\(\int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx\) [210]

Optimal result

Integrand size = 21, antiderivative size = 136 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}-\frac {2 b F^{a+b (c+d x)^2} \log (F)}{15 d (c+d x)^3}-\frac {4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{15 d (c+d x)}+\frac {4 b^{5/2} F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {5}{2}}(F)}{15 d} \] Output:

-1/5*F^(a+b*(d*x+c)^2)/d/(d*x+c)^5-2/15*b*F^(a+b*(d*x+c)^2)*ln(F)/d/(d*x+c 
)^3-4/15*b^2*F^(a+b*(d*x+c)^2)*ln(F)^2/d/(d*x+c)+4/15*b^(5/2)*F^a*Pi^(1/2) 
*erfi(b^(1/2)*(d*x+c)*ln(F)^(1/2))*ln(F)^(5/2)/d
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.71 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=\frac {F^a \left (4 b^{5/2} \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right ) \log ^{\frac {5}{2}}(F)-\frac {F^{b (c+d x)^2} \left (3+2 b (c+d x)^2 \log (F)+4 b^2 (c+d x)^4 \log ^2(F)\right )}{(c+d x)^5}\right )}{15 d} \] Input:

Integrate[F^(a + b*(c + d*x)^2)/(c + d*x)^6,x]
 

Output:

(F^a*(4*b^(5/2)*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Log[F]^(5/2) 
 - (F^(b*(c + d*x)^2)*(3 + 2*b*(c + d*x)^2*Log[F] + 4*b^2*(c + d*x)^4*Log[ 
F]^2))/(c + d*x)^5))/(15*d)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2643, 2643, 2643, 2633}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {2}{5} b \log (F) \int \frac {F^{b (c+d x)^2+a}}{(c+d x)^4}dx-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {2}{5} b \log (F) \left (\frac {2}{3} b \log (F) \int \frac {F^{b (c+d x)^2+a}}{(c+d x)^2}dx-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}\right )-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {2}{5} b \log (F) \left (\frac {2}{3} b \log (F) \left (2 b \log (F) \int F^{b (c+d x)^2+a}dx-\frac {F^{a+b (c+d x)^2}}{d (c+d x)}\right )-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}\right )-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {2}{5} b \log (F) \left (\frac {2}{3} b \log (F) \left (\frac {\sqrt {\pi } \sqrt {b} F^a \sqrt {\log (F)} \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{d}-\frac {F^{a+b (c+d x)^2}}{d (c+d x)}\right )-\frac {F^{a+b (c+d x)^2}}{3 d (c+d x)^3}\right )-\frac {F^{a+b (c+d x)^2}}{5 d (c+d x)^5}\)

Input:

Int[F^(a + b*(c + d*x)^2)/(c + d*x)^6,x]
 

Output:

-1/5*F^(a + b*(c + d*x)^2)/(d*(c + d*x)^5) + (2*b*Log[F]*(-1/3*F^(a + b*(c 
 + d*x)^2)/(d*(c + d*x)^3) + (2*b*(-(F^(a + b*(c + d*x)^2)/(d*(c + d*x))) 
+ (Sqrt[b]*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*(c + d*x)*Sqrt[Log[F]]]*Sqrt[Log[F]]) 
/d)*Log[F])/3))/5
 

Defintions of rubi rules used

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2643

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) 
, x] - Simp[b*n*(Log[F]/(m + 1))   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[ 
-4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 
0] && LeQ[-n, m + 1]))
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {F^{b \left (d x +c \right )^{2}} F^{a}}{5 d \left (d x +c \right )^{5}}-\frac {2 b \ln \left (F \right ) F^{b \left (d x +c \right )^{2}} F^{a}}{15 d \left (d x +c \right )^{3}}-\frac {4 b^{2} \ln \left (F \right )^{2} F^{b \left (d x +c \right )^{2}} F^{a}}{15 d \left (d x +c \right )}+\frac {4 b^{3} \ln \left (F \right )^{3} \sqrt {\pi }\, F^{a} \operatorname {erf}\left (\sqrt {-b \ln \left (F \right )}\, \left (d x +c \right )\right )}{15 d \sqrt {-b \ln \left (F \right )}}\)	\(129\)

