ) + F^{a} b^{2} \log \left (F\right )^{2} + 2 \,{\left (6 \, F^{a} c^{2} d^{2} - F^{a} b d^{2} \log \left (F\right )\right )} x^{2} + 4 \,{\left (2 \, F^{a} c^{3} d - F^{a} b c d \log \left (F\right )\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{3} d^{5} x^{4} \log \left (F\right )^{3} + 4 \, b^{3} c d^{4} x^{3} \log \left (F\right )^{3} + 6 \, b^{3} c^{2} d^{3} x^{2} \log \left (F\right )^{3} + 4 \, b^{3} c^{3} d^{2} x \log \left (F\right )^{3} + b^{3} c^{4} d \log \left (F\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^7,x, algorithm="maxima")

[Out]

-1/2*(2*F^a*d^4*x^4 + 8*F^a*c*d^3*x^3 + 2*F^a*c^4 - 2*F^a*b*c^2*log(F) + F^a*b^2*log(F)^2 + 2*(6*F^a*c^2*d^2 -
F^a*b*d^2*log(F))*x^2 + 4*(2*F^a*c^3*d - F^a*b*c*d*log(F))*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^3*d^5*x^4*lo
g(F)^3 + 4*b^3*c*d^4*x^3*log(F)^3 + 6*b^3*c^2*d^3*x^2*log(F)^3 + 4*b^3*c^3*d^2*x*log(F)^3 + b^3*c^4*d*log(F)^3
)

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Fricas [B]  time = 1.81084, size = 386, normalized size = 4.24 \begin{align*} -\frac{{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} + 8 \, c^{3} d x + 2 \, c^{4} + b^{2} \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 )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \,{\left (b^{3} d^{5} x^{4} + 4 \, b^{3} c d^{4} x^{3} + 6 \, b^{3} c^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} d^{2} x + b^{3} c^{4} d\right )} \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)^7,x, algorithm="fricas")

[Out]

-1/2*(2*d^4*x^4 + 8*c*d^3*x^3 + 12*c^2*d^2*x^2 + 8*c^3*d*x + 2*c^4 + b^2*log(F)^2 - 2*(b*d^2*x^2 + 2*b*c*d*x +
b*c^2)*log(F))*F^((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))/((b^3*d^5*x^4 + 4*b^3*c*d^4*
x^3 + 6*b^3*c^2*d^3*x^2 + 4*b^3*c^3*d^2*x + b^3*c^4*d)*log(F)^3)

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Sympy [B]  time = 0.351453, size = 189, normalized size = 2.08 \begin{align*} \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}} \left (- b^{2} \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 c^{4} - 8 c^{3} d x - 12 c^{2} d^{2} x^{2} - 8 c d^{3} x^{3} - 2 d^{4} x^{4}\right )}{2 b^{3} c^{4} d \log{\left (F \right )}^{3} + 8 b^{3} c^{3} d^{2} x \log{\left (F \right )}^{3} + 12 b^{3} c^{2} d^{3} x^{2} \log{\left (F \right )}^{3} + 8 b^{3} c d^{4} x^{3} \log{\left (F \right )}^{3} + 2 b^{3} d^{5} x^{4} \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)**7,x)

[Out]

F**(a + b/(c + d*x)**2)*(-b**2*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*c**4
- 8*c**3*d*x - 12*c**2*d**2*x**2 - 8*c*d**3*x**3 - 2*d**4*x**4)/(2*b**3*c**4*d*log(F)**3 + 8*b**3*c**3*d**2*x*
log(F)**3 + 12*b**3*c**2*d**3*x**2*log(F)**3 + 8*b**3*c*d**4*x**3*log(F)**3 + 2*b**3*d**5*x**4*log(F)**3)

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Giac [B]  time = 1.34182, size = 2236, normalized size = 24.57 \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)^7,x, algorithm="giac")

[Out]

