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

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

[Out]

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

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Rubi [A]  time = 0.0754626, 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 \[ -\frac{b^5 F^a \log ^5(F) \text{Gamma}\left (-5,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.35865, size = 32, normalized size = 1.03 \[ - \frac{F^{a} b^{5} \Gamma{\left (-5,- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}} \right )} \log{\left (F \right )}^{5}}{2 d} \]

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

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Mathematica [B]  time = 0.12449, size = 112, normalized size = 3.61 \[ \frac{F^a \left ((c+d x)^2 F^{\frac{b}{(c+d x)^2}} \left (b^4 \log ^4(F)+b^3 \log ^3(F) (c+d x)^2+2 b^2 \log ^2(F) (c+d x)^4+6 b \log (F) (c+d x)^6+24 (c+d x)^8\right )-b^5 \log ^5(F) \text{ExpIntegralEi}\left (\frac{b \log (F)}{(c+d x)^2}\right )\right )}{240 d} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(-(b^5*ExpIntegralEi[(b*Log[F])/(c + d*x)^2]*Log[F]^5) + F^(b/(c + d*x)^2)*
(c + d*x)^2*(24*(c + d*x)^8 + 6*b*(c + d*x)^6*Log[F] + 2*b^2*(c + d*x)^4*Log[F]^
2 + b^3*(c + d*x)^2*Log[F]^3 + b^4*Log[F]^4)))/(240*d)

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Maple [B]  time = 0.09, size = 961, normalized size = 31. \[ \text{result too large to display} \]

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]

1/10*F^a/d*F^(b/(d*x+c)^2)*c^10+1/10*F^a*d^9*F^(b/(d*x+c)^2)*x^10+F^a*F^(b/(d*x+
c)^2)*c^9*x+1/6*F^a*d^2*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^3*x^3+1/8*F^a*d*b^2*ln(F)^
2*F^(b/(d*x+c)^2)*c^4*x^2+1/60*F^a*d^2*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x^3+1/40*F^
a*d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2*x^2+1/5*F^a*d^6*b*ln(F)*F^(b/(d*x+c)^2)*c*x^
7+7/10*F^a*d^5*b*ln(F)*F^(b/(d*x+c)^2)*c^2*x^6+7/5*F^a*d^4*b*ln(F)*F^(b/(d*x+c)^
2)*c^3*x^5+7/4*F^a*d^3*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^4+7/5*F^a*d^2*b*ln(F)*F^(b/
(d*x+c)^2)*c^5*x^3+7/10*F^a*d*b*ln(F)*F^(b/(d*x+c)^2)*c^6*x^2+1/20*F^a*d^4*b^2*l
n(F)^2*F^(b/(d*x+c)^2)*c*x^5+1/8*F^a*d^3*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^2*x^4+F^a
*d^8*F^(b/(d*x+c)^2)*c*x^9+9/2*F^a*d^7*F^(b/(d*x+c)^2)*c^2*x^8+12*F^a*d^6*F^(b/(
d*x+c)^2)*c^3*x^7+21*F^a*d^5*F^(b/(d*x+c)^2)*c^4*x^6+126/5*F^a*d^4*F^(b/(d*x+c)^
2)*c^5*x^5+21*F^a*d^3*F^(b/(d*x+c)^2)*c^6*x^4+12*F^a*d^2*F^(b/(d*x+c)^2)*c^7*x^3
+9/2*F^a*d*F^(b/(d*x+c)^2)*c^8*x^2+1/240*F^a/d*b^5*ln(F)^5*Ei(1,-b*ln(F)/(d*x+c)
^2)+1/20*F^a*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^5*x+1/60*F^a*b^3*ln(F)^3*F^(b/(d*x+c)
^2)*c^3*x+1/120*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c*x+1/5*F^a*b*ln(F)*F^(b/(d*x+c)
^2)*c^7*x+1/240*F^a/d*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c^2+1/40*F^a/d*b*ln(F)*F^(b/(d
*x+c)^2)*c^8+1/120*F^a/d*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^6+1/120*F^a*d^5*b^2*ln(F)
^2*F^(b/(d*x+c)^2)*x^6+1/240*F^a*d^3*b^3*ln(F)^3*F^(b/(d*x+c)^2)*x^4+1/240*F^a*d
*b^4*ln(F)^4*F^(b/(d*x+c)^2)*x^2+1/40*F^a*d^7*b*ln(F)*F^(b/(d*x+c)^2)*x^8+1/240*
F^a/d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{240} \,{\left (24 \, F^{a} d^{9} x^{10} + 240 \, F^{a} c d^{8} x^{9} + 6 \,{\left (180 \, F^{a} c^{2} d^{7} + F^{a} b d^{7} \log \left (F\right )\right )} x^{8} + 48 \,{\left (60 \, F^{a} c^{3} d^{6} + F^{a} b c d^{6} \log \left (F\right )\right )} x^{7} + 2 \,{\left (2520 \, F^{a} c^{4} d^{5} + 84 \, F^{a} b c^{2} d^{5} \log \left (F\right ) + F^{a} b^{2} d^{5} \log \left (F\right )^{2}\right )} x^{6} + 12 \,{\left (504 \, F^{a} c^{5} d^{4} + 28 \, F^{a} b c^{3} d^{4} \log \left (F\right ) + F^{a} b^{2} c d^{4} \log \left (F\right )^{2}\right )} x^{5} +{\left (5040 \, F^{a} c^{6} d^{3} + 420 \, F^{a} b c^{4} d^{3} \log \left (F\right ) + 30 \, F^{a} b^{2} c^{2} d^{3} \log \left (F\right )^{2} + F^{a} b^{3} d^{3} \log \left (F\right )^{3}\right )} x^{4} + 4 \,{\left (720 \, F^{a} c^{7} d^{2} + 84 \, F^{a} b c^{5} d^{2} \log \left (F\right ) + 10 \, F^{a} b^{2} c^{3} d^{2} \log \left (F\right )^{2} + F^{a} b^{3} c d^{2} \log \left (F\right )^{3}\right )} x^{3} +{\left (1080 \, F^{a} c^{8} d + 168 \, F^{a} b c^{6} d \log \left (F\right ) + 30 \, F^{a} b^{2} c^{4} d \log \left (F\right )^{2} + 6 \, F^{a} b^{3} c^{2} d \log \left (F\right )^{3} + F^{a} b^{4} d \log \left (F\right )^{4}\right )} x^{2} + 2 \,{\left (120 \, F^{a} c^{9} + 24 \, F^{a} b c^{7} \log \left (F\right ) + 6 \, F^{a} b^{2} c^{5} \log \left (F\right )^{2} + 2 \, F^{a} b^{3} c^{3} \log \left (F\right )^{3} + F^{a} b^{4} c \log \left (F\right )^{4}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac{{\left (F^{a} b^{5} d^{2} x^{2} \log \left (F\right )^{5} + 2 \, F^{a} b^{5} c d x \log \left (F\right )^{5} - 24 \, F^{a} b c^{10} \log \left (F\right ) - 6 \, F^{a} b^{2} c^{8} \log \left (F\right )^{2} - 2 \, F^{a} b^{3} c^{6} \log \left (F\right )^{3} - F^{a} b^{4} c^{4} \log \left (F\right )^{4}\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{120 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(24*F^a*d^9*x^10 + 240*F^a*c*d^8*x^9 + 6*(180*F^a*c^2*d^7 + F^a*b*d^7*log(
F))*x^8 + 48*(60*F^a*c^3*d^6 + F^a*b*c*d^6*log(F))*x^7 + 2*(2520*F^a*c^4*d^5 + 8
4*F^a*b*c^2*d^5*log(F) + F^a*b^2*d^5*log(F)^2)*x^6 + 12*(504*F^a*c^5*d^4 + 28*F^
a*b*c^3*d^4*log(F) + F^a*b^2*c*d^4*log(F)^2)*x^5 + (5040*F^a*c^6*d^3 + 420*F^a*b
*c^4*d^3*log(F) + 30*F^a*b^2*c^2*d^3*log(F)^2 + F^a*b^3*d^3*log(F)^3)*x^4 + 4*(7
20*F^a*c^7*d^2 + 84*F^a*b*c^5*d^2*log(F) + 10*F^a*b^2*c^3*d^2*log(F)^2 + F^a*b^3
*c*d^2*log(F)^3)*x^3 + (1080*F^a*c^8*d + 168*F^a*b*c^6*d*log(F) + 30*F^a*b^2*c^4
*d*log(F)^2 + 6*F^a*b^3*c^2*d*log(F)^3 + F^a*b^4*d*log(F)^4)*x^2 + 2*(120*F^a*c^
9 + 24*F^a*b*c^7*log(F) + 6*F^a*b^2*c^5*log(F)^2 + 2*F^a*b^3*c^3*log(F)^3 + F^a*
b^4*c*log(F)^4)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2)) + integrate(1/120*(F^a*b^5*d^
2*x^2*log(F)^5 + 2*F^a*b^5*c*d*x*log(F)^5 - 24*F^a*b*c^10*log(F) - 6*F^a*b^2*c^8
*log(F)^2 - 2*F^a*b^3*c^6*log(F)^3 - F^a*b^4*c^4*log(F)^4)*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)

