3.256 \(\int F^{a+b (c+d x)^2} (c+d x)^9 \, dx\)

Optimal. Leaf size=31 \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^2\right )}{2 b^5 d \log ^5(F)} \]

[Out]

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

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Rubi [A]  time = 0.110374, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^2\right )}{2 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.87305, size = 29, normalized size = 0.94 \[ \frac{F^{a} \Gamma{\left (5,- b \left (c + d x\right )^{2} \log{\left (F \right )} \right )}}{2 b^{5} d \log{\left (F \right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**9,x)

[Out]

F**a*Gamma(5, -b*(c + d*x)**2*log(F))/(2*b**5*d*log(F)**5)

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Mathematica [B]  time = 0.0637502, size = 88, normalized size = 2.84 \[ \frac{F^{a+b (c+d x)^2} \left (b^4 \log ^4(F) (c+d x)^8-4 b^3 \log ^3(F) (c+d x)^6+12 b^2 \log ^2(F) (c+d x)^4-24 b \log (F) (c+d x)^2+24\right )}{2 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^9,x]

[Out]

(F^(a + b*(c + d*x)^2)*(24 - 24*b*(c + d*x)^2*Log[F] + 12*b^2*(c + d*x)^4*Log[F]
^2 - 4*b^3*(c + d*x)^6*Log[F]^3 + b^4*(c + d*x)^8*Log[F]^4))/(2*b^5*d*Log[F]^5)

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Maple [B]  time = 0.022, size = 396, normalized size = 12.8 \[{\frac{ \left ( 24-24\,\ln \left ( F \right ) b{c}^{2}-24\,\ln \left ( F \right ) b{d}^{2}{x}^{2}+ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{8}-48\,\ln \left ( F \right ) bcdx+{d}^{8}{x}^{8}{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}-4\,{d}^{6}{x}^{6}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+12\,{d}^{4}{x}^{4}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+12\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{4}-4\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{6}-80\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{3}{x}^{3}-60\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{4}{d}^{2}{x}^{2}-24\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{5}dx+48\,{d}^{3}c{x}^{3}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}+72\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{d}^{2}{x}^{2}+48\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{3}dx+8\,c{d}^{7}{x}^{7}{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}+28\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{2}{d}^{6}{x}^{6}+56\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{3}{d}^{5}{x}^{5}+70\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{4}{d}^{4}{x}^{4}+56\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{5}{d}^{3}{x}^{3}+28\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{6}{d}^{2}{x}^{2}+8\, \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}{c}^{7}dx-24\,c{d}^{5}{x}^{5}{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}-60\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{2}{d}^{4}{x}^{4} \right ){F}^{b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a}}{2\,{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*(d*x+c)^2)*(d*x+c)^9,x)

[Out]

1/2*(24-24*ln(F)*b*c^2-24*ln(F)*b*d^2*x^2+ln(F)^4*b^4*c^8-48*ln(F)*b*c*d*x+d^8*x
^8*b^4*ln(F)^4-4*d^6*x^6*b^3*ln(F)^3+12*d^4*x^4*b^2*ln(F)^2+12*ln(F)^2*b^2*c^4-4
*ln(F)^3*b^3*c^6-80*ln(F)^3*b^3*c^3*d^3*x^3-60*ln(F)^3*b^3*c^4*d^2*x^2-24*ln(F)^
3*b^3*c^5*d*x+48*d^3*c*x^3*b^2*ln(F)^2+72*ln(F)^2*b^2*c^2*d^2*x^2+48*ln(F)^2*b^2
*c^3*d*x+8*c*d^7*x^7*b^4*ln(F)^4+28*ln(F)^4*b^4*c^2*d^6*x^6+56*ln(F)^4*b^4*c^3*d
^5*x^5+70*ln(F)^4*b^4*c^4*d^4*x^4+56*ln(F)^4*b^4*c^5*d^3*x^3+28*ln(F)^4*b^4*c^6*
d^2*x^2+8*ln(F)^4*b^4*c^7*d*x-24*c*d^5*x^5*b^3*ln(F)^3-60*ln(F)^3*b^3*c^2*d^4*x^
4)*F^(b*d^2*x^2+2*b*c*d*x+b*c^2+a)/b^5/ln(F)^5/d

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Maxima [A]  time = 1.48098, size = 6249, normalized size = 201.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^9*F^((d*x + c)^2*b + a),x, algorithm="maxima")

[Out]