Input:

int(F^(a+b*(d*x+c)^2)/(d*x+c)^6,x,method=_RETURNVERBOSE)
 

Output:

-1/5/d/(d*x+c)^5*F^(b*(d*x+c)^2)*F^a-2/15/d*b*ln(F)/(d*x+c)^3*F^(b*(d*x+c) 
^2)*F^a-4/15/d*b^2*ln(F)^2/(d*x+c)*F^(b*(d*x+c)^2)*F^a+4/15/d*b^3*ln(F)^3* 
Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)*(d*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (118) = 236\).

Time = 0.08 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.12 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=-\frac {4 \, \sqrt {\pi } {\left (b^{2} d^{5} x^{5} + 5 \, b^{2} c d^{4} x^{4} + 10 \, b^{2} c^{2} d^{3} x^{3} + 10 \, b^{2} c^{3} d^{2} x^{2} + 5 \, b^{2} c^{4} d x + b^{2} c^{5}\right )} \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) \log \left (F\right )^{2} + {\left (4 \, {\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x + b^{2} c^{4} d\right )} \log \left (F\right )^{2} + 2 \, {\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \left (F\right ) + 3 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{15 \, {\left (d^{7} x^{5} + 5 \, c d^{6} x^{4} + 10 \, c^{2} d^{5} x^{3} + 10 \, c^{3} d^{4} x^{2} + 5 \, c^{4} d^{3} x + c^{5} d^{2}\right )}} \] Input:

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^6,x, algorithm="fricas")
 

Output:

-1/15*(4*sqrt(pi)*(b^2*d^5*x^5 + 5*b^2*c*d^4*x^4 + 10*b^2*c^2*d^3*x^3 + 10 
*b^2*c^3*d^2*x^2 + 5*b^2*c^4*d*x + b^2*c^5)*sqrt(-b*d^2*log(F))*F^a*erf(sq 
rt(-b*d^2*log(F))*(d*x + c)/d)*log(F)^2 + (4*(b^2*d^5*x^4 + 4*b^2*c*d^4*x^ 
3 + 6*b^2*c^2*d^3*x^2 + 4*b^2*c^3*d^2*x + b^2*c^4*d)*log(F)^2 + 2*(b*d^3*x 
^2 + 2*b*c*d^2*x + b*c^2*d)*log(F) + 3*d)*F^(b*d^2*x^2 + 2*b*c*d*x + b*c^2 
 + a))/(d^7*x^5 + 5*c*d^6*x^4 + 10*c^2*d^5*x^3 + 10*c^3*d^4*x^2 + 5*c^4*d^ 
3*x + c^5*d^2)
 

Sympy [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{6}}\, dx \] Input:

integrate(F**(a+b*(d*x+c)**2)/(d*x+c)**6,x)
 

Output:

Integral(F**(a + b*(c + d*x)**2)/(c + d*x)**6, x)
 

Maxima [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{6}} \,d x } \] Input:

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^6,x, algorithm="maxima")
 

Output:

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^6, x)
 

Giac [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{6}} \,d x } \] Input:

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^6,x, algorithm="giac")
 

Output:

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^6, x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.24 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=\frac {4\,F^a\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2}\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2\right )}^{5/2}}{15\,d\,{\left (c+d\,x\right )}^5}-\frac {4\,F^a\,\sqrt {\pi }\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2\right )}^{5/2}}{15\,d\,{\left (c+d\,x\right )}^5}-\frac {4\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^2\,{\ln \left (F\right )}^2}{15\,d\,\left (c+d\,x\right )}-\frac {2\,F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b\,\ln \left (F\right )}{15\,d\,{\left (c+d\,x\right )}^3}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}}{5\,d\,{\left (c+d\,x\right )}^5} \] Input:

int(F^(a + b*(c + d*x)^2)/(c + d*x)^6,x)
 