-1/2*((2*(pi^3*b^3*d^3*sgn(F) - 3*pi*b^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d^3*log(abs(F))^2)
*(pi*b*d^2*sgn(F)/(d*x + c)^2 - pi*b^2*d^2*log(abs(F))*sgn(F)/(d*x + c)^4 - pi*b*d^2/(d*x + c)^2 + pi*b^2*d^2*
log(abs(F))/(d*x + c)^4)/((pi^3*b^3*d^3*sgn(F) - 3*pi*b^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d
^3*log(abs(F))^2)^2 + (3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^
3)^2) + (3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^3)*(pi^2*b^2*d
^2*sgn(F)/(d*x + c)^4 - pi^2*b^2*d^2/(d*x + c)^4 + 4*d^2 - 4*b*d^2*log(abs(F))/(d*x + c)^2 + 2*b^2*d^2*log(abs
(F))^2/(d*x + c)^4)/((pi^3*b^3*d^3*sgn(F) - 3*pi*b^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d^3*lo
g(abs(F))^2)^2 + (3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^3)^2)
)*cos(-1/2*pi*a*sgn(F) + 1/2*pi*a - 1/2*pi*b*sgn(F)/(d^2*x^2 + 2*c*d*x + c^2) + 1/2*pi*b/(d^2*x^2 + 2*c*d*x +
c^2)) - (2*(3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^3)*(pi*b*d^
2*sgn(F)/(d*x + c)^2 - pi*b^2*d^2*log(abs(F))*sgn(F)/(d*x + c)^4 - pi*b*d^2/(d*x + c)^2 + pi*b^2*d^2*log(abs(F
))/(d*x + c)^4)/((pi^3*b^3*d^3*sgn(F) - 3*pi*b^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d^3*log(ab
s(F))^2)^2 + (3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^3)^2) - (
pi^3*b^3*d^3*sgn(F) - 3*pi*b^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d^3*log(abs(F))^2)*(pi^2*b^2
*d^2*sgn(F)/(d*x + c)^4 - pi^2*b^2*d^2/(d*x + c)^4 + 4*d^2 - 4*b*d^2*log(abs(F))/(d*x + c)^2 + 2*b^2*d^2*log(a
bs(F))^2/(d*x + c)^4)/((pi^3*b^3*d^3*sgn(F) - 3*pi*b^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3 + 3*pi*b^3*d^3*
log(abs(F))^2)^2 + (3*pi^2*b^3*d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^3)^
2))*sin(-1/2*pi*a*sgn(F) + 1/2*pi*a - 1/2*pi*b*sgn(F)/(d^2*x^2 + 2*c*d*x + c^2) + 1/2*pi*b/(d^2*x^2 + 2*c*d*x
+ c^2)))*e^(a*log(abs(F)) + b*log(abs(F))/(d*x + c)^2) - 1/4*((pi^2*b^2*d^2*i*sgn(F)/(d*x + c)^4 - pi^2*b^2*d^
2*i/(d*x + c)^4 + 4*d^2*i - 4*b*d^2*i*log(abs(F))/(d*x + c)^2 + 2*b^2*d^2*i*log(abs(F))^2/(d*x + c)^4 + 2*pi*b
*d^2*sgn(F)/(d*x + c)^2 - 2*pi*b^2*d^2*log(abs(F))*sgn(F)/(d*x + c)^4 - 2*pi*b*d^2/(d*x + c)^2 + 2*pi*b^2*d^2*
log(abs(F))/(d*x + c)^4)*e^(1/2*(pi*a*(sgn(F) - 1) + pi*b*(sgn(F) - 1)/(d*x + c)^2)*i)/(pi^3*b^3*d^3*i*sgn(F)
- 3*pi*b^3*d^3*i*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*i + 3*pi*b^3*d^3*i*log(abs(F))^2 - 3*pi^2*b^3*d^3*log(abs
(F))*sgn(F) + 3*pi^2*b^3*d^3*log(abs(F)) - 2*b^3*d^3*log(abs(F))^3) + (pi^2*b^2*d^2*i*sgn(F)/(d*x + c)^4 - pi^
2*b^2*d^2*i/(d*x + c)^4 + 4*d^2*i - 4*b*d^2*i*log(abs(F))/(d*x + c)^2 + 2*b^2*d^2*i*log(abs(F))^2/(d*x + c)^4
- 2*pi*b*d^2*sgn(F)/(d*x + c)^2 + 2*pi*b^2*d^2*log(abs(F))*sgn(F)/(d*x + c)^4 + 2*pi*b*d^2/(d*x + c)^2 - 2*pi*
b^2*d^2*log(abs(F))/(d*x + c)^4)*e^(-1/2*(pi*a*(sgn(F) - 1) + pi*b*(sgn(F) - 1)/(d*x + c)^2)*i)/(pi^3*b^3*d^3*
i*sgn(F) - 3*pi*b^3*d^3*i*log(abs(F))^2*sgn(F) - pi^3*b^3*d^3*i + 3*pi*b^3*d^3*i*log(abs(F))^2 + 3*pi^2*b^3*d^
3*log(abs(F))*sgn(F) - 3*pi^2*b^3*d^3*log(abs(F)) + 2*b^3*d^3*log(abs(F))^3))*e^(a*log(abs(F)) + b*log(abs(F))
/(d*x + c)^2)/i