\(\int F^{c (a+b x)} (d+e x)^{7/2} \, dx\) [69]

Optimal result

Integrand size = 19, antiderivative size = 208 \[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=\frac {105 e^{7/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 e^3 F^{c (a+b x)} \sqrt {d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac {35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac {7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)} \] Output:

105/16*e^(7/2)*F^(c*(a-b*d/e))*Pi^(1/2)*erfi(b^(1/2)*c^(1/2)*(e*x+d)^(1/2) 
*ln(F)^(1/2)/e^(1/2))/b^(9/2)/c^(9/2)/ln(F)^(9/2)-105/8*e^3*F^(c*(b*x+a))* 
(e*x+d)^(1/2)/b^4/c^4/ln(F)^4+35/4*e^2*F^(c*(b*x+a))*(e*x+d)^(3/2)/b^3/c^3 
/ln(F)^3-7/2*e*F^(c*(b*x+a))*(e*x+d)^(5/2)/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*( 
e*x+d)^(7/2)/b/c/ln(F)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.35 \[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=\frac {e^4 F^{c \left (a-\frac {b d}{e}\right )} \Gamma \left (\frac {9}{2},-\frac {b c (d+e x) \log (F)}{e}\right ) \sqrt {-\frac {b c (d+e x) \log (F)}{e}}}{b^5 c^5 \sqrt {d+e x} \log ^5(F)} \] Input:

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

Output:

(e^4*F^(c*(a - (b*d)/e))*Gamma[9/2, -((b*c*(d + e*x)*Log[F])/e)]*Sqrt[-((b 
*c*(d + e*x)*Log[F])/e)])/(b^5*c^5*Sqrt[d + e*x]*Log[F]^5)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2607, 2607, 2607, 2607, 2611, 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.

\(\displaystyle \int (d+e x)^{7/2} F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)}-\frac {7 e \int F^{c (a+b x)} (d+e x)^{5/2}dx}{2 b c \log (F)}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)}-\frac {7 e \left (\frac {(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)}-\frac {5 e \int F^{c (a+b x)} (d+e x)^{3/2}dx}{2 b c \log (F)}\right )}{2 b c \log (F)}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)}-\frac {7 e \left (\frac {(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)}-\frac {5 e \left (\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \int F^{c (a+b x)} \sqrt {d+e x}dx}{2 b c \log (F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)}-\frac {7 e \left (\frac {(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)}-\frac {5 e \left (\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {e \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}}dx}{2 b c \log (F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)}-\frac {7 e \left (\frac {(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)}-\frac {5 e \left (\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\int F^{c \left (a-\frac {b d}{e}\right )+\frac {b c (d+e x)}{e}}d\sqrt {d+e x}}{b c \log (F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)}-\frac {7 e \left (\frac {(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)}-\frac {5 e \left (\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\sqrt {\pi } \sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\right )}{2 b c \log (F)}\)

Input:

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

Output:

(F^(c*(a + b*x))*(d + e*x)^(7/2))/(b*c*Log[F]) - (7*e*((F^(c*(a + b*x))*(d 
 + e*x)^(5/2))/(b*c*Log[F]) - (5*e*((F^(c*(a + b*x))*(d + e*x)^(3/2))/(b*c 
*Log[F]) - (3*e*(-1/2*(Sqrt[e]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]* 
Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(b^(3/2)*c^(3/2)*Log[F]^(3/2 
)) + (F^(c*(a + b*x))*Sqrt[d + e*x])/(b*c*Log[F])))/(2*b*c*Log[F])))/(2*b* 
c*Log[F])))/(2*b*c*Log[F])
 

Defintions of rubi rules used

rule 2607

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), 
x] - Simp[d*(m/(f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ 
n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* 
m] &&  !TrueQ[$UseGamma]
 

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.11 \[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=-\frac {\frac {105 \, \sqrt {\pi } \sqrt {-\frac {b c \log \left (F\right )}{e}} e^{4} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right )}{F^{\frac {b c d - a c e}{e}}} + 2 \, {\left (105 \, b c e^{3} \log \left (F\right ) - 8 \, {\left (b^{4} c^{4} e^{3} x^{3} + 3 \, b^{4} c^{4} d e^{2} x^{2} + 3 \, b^{4} c^{4} d^{2} e x + b^{4} c^{4} d^{3}\right )} \log \left (F\right )^{4} + 28 \, {\left (b^{3} c^{3} e^{3} x^{2} + 2 \, b^{3} c^{3} d e^{2} x + b^{3} c^{3} d^{2} e\right )} \log \left (F\right )^{3} - 70 \, {\left (b^{2} c^{2} e^{3} x + b^{2} c^{2} d e^{2}\right )} \log \left (F\right )^{2}\right )} \sqrt {e x + d} F^{b c x + a c}}{16 \, b^{5} c^{5} \log \left (F\right )^{5}} \] Input:

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

Output:

-1/16*(105*sqrt(pi)*sqrt(-b*c*log(F)/e)*e^4*erf(sqrt(e*x + d)*sqrt(-b*c*lo 
g(F)/e))/F^((b*c*d - a*c*e)/e) + 2*(105*b*c*e^3*log(F) - 8*(b^4*c^4*e^3*x^ 
3 + 3*b^4*c^4*d*e^2*x^2 + 3*b^4*c^4*d^2*e*x + b^4*c^4*d^3)*log(F)^4 + 28*( 
b^3*c^3*e^3*x^2 + 2*b^3*c^3*d*e^2*x + b^3*c^3*d^2*e)*log(F)^3 - 70*(b^2*c^ 
2*e^3*x + b^2*c^2*d*e^2)*log(F)^2)*sqrt(e*x + d)*F^(b*c*x + a*c))/(b^5*c^5 
*log(F)^5)
 

Sympy [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (172) = 344\).