-9/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b*c*d*(erf(sqrt(-(b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)/((b*d^2*log(F))^(3/2)*sqrt(-(b*d^
2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - b*d^2*e^((b*d^2*x*log(F) + b*c*d
*log(F))^2/(b*d^2*log(F)))*log(F)/(b*d^2*log(F))^(3/2))*F^(b*c^2 + a)*c^8*d/(sqr
t(b*d^2*log(F))*F^(b*c^2)) + 18*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^2*c^
2*d^2*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^
2/((b*d^2*log(F))^(5/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F))))
- 2*b^2*c*d^3*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/(b*
d^2*log(F))^(5/2) - (b*d^2*x*log(F) + b*c*d*log(F))^3*gamma(3/2, -(b*d^2*x*log(F
) + b*c*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(5/2)*(-(b*d^2*x*log(F) + b*
c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*c^7*d^2/(sqrt(b*d^2*log(F))*
F^(b*c^2)) - 42*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^3*c^3*d^3*(erf(sqrt(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^3/((b*d^2*log(F)
)^(7/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 3*b^3*c^2*d^4
*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F))^(7
/2) + b^2*d^4*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)
^2/(b*d^2*log(F))^(7/2) - 3*(b*d^2*x*log(F) + b*c*d*log(F))^3*b*c*d*gamma(3/2, -
(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(7/2)*(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)))*F^(b*c^2 + a)*c^6*d^3
/(sqrt(b*d^2*log(F))*F^(b*c^2)) + 63*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b
^4*c^4*d^4*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*lo
g(F)^4/((b*d^2*log(F))^(9/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(
F)))) - 4*b^4*c^3*d^5*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F
)^4/(b*d^2*log(F))^(9/2) + 4*b^3*c*d^5*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))
^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log(F))^(9/2) - 6*(b*d^2*x*log(F) + b*c*d*log
(F))^3*b^2*c^2*d^2*gamma(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))
*log(F)^2/((b*d^2*log(F))^(9/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F
)))^(3/2)) - (b*d^2*x*log(F) + b*c*d*log(F))^5*gamma(5/2, -(b*d^2*x*log(F) + b*c
*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F))^(9/2)*(-(b*d^2*x*log(F) + b*c*d*log
(F))^2/(b*d^2*log(F)))^(5/2)))*F^(b*c^2 + a)*c^5*d^4/(sqrt(b*d^2*log(F))*F^(b*c^
2)) - 63*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^5*c^5*d^5*(erf(sqrt(-(b*d^2
*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^5/((b*d^2*log(F))^(11/2
)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 5*b^5*c^4*d^6*e^((b
*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/(b*d^2*log(F))^(11/2) +
10*b^4*c^2*d^6*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(
F)^4/(b*d^2*log(F))^(11/2) - 10*(b*d^2*x*log(F) + b*c*d*log(F))^3*b^3*c^3*d^3*ga
mma(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/((b*d^2*log
(F))^(11/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)) - b^3*d^6
*gamma(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/(b*d^2*log
(F))^(11/2) - 5*(b*d^2*x*log(F) + b*c*d*log(F))^5*b*c*d*gamma(5/2, -(b*d^2*x*log
(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(11/2)*(-(b*d^2*x*l
og(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)))*F^(b*c^2 + a)*c^4*d^5/(sqrt(b*d^
2*log(F))*F^(b*c^2)) + 42*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^6*c^6*d^6*
(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^6/((b*
d^2*log(F))^(13/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 6*
b^6*c^5*d^7*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2
*log(F))^(13/2) + 20*b^5*c^3*d^7*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*
d^2*log(F)))*log(F)^5/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x*log(F) + b*c*d*log(F))
^3*b^4*c^4*d^4*gamma(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log
(F)^4/((b*d^2*log(F))^(13/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))
^(3/2)) - 6*b^4*c*d^7*gamma(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F))
)*log(F)^4/(b*d^2*log(F))^(13/2) - 15*(b*d^2*x*log(F) + b*c*d*log(F))^5*b^2*c^2*
d^2*gamma(5/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/((b*d
^2*log(F))^(13/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)) - (
b*d^2*x*log(F) + b*c*d*log(F))^7*gamma(7/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(
b*d^2*log(F)))/((b*d^2*log(F))^(13/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2
*log(F)))^(7/2)))*F^(b*c^2 + a)*c^3*d^6/(sqrt(b*d^2*log(F))*F^(b*c^2)) - 18*(sqr
