3.348 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^7} \, dx\)

Optimal. Leaf size=62 \[ \frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0867742, antiderivative size = 62, normalized size of antiderivative = 1., 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 = {2212, 2209} \[ \frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b^2 d \log ^2(F)}-\frac{F^{a+\frac{b}{(c+d x)^3}}}{3 b d \log (F) (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.0289271, size = 47, normalized size = 0.76 \[ \frac{F^{a+\frac{b}{(c+d x)^3}} \left ((c+d x)^3-b \log (F)\right )}{3 b^2 d \log ^2(F) (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^7,x]

[Out]

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

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Maple [B]  time = 0.061, size = 261, normalized size = 4.2 \begin{align*}{\frac{1}{ \left ( dx+c \right ) ^{6}} \left ({\frac{{d}^{5}{x}^{6}}{3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \ln \left ( F \right ) }}}-{\frac{{c}^{2} \left ( -2\,{c}^{3}+b\ln \left ( F \right ) \right ) x}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \ln \left ( F \right ) }}}-{\frac{{c}^{3} \left ( -{c}^{3}+b\ln \left ( F \right ) \right ) }{3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}d}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \ln \left ( F \right ) }}}-{\frac{{d}^{2} \left ( -20\,{c}^{3}+b\ln \left ( F \right ) \right ){x}^{3}}{3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \ln \left ( F \right ) }}}+5\,{\frac{{c}^{2}{d}^{3}{x}^{4}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \ln \left ( F \right ) }}}+2\,{\frac{c{d}^{4}{x}^{5}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \ln \left ( F \right ) }}}-{\frac{cd \left ( -5\,{c}^{3}+b\ln \left ( F \right ) \right ){x}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \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)^3)/(d*x+c)^7,x)

[Out]

(1/3/ln(F)^2/b^2*d^5*x^6*exp((a+b/(d*x+c)^3)*ln(F))-c^2*(-2*c^3+b*ln(F))/ln(F)^2/b^2*x*exp((a+b/(d*x+c)^3)*ln(
F))-1/3*c^3*(-c^3+b*ln(F))/d/ln(F)^2/b^2*exp((a+b/(d*x+c)^3)*ln(F))-1/3*d^2*(-20*c^3+b*ln(F))/ln(F)^2/b^2*x^3*
exp((a+b/(d*x+c)^3)*ln(F))+5*d^3*c^2/ln(F)^2/b^2*x^4*exp((a+b/(d*x+c)^3)*ln(F))+2*d^4*c/ln(F)^2/b^2*x^5*exp((a
+b/(d*x+c)^3)*ln(F))-c*d*(-5*c^3+b*ln(F))/ln(F)^2/b^2*x^2*exp((a+b/(d*x+c)^3)*ln(F)))/(d*x+c)^6

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Maxima [B]  time = 1.03431, size = 194, normalized size = 3.13 \begin{align*} \frac{{\left (F^{a} d^{3} x^{3} + 3 \, F^{a} c d^{2} x^{2} + 3 \, F^{a} c^{2} d x + F^{a} c^{3} - F^{a} b \log \left (F\right )\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \,{\left (b^{2} d^{4} x^{3} \log \left (F\right )^{2} + 3 \, b^{2} c d^{3} x^{2} \log \left (F\right )^{2} + 3 \, b^{2} c^{2} d^{2} x \log \left (F\right )^{2} + b^{2} c^{3} d \log \left (F\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^7,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.59322, size = 312, normalized size = 5.03 \begin{align*} \frac{{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3} - b \log \left (F\right )\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^7,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.339172, size = 114, normalized size = 1.84 \begin{align*} \frac{F^{a + \frac{b}{\left (c + d x\right )^{3}}} \left (- b \log{\left (F \right )} + c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}\right )}{3 b^{2} c^{3} d \log{\left (F \right )}^{2} + 9 b^{2} c^{2} d^{2} x \log{\left (F \right )}^{2} + 9 b^{2} c d^{3} x^{2} \log{\left (F \right )}^{2} + 3 b^{2} d^{4} x^{3} \log{\left (F \right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**7,x)

[Out]

F**(a + b/(c + d*x)**3)*(-b*log(F) + c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3)/(3*b**2*c**3*d*log(F)**2 +
9*b**2*c**2*d**2*x*log(F)**2 + 9*b**2*c*d**3*x**2*log(F)**2 + 3*b**2*d**4*x**3*log(F)**2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^7,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^7, x)