Time = 0.20 (sec) , antiderivative size = 1023, normalized size of antiderivative = 4.92 \[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/16*(16*sqrt(pi)*d^4*e*erf(-sqrt(-b*c*e*log(F))*sqrt(e*x + d)/e)*e^(-(b* 
c*d*log(F) - a*c*e*log(F))/e)/sqrt(-b*c*e*log(F)) - 32*d^3*(sqrt(pi)*(2*b* 
c*d*log(F) + e)*e*erf(-sqrt(-b*c*e*log(F))*sqrt(e*x + d)/e)*e^(-(b*c*d*log 
(F) - a*c*e*log(F))/e)/(sqrt(-b*c*e*log(F))*b*c*log(F)) + 2*sqrt(e*x + d)* 
e*e^(((e*x + d)*b*c*log(F) - b*c*d*log(F) + a*c*e*log(F))/e)/(b*c*log(F))) 
 + 24*d^2*(sqrt(pi)*(4*b^2*c^2*d^2*log(F)^2 + 4*b*c*d*e*log(F) + 3*e^2)*e* 
erf(-sqrt(-b*c*e*log(F))*sqrt(e*x + d)/e)*e^(-(b*c*d*log(F) - a*c*e*log(F) 
)/e)/(sqrt(-b*c*e*log(F))*b^2*c^2*log(F)^2) - 2*(2*(e*x + d)^(3/2)*b*c*e*l 
og(F) - 4*sqrt(e*x + d)*b*c*d*e*log(F) - 3*sqrt(e*x + d)*e^2)*e^(((e*x + d 
)*b*c*log(F) - b*c*d*log(F) + a*c*e*log(F))/e)/(b^2*c^2*log(F)^2)) - 8*d*( 
sqrt(pi)*(8*b^3*c^3*d^3*log(F)^3 + 12*b^2*c^2*d^2*e*log(F)^2 + 18*b*c*d*e^ 
2*log(F) + 15*e^3)*e*erf(-sqrt(-b*c*e*log(F))*sqrt(e*x + d)/e)*e^(-(b*c*d* 
log(F) - a*c*e*log(F))/e)/(sqrt(-b*c*e*log(F))*b^3*c^3*log(F)^3) + 2*(4*(e 
*x + d)^(5/2)*b^2*c^2*e*log(F)^2 - 12*(e*x + d)^(3/2)*b^2*c^2*d*e*log(F)^2 
 + 12*sqrt(e*x + d)*b^2*c^2*d^2*e*log(F)^2 - 10*(e*x + d)^(3/2)*b*c*e^2*lo 
g(F) + 18*sqrt(e*x + d)*b*c*d*e^2*log(F) + 15*sqrt(e*x + d)*e^3)*e^(((e*x 
+ d)*b*c*log(F) - b*c*d*log(F) + a*c*e*log(F))/e)/(b^3*c^3*log(F)^3)) + sq 
rt(pi)*(16*b^4*c^4*d^4*log(F)^4 + 32*b^3*c^3*d^3*e*log(F)^3 + 72*b^2*c^2*d 
^2*e^2*log(F)^2 + 120*b*c*d*e^3*log(F) + 105*e^4)*e*erf(-sqrt(-b*c*e*log(F 
))*sqrt(e*x + d)/e)*e^(-(b*c*d*log(F) - a*c*e*log(F))/e)/(sqrt(-b*c*e*l...
 

Mupad [F(-1)]

Timed out. \[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=\int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^{7/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int F^{c (a+b x)} (d+e x)^{7/2} \, dx=f^{a c} \left (\left (\int f^{b c x} \sqrt {e x +d}\, x^{3}d x \right ) e^{3}+3 \left (\int f^{b c x} \sqrt {e x +d}\, x^{2}d x \right ) d \,e^{2}+3 \left (\int f^{b c x} \sqrt {e x +d}\, x d x \right ) d^{2} e +\left (\int f^{b c x} \sqrt {e x +d}d x \right ) d^{3}\right ) \] Input:

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

Output:

f**(a*c)*(int(f**(b*c*x)*sqrt(d + e*x)*x**3,x)*e**3 + 3*int(f**(b*c*x)*sqr 
t(d + e*x)*x**2,x)*d*e**2 + 3*int(f**(b*c*x)*sqrt(d + e*x)*x,x)*d**2*e + i 
nt(f**(b*c*x)*sqrt(d + e*x),x)*d**3)