t(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^7*c^7*d^7*(erf(sqrt(-(b*d^2*x*log(F) + b
*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^7/((b*d^2*log(F))^(15/2)*sqrt(-(b*d^
2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 7*b^7*c^6*d^8*e^((b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^7/(b*d^2*log(F))^(15/2) + 35*b^6*c^4*d
^8*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2*l
og(F))^(15/2) - 21*(b*d^2*x*log(F) + b*c*d*log(F))^3*b^5*c^5*d^5*gamma(3/2, -(b*
d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/((b*d^2*log(F))^(15/2)*(
-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)) - 21*b^5*c^2*d^8*gamma
(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/(b*d^2*log(F))^(
15/2) + b^4*d^8*gamma(4, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(
F)^4/(b*d^2*log(F))^(15/2) - 35*(b*d^2*x*log(F) + b*c*d*log(F))^5*b^3*c^3*d^3*ga
mma(5/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/((b*d^2*log
(F))^(15/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)) - 7*(b*d^
2*x*log(F) + b*c*d*log(F))^7*b*c*d*gamma(7/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2
/(b*d^2*log(F)))*log(F)/((b*d^2*log(F))^(15/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))
^2/(b*d^2*log(F)))^(7/2)))*F^(b*c^2 + a)*c^2*d^7/(sqrt(b*d^2*log(F))*F^(b*c^2))
+ 9/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d*log(F))*b^8*c^8*d^8*(erf(sqrt(-(b*d^2*x*
log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 1)*log(F)^8/((b*d^2*log(F))^(17/2)*s
qrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))) - 8*b^8*c^7*d^9*e^((b*d^
2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^8/(b*d^2*log(F))^(17/2) + 56
*b^7*c^5*d^9*gamma(2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^
7/(b*d^2*log(F))^(17/2) - 28*(b*d^2*x*log(F) + b*c*d*log(F))^3*b^6*c^6*d^6*gamma
(3/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/((b*d^2*log(F)
)^(17/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(3/2)) - 56*b^6*c^3
*d^9*gamma(3, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2
*log(F))^(17/2) + 8*b^5*c*d^9*gamma(4, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2
*log(F)))*log(F)^5/(b*d^2*log(F))^(17/2) - 70*(b*d^2*x*log(F) + b*c*d*log(F))^5*
b^4*c^4*d^4*gamma(5/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)
^4/((b*d^2*log(F))^(17/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5
/2)) - 28*(b*d^2*x*log(F) + b*c*d*log(F))^7*b^2*c^2*d^2*gamma(7/2, -(b*d^2*x*log
(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^2/((b*d^2*log(F))^(17/2)*(-(b*d^2*x
*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(7/2)) - (b*d^2*x*log(F) + b*c*d*log(F
))^9*gamma(9/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))/((b*d^2*log(F
))^(17/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(9/2)))*F^(b*c^2 +
a)*c*d^8/(sqrt(b*d^2*log(F))*F^(b*c^2)) - 1/2*(sqrt(pi)*(b*d^2*x*log(F) + b*c*d
*log(F))*b^9*c^9*d^9*(erf(sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F))
)) - 1)*log(F)^9/((b*d^2*log(F))^(19/2)*sqrt(-(b*d^2*x*log(F) + b*c*d*log(F))^2/
(b*d^2*log(F)))) - 9*b^9*c^8*d^10*e^((b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*lo
g(F)))*log(F)^9/(b*d^2*log(F))^(19/2) + 84*b^8*c^6*d^10*gamma(2, -(b*d^2*x*log(F
) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^8/(b*d^2*log(F))^(19/2) - 36*(b*d^2*x
*log(F) + b*c*d*log(F))^3*b^7*c^7*d^7*gamma(3/2, -(b*d^2*x*log(F) + b*c*d*log(F)
)^2/(b*d^2*log(F)))*log(F)^7/((b*d^2*log(F))^(19/2)*(-(b*d^2*x*log(F) + b*c*d*lo
g(F))^2/(b*d^2*log(F)))^(3/2)) - 126*b^7*c^4*d^10*gamma(3, -(b*d^2*x*log(F) + b*
c*d*log(F))^2/(b*d^2*log(F)))*log(F)^7/(b*d^2*log(F))^(19/2) + 36*b^6*c^2*d^10*g
amma(4, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^6/(b*d^2*log(F
))^(19/2) - 126*(b*d^2*x*log(F) + b*c*d*log(F))^5*b^5*c^5*d^5*gamma(5/2, -(b*d^2
*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/((b*d^2*log(F))^(19/2)*(-(b
*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(5/2)) - b^5*d^10*gamma(5, -(b*d
^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)^5/(b*d^2*log(F))^(19/2) - 8
4*(b*d^2*x*log(F) + b*c*d*log(F))^7*b^3*c^3*d^3*gamma(7/2, -(b*d^2*x*log(F) + b*
c*d*log(F))^2/(b*d^2*log(F)))*log(F)^3/((b*d^2*log(F))^(19/2)*(-(b*d^2*x*log(F)
+ b*c*d*log(F))^2/(b*d^2*log(F)))^(7/2)) - 9*(b*d^2*x*log(F) + b*c*d*log(F))^9*b
*c*d*gamma(9/2, -(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))*log(F)/((b*d^
2*log(F))^(19/2)*(-(b*d^2*x*log(F) + b*c*d*log(F))^2/(b*d^2*log(F)))^(9/2)))*F^(
b*c^2 + a)*d^9/(sqrt(b*d^2*log(F))*F^(b*c^2)) + 1/2*sqrt(pi)*F^(b*c^2 + a)*c^9*e
rf(sqrt(-b*log(F))*d*x - b*c*log(F)/sqrt(-b*log(F)))/(sqrt(-b*log(F))*F^(b*c^2)*
d)

