\(\int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx\) [316]

Optimal result

Integrand size = 21, antiderivative size = 31 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^2}\right ) \log ^4(F)}{2 d} \]

[Out]

1/2*F^a*(d*x+c)^8*Ei(5,-b*ln(F)/(d*x+c)^2)/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250} \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\frac {b^4 F^a \log ^4(F) \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[In]

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

[Out]

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

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^2}\right ) \log ^4(F)}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\frac {b^4 F^a \Gamma \left (-4,-\frac {b \log (F)}{(c+d x)^2}\right ) \log ^4(F)}{2 d} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(645\) vs. \(2(29)=58\).

Time = 1.98 (sec) , antiderivative size = 646, normalized size of antiderivative = 20.84

method	result	size
risch	\(\frac {5 F^{a} d^{3} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{4}}{8}+\frac {5 F^{a} d^{2} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{3}}{6}+\frac {5 F^{a} d b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{2}}{8}+\frac {F^{a} d^{2} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{3}}{12}+\frac {F^{a} d \,b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{2}}{8}+\frac {F^{a} d^{4} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{5}}{4}+\frac {F^{a} d^{5} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} x^{6}}{24}+\frac {F^{a} d^{3} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{4}}{48}+\frac {F^{a} d \,b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{2}}{48}+\frac {F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6}}{24 d}+\frac {F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4}}{48 d}+\frac {F^{a} b \ln \left (F \right ) F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x}{4}+\frac {F^{a} b^{2} \ln \left (F \right )^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x}{12}+\frac {F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c x}{24}+\frac {F^{a} b^{3} \ln \left (F \right )^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2}}{48 d}+\frac {F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{8}}{8 d}+\frac {F^{a} d^{7} F^{\frac {b}{\left (d x +c \right )^{2}}} x^{8}}{8}+F^{a} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{7} x +\frac {F^{a} b^{4} \ln \left (F \right )^{4} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{\left (d x +c \right )^{2}}\right )}{48 d}+F^{a} d^{6} F^{\frac {b}{\left (d x +c \right )^{2}}} c \,x^{7}+\frac {7 F^{a} d^{5} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{2} x^{6}}{2}+7 F^{a} d^{4} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{3} x^{5}+\frac {35 F^{a} d^{3} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{4} x^{4}}{4}+7 F^{a} d^{2} F^{\frac {b}{\left (d x +c \right )^{2}}} c^{5} x^{3}+\frac {7 F^{a} d \,F^{\frac {b}{\left (d x +c \right )^{2}}} c^{6} x^{2}}{2}\)	\(646\)

[In]

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

[Out]

5/8*F^a*d^3*b*ln(F)*F^(b/(d*x+c)^2)*c^2*x^4+5/6*F^a*d^2*b*ln(F)*F^(b/(d*x+c)^2)*c^3*x^3+5/8*F^a*d*b*ln(F)*F^(b
/(d*x+c)^2)*c^4*x^2+1/12*F^a*d^2*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^3+1/8*F^a*d*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^2*x
^2+1/4*F^a*d^4*b*ln(F)*F^(b/(d*x+c)^2)*c*x^5+1/24*F^a*d^5*b*ln(F)*F^(b/(d*x+c)^2)*x^6+1/48*F^a*d^3*b^2*ln(F)^2
*F^(b/(d*x+c)^2)*x^4+1/48*F^a*d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*x^2+1/24*F^a/d*b*ln(F)*F^(b/(d*x+c)^2)*c^6+1/48*F^
a/d*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^4+1/4*F^a*b*ln(F)*F^(b/(d*x+c)^2)*c^5*x+1/12*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)
*c^3*x+1/24*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x+1/48*F^a/d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2+1/8*F^a/d*F^(b/(d*x
+c)^2)*c^8+1/8*F^a*d^7*F^(b/(d*x+c)^2)*x^8+F^a*F^(b/(d*x+c)^2)*c^7*x+1/48*F^a/d*b^4*ln(F)^4*Ei(1,-b*ln(F)/(d*x
+c)^2)+F^a*d^6*F^(b/(d*x+c)^2)*c*x^7+7/2*F^a*d^5*F^(b/(d*x+c)^2)*c^2*x^6+7*F^a*d^4*F^(b/(d*x+c)^2)*c^3*x^5+35/
4*F^a*d^3*F^(b/(d*x+c)^2)*c^4*x^4+7*F^a*d^2*F^(b/(d*x+c)^2)*c^5*x^3+7/2*F^a*d*F^(b/(d*x+c)^2)*c^6*x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (29) = 58\).