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Fricas [A]  time = 0.260159, size = 628, normalized size = 20.26 \[ -\frac{F^{a} b^{5}{\rm Ei}\left (\frac{b \log \left (F\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) \log \left (F\right )^{5} -{\left (24 \, d^{10} x^{10} + 240 \, c d^{9} x^{9} + 1080 \, c^{2} d^{8} x^{8} + 2880 \, c^{3} d^{7} x^{7} + 5040 \, c^{4} d^{6} x^{6} + 6048 \, c^{5} d^{5} x^{5} + 5040 \, c^{6} d^{4} x^{4} + 2880 \, c^{7} d^{3} x^{3} + 1080 \, c^{8} d^{2} x^{2} + 240 \, c^{9} d x + 24 \, c^{10} +{\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \left (F\right )^{4} +{\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \left (F\right )^{3} + 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 6 \,{\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/240*(F^a*b^5*Ei(b*log(F)/(d^2*x^2 + 2*c*d*x + c^2))*log(F)^5 - (24*d^10*x^10
+ 240*c*d^9*x^9 + 1080*c^2*d^8*x^8 + 2880*c^3*d^7*x^7 + 5040*c^4*d^6*x^6 + 6048*
c^5*d^5*x^5 + 5040*c^6*d^4*x^4 + 2880*c^7*d^3*x^3 + 1080*c^8*d^2*x^2 + 240*c^9*d
*x + 24*c^10 + (b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)*log(F)^4 + (b^3*d^4*x^4 + 4
*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*log(F)^3 + 2*(b^2*
d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4
*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 + 6*(b*d^8*x^8 + 8*b*c*d^7*x^7 + 28
*b*c^2*d^6*x^6 + 56*b*c^3*d^5*x^5 + 70*b*c^4*d^4*x^4 + 56*b*c^5*d^3*x^3 + 28*b*c
^6*d^2*x^2 + 8*b*c^7*d*x + b*c^8)*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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{9} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^9*F^(a + b/(d*x + c)^2), x)