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Fricas [A]  time = 0.25177, size = 437, normalized size = 14.1 \[ \frac{{\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} - 4 \,{\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} - 24 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{5} d \log \left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^9*F^((d*x + c)^2*b + a),x, algorithm="fricas")

[Out]

1/2*((b^4*d^8*x^8 + 8*b^4*c*d^7*x^7 + 28*b^4*c^2*d^6*x^6 + 56*b^4*c^3*d^5*x^5 +
70*b^4*c^4*d^4*x^4 + 56*b^4*c^5*d^3*x^3 + 28*b^4*c^6*d^2*x^2 + 8*b^4*c^7*d*x + b
^4*c^8)*log(F)^4 - 4*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + 15*b^3*c^2*d^4*x^4 + 20*b^
3*c^3*d^3*x^3 + 15*b^3*c^4*d^2*x^2 + 6*b^3*c^5*d*x + b^3*c^6)*log(F)^3 + 12*(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 - 24*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) + 24)*F^(b*d^2*x^2 + 2*b*c*d*x +
b*c^2 + a)/(b^5*d*log(F)^5)

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Sympy [A]  time = 0.997536, size = 558, normalized size = 18. \[ \begin{cases} \frac{F^{a + b \left (c + d x\right )^{2}} \left (b^{4} c^{8} \log{\left (F \right )}^{4} + 8 b^{4} c^{7} d x \log{\left (F \right )}^{4} + 28 b^{4} c^{6} d^{2} x^{2} \log{\left (F \right )}^{4} + 56 b^{4} c^{5} d^{3} x^{3} \log{\left (F \right )}^{4} + 70 b^{4} c^{4} d^{4} x^{4} \log{\left (F \right )}^{4} + 56 b^{4} c^{3} d^{5} x^{5} \log{\left (F \right )}^{4} + 28 b^{4} c^{2} d^{6} x^{6} \log{\left (F \right )}^{4} + 8 b^{4} c d^{7} x^{7} \log{\left (F \right )}^{4} + b^{4} d^{8} x^{8} \log{\left (F \right )}^{4} - 4 b^{3} c^{6} \log{\left (F \right )}^{3} - 24 b^{3} c^{5} d x \log{\left (F \right )}^{3} - 60 b^{3} c^{4} d^{2} x^{2} \log{\left (F \right )}^{3} - 80 b^{3} c^{3} d^{3} x^{3} \log{\left (F \right )}^{3} - 60 b^{3} c^{2} d^{4} x^{4} \log{\left (F \right )}^{3} - 24 b^{3} c d^{5} x^{5} \log{\left (F \right )}^{3} - 4 b^{3} d^{6} x^{6} \log{\left (F \right )}^{3} + 12 b^{2} c^{4} \log{\left (F \right )}^{2} + 48 b^{2} c^{3} d x \log{\left (F \right )}^{2} + 72 b^{2} c^{2} d^{2} x^{2} \log{\left (F \right )}^{2} + 48 b^{2} c d^{3} x^{3} \log{\left (F \right )}^{2} + 12 b^{2} d^{4} x^{4} \log{\left (F \right )}^{2} - 24 b c^{2} \log{\left (F \right )} - 48 b c d x \log{\left (F \right )} - 24 b d^{2} x^{2} \log{\left (F \right )} + 24\right )}{2 b^{5} d \log{\left (F \right )}^{5}} & \text{for}\: 2 b^{5} d \log{\left (F \right )}^{5} \neq 0 \\c^{9} x + \frac{9 c^{8} d x^{2}}{2} + 12 c^{7} d^{2} x^{3} + 21 c^{6} d^{3} x^{4} + \frac{126 c^{5} d^{4} x^{5}}{5} + 21 c^{4} d^{5} x^{6} + 12 c^{3} d^{6} x^{7} + \frac{9 c^{2} d^{7} x^{8}}{2} + c d^{8} x^{9} + \frac{d^{9} x^{10}}{10} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**9,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**2)*(b**4*c**8*log(F)**4 + 8*b**4*c**7*d*x*log(F)
**4 + 28*b**4*c**6*d**2*x**2*log(F)**4 + 56*b**4*c**5*d**3*x**3*log(F)**4 + 70*b
**4*c**4*d**4*x**4*log(F)**4 + 56*b**4*c**3*d**5*x**5*log(F)**4 + 28*b**4*c**2*d
**6*x**6*log(F)**4 + 8*b**4*c*d**7*x**7*log(F)**4 + b**4*d**8*x**8*log(F)**4 - 4
*b**3*c**6*log(F)**3 - 24*b**3*c**5*d*x*log(F)**3 - 60*b**3*c**4*d**2*x**2*log(F
)**3 - 80*b**3*c**3*d**3*x**3*log(F)**3 - 60*b**3*c**2*d**4*x**4*log(F)**3 - 24*
b**3*c*d**5*x**5*log(F)**3 - 4*b**3*d**6*x**6*log(F)**3 + 12*b**2*c**4*log(F)**2
+ 48*b**2*c**3*d*x*log(F)**2 + 72*b**2*c**2*d**2*x**2*log(F)**2 + 48*b**2*c*d**
3*x**3*log(F)**2 + 12*b**2*d**4*x**4*log(F)**2 - 24*b*c**2*log(F) - 48*b*c*d*x*l
og(F) - 24*b*d**2*x**2*log(F) + 24)/(2*b**5*d*log(F)**5), Ne(2*b**5*d*log(F)**5,
0)), (c**9*x + 9*c**8*d*x**2/2 + 12*c**7*d**2*x**3 + 21*c**6*d**3*x**4 + 126*c*
*5*d**4*x**5/5 + 21*c**4*d**5*x**6 + 12*c**3*d**6*x**7 + 9*c**2*d**7*x**8/2 + c*
d**8*x**9 + d**9*x**10/10, True))

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GIAC/XCAS [A]  time = 0.226636, size = 167, normalized size = 5.39 \[ \frac{{\left (b^{4} d^{8}{\left (x + \frac{c}{d}\right )}^{8}{\rm ln}\left (F\right )^{4} - 4 \, b^{3} d^{6}{\left (x + \frac{c}{d}\right )}^{6}{\rm ln}\left (F\right )^{3} + 12 \, b^{2} d^{4}{\left (x + \frac{c}{d}\right )}^{4}{\rm ln}\left (F\right )^{2} - 24 \, b d^{2}{\left (x + \frac{c}{d}\right )}^{2}{\rm ln}\left (F\right ) + 24\right )} e^{\left (b d^{2} x^{2}{\rm ln}\left (F\right ) + 2 \, b c d x{\rm ln}\left (F\right ) + b c^{2}{\rm ln}\left (F\right ) + a{\rm ln}\left (F\right )\right )}}{2 \, b^{5} d{\rm ln}\left (F\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^9*F^((d*x + c)^2*b + a),x, algorithm="giac")

[Out]

1/2*(b^4*d^8*(x + c/d)^8*ln(F)^4 - 4*b^3*d^6*(x + c/d)^6*ln(F)^3 + 12*b^2*d^4*(x
+ c/d)^4*ln(F)^2 - 24*b*d^2*(x + c/d)^2*ln(F) + 24)*e^(b*d^2*x^2*ln(F) + 2*b*c*
d*x*ln(F) + b*c^2*ln(F) + a*ln(F))/(b^5*d*ln(F)^5)