Time = 0.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 10.68 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=-\frac {F^{a} b^{4} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \left (F\right )^{4} - {\left (6 \, d^{8} x^{8} + 48 \, c d^{7} x^{7} + 168 \, c^{2} d^{6} x^{6} + 336 \, c^{3} d^{5} x^{5} + 420 \, c^{4} d^{4} x^{4} + 336 \, c^{5} d^{3} x^{3} + 168 \, c^{6} d^{2} x^{2} + 48 \, c^{7} d x + 6 \, c^{8} + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2}\right )} \log \left (F\right )^{3} + {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{6} x^{6} + 6 \, b c d^{5} x^{5} + 15 \, b c^{2} d^{4} x^{4} + 20 \, b c^{3} d^{3} x^{3} + 15 \, b c^{4} d^{2} x^{2} + 6 \, b c^{5} d x + b c^{6}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{48 \, d} \]

[In]

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

[Out]

-1/48*(F^a*b^4*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))*log(F)^4 - (6*d^8*x^8 + 48*c*d^7*x^7 + 168*c^2*d^6*x^6 +
336*c^3*d^5*x^5 + 420*c^4*d^4*x^4 + 336*c^5*d^3*x^3 + 168*c^6*d^2*x^2 + 48*c^7*d*x + 6*c^8 + (b^3*d^2*x^2 + 2
*b^3*c*d*x + b^3*c^2)*log(F)^3 + (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^6*x^6 + 6*b*c*d^5*x^5 + 15*b*c^2*d^4*x^4 + 20*b*c^3*d^3*x^3 + 15*b*c^4*d^2*x^2 + 6*b*c^5*d*
x + b*c^6)*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)))/d

Sympy [F]

\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (c + d x\right )^{7}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\int { {\left (d x + c\right )}^{7} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]

[In]

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

[Out]

1/48*(6*F^a*d^7*x^8 + 48*F^a*c*d^6*x^7 + 2*(84*F^a*c^2*d^5 + F^a*b*d^5*log(F))*x^6 + 12*(28*F^a*c^3*d^4 + F^a*
b*c*d^4*log(F))*x^5 + (420*F^a*c^4*d^3 + 30*F^a*b*c^2*d^3*log(F) + F^a*b^2*d^3*log(F)^2)*x^4 + 4*(84*F^a*c^5*d
^2 + 10*F^a*b*c^3*d^2*log(F) + F^a*b^2*c*d^2*log(F)^2)*x^3 + (168*F^a*c^6*d + 30*F^a*b*c^4*d*log(F) + 6*F^a*b^
2*c^2*d*log(F)^2 + F^a*b^3*d*log(F)^3)*x^2 + 2*(24*F^a*c^7 + 6*F^a*b*c^5*log(F) + 2*F^a*b^2*c^3*log(F)^2 + F^a
*b^3*c*log(F)^3)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2)) + integrate(1/24*(F^a*b^4*d^2*x^2*log(F)^4 + 2*F^a*b^4*c*d
*x*log(F)^4 - 6*F^a*b*c^8*log(F) - 2*F^a*b^2*c^6*log(F)^2 - F^a*b^3*c^4*log(F)^3)*F^(b/(d^2*x^2 + 2*c*d*x + c^
2))/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

Giac [F]

\[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\int { {\left (d x + c\right )}^{7} F^{a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.87 \[ \int F^{a+\frac {b}{(c+d x)^2}} (c+d x)^7 \, dx=\frac {F^a\,b^4\,{\ln \left (F\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^2}\right )}{48\,d}+\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^2}}\,b^4\,{\ln \left (F\right )}^4\,\left (\frac {{\left (c+d\,x\right )}^2}{24\,b\,\ln \left (F\right )}+\frac {{\left (c+d\,x\right )}^4}{24\,b^2\,{\ln \left (F\right )}^2}+\frac {{\left (c+d\,x\right )}^6}{12\,b^3\,{\ln \left (F\right )}^3}+\frac {{\left (c+d\,x\right )}^8}{4\,b^4\,{\ln \left (F\right )}^4}\right )}{2\,d} \]

[In]

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

[Out]

(F^a*b^4*log(F)^4*expint(-(b*log(F))/(c + d*x)^2))/(48*d) + (F^a*F^(b/(c + d*x)^2)*b^4*log(F)^4*((c + d*x)^2/(
24*b*log(F)) + (c + d*x)^4/(24*b^2*log(F)^2) + (c + d*x)^6/(12*b^3*log(F)^3) + (c + d*x)^8/(4*b^4*log(F)^4)))/
(2*d)