Output:

(4*F^a*pi^(1/2)*erfc((-b*log(F)*(c + d*x)^2)^(1/2))*(-b*log(F)*(c + d*x)^2 
)^(5/2))/(15*d*(c + d*x)^5) - (4*F^a*pi^(1/2)*(-b*log(F)*(c + d*x)^2)^(5/2 
))/(15*d*(c + d*x)^5) - (4*F^a*F^(b*(c + d*x)^2)*b^2*log(F)^2)/(15*d*(c + 
d*x)) - (2*F^a*F^(b*(c + d*x)^2)*b*log(F))/(15*d*(c + d*x)^3) - (F^a*F^(b* 
(c + d*x)^2))/(5*d*(c + d*x)^5)
 

Reduce [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx=\text {too large to display} \] Input:

int(F^(a+b*(d*x+c)^2)/(d*x+c)^6,x)
 

Output:

(f**(a + b*c**2)*( - f**(2*b*c*d*x + b*d**2*x**2) + 8*int(f**(2*b*c*d*x + 
b*d**2*x**2)/(2*log(f)*b*c**8 + 12*log(f)*b*c**7*d*x + 30*log(f)*b*c**6*d* 
*2*x**2 + 40*log(f)*b*c**5*d**3*x**3 + 30*log(f)*b*c**4*d**4*x**4 + 12*log 
(f)*b*c**3*d**5*x**5 + 2*log(f)*b*c**2*d**6*x**6 - 5*c**6 - 30*c**5*d*x - 
75*c**4*d**2*x**2 - 100*c**3*d**3*x**3 - 75*c**2*d**4*x**4 - 30*c*d**5*x** 
5 - 5*d**6*x**6),x)*log(f)**2*b**2*c**9*d + 40*int(f**(2*b*c*d*x + b*d**2* 
x**2)/(2*log(f)*b*c**8 + 12*log(f)*b*c**7*d*x + 30*log(f)*b*c**6*d**2*x**2 
 + 40*log(f)*b*c**5*d**3*x**3 + 30*log(f)*b*c**4*d**4*x**4 + 12*log(f)*b*c 
**3*d**5*x**5 + 2*log(f)*b*c**2*d**6*x**6 - 5*c**6 - 30*c**5*d*x - 75*c**4 
*d**2*x**2 - 100*c**3*d**3*x**3 - 75*c**2*d**4*x**4 - 30*c*d**5*x**5 - 5*d 
**6*x**6),x)*log(f)**2*b**2*c**8*d**2*x + 80*int(f**(2*b*c*d*x + b*d**2*x* 
*2)/(2*log(f)*b*c**8 + 12*log(f)*b*c**7*d*x + 30*log(f)*b*c**6*d**2*x**2 + 
 40*log(f)*b*c**5*d**3*x**3 + 30*log(f)*b*c**4*d**4*x**4 + 12*log(f)*b*c** 
3*d**5*x**5 + 2*log(f)*b*c**2*d**6*x**6 - 5*c**6 - 30*c**5*d*x - 75*c**4*d 
**2*x**2 - 100*c**3*d**3*x**3 - 75*c**2*d**4*x**4 - 30*c*d**5*x**5 - 5*d** 
6*x**6),x)*log(f)**2*b**2*c**7*d**3*x**2 + 80*int(f**(2*b*c*d*x + b*d**2*x 
**2)/(2*log(f)*b*c**8 + 12*log(f)*b*c**7*d*x + 30*log(f)*b*c**6*d**2*x**2 
+ 40*log(f)*b*c**5*d**3*x**3 + 30*log(f)*b*c**4*d**4*x**4 + 12*log(f)*b*c* 
*3*d**5*x**5 + 2*log(f)*b*c**2*d**6*x**6 - 5*c**6 - 30*c**5*d*x - 75*c**4* 
d**2*x**2 - 100*c**3*d**3*x**3 - 75*c**2*d**4*x**4 - 30*c*d**5*x**5 - 5...