3.328 \(\int F^{a+\frac{b}{(c+d x)^2}} (c+d x)^8 \, dx\)

Optimal. Leaf size=49 \[ \frac{F^a (c+d x)^9 \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0788528, antiderivative size = 49, 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 \[ \frac{F^a (c+d x)^9 \left (-\frac{b \log (F)}{(c+d x)^2}\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-\frac{b \log (F)}{(c+d x)^2}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b/(c + d*x)^2)*(c + d*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.72121, size = 48, normalized size = 0.98 \[ \frac{F^{a} \left (- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}}\right )^{\frac{9}{2}} \left (c + d x\right )^{9} \Gamma{\left (- \frac{9}{2},- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{2}} \right )}}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b/(d*x+c)**2)*(d*x+c)**8,x)

[Out]

F**a*(-b*log(F)/(c + d*x)**2)**(9/2)*(c + d*x)**9*Gamma(-9/2, -b*log(F)/(c + d*x
)**2)/(2*d)

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Mathematica [B]  time = 0.226479, size = 129, normalized size = 2.63 \[ \frac{F^a \left ((c+d x) F^{\frac{b}{(c+d x)^2}} \left (16 b^4 \log ^4(F)+8 b^3 \log ^3(F) (c+d x)^2+12 b^2 \log ^2(F) (c+d x)^4+30 b \log (F) (c+d x)^6+105 (c+d x)^8\right )-16 \sqrt{\pi } b^{9/2} \log ^{\frac{9}{2}}(F) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )\right )}{945 d} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b/(c + d*x)^2)*(c + d*x)^8,x]

[Out]

(F^a*(-16*b^(9/2)*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[Log[F]])/(c + d*x)]*Log[F]^(9/2) +
F^(b/(c + d*x)^2)*(c + d*x)*(105*(c + d*x)^8 + 30*b*(c + d*x)^6*Log[F] + 12*b^2
*(c + d*x)^4*Log[F]^2 + 8*b^3*(c + d*x)^2*Log[F]^3 + 16*b^4*Log[F]^4)))/(945*d)

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Maple [B]  time = 0.079, size = 826, normalized size = 16.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b/(d*x+c)^2)*(d*x+c)^8,x)

[Out]

1/9*F^a/d*F^(b/(d*x+c)^2)*c^9+1/9*F^a*d^8*F^(b/(d*x+c)^2)*x^9+F^a*F^(b/(d*x+c)^2
)*c^8*x+8/315*F^a*d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c*x^2-16/945*F^a/d*b^5*ln(F)^5*P
i^(1/2)/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)/(d*x+c))+10/9*F^a*d^3*b*ln(F)*F^(b
/(d*x+c)^2)*c^3*x^4+10/9*F^a*d^2*b*ln(F)*F^(b/(d*x+c)^2)*c^4*x^3+2/3*F^a*d*b*ln(
F)*F^(b/(d*x+c)^2)*c^5*x^2+4/63*F^a*d^3*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c*x^4+8/63*F
^a*d^2*b^2*ln(F)^2*F^(b/(d*x+c)^2)*c^2*x^3+8/63*F^a*d*b^2*ln(F)^2*F^(b/(d*x+c)^2
)*c^3*x^2+2/9*F^a*d^5*b*ln(F)*F^(b/(d*x+c)^2)*c*x^6+2/3*F^a*d^4*b*ln(F)*F^(b/(d*
x+c)^2)*c^2*x^5+16/945*F^a*b^4*ln(F)^4*F^(b/(d*x+c)^2)*x+F^a*d^7*F^(b/(d*x+c)^2)
*c*x^8+4*F^a*d^6*F^(b/(d*x+c)^2)*c^2*x^7+28/3*F^a*d^5*F^(b/(d*x+c)^2)*c^3*x^6+14
*F^a*d^4*F^(b/(d*x+c)^2)*c^4*x^5+14*F^a*d^3*F^(b/(d*x+c)^2)*c^5*x^4+28/3*F^a*d^2
*F^(b/(d*x+c)^2)*c^6*x^3+4*F^a*d*F^(b/(d*x+c)^2)*c^7*x^2+4/63*F^a*b^2*ln(F)^2*F^
(b/(d*x+c)^2)*c^4*x+8/315*F^a*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^2*x+2/9*F^a*b*ln(F)*
F^(b/(d*x+c)^2)*c^6*x+2/63*F^a/d*b*ln(F)*F^(b/(d*x+c)^2)*c^7+4/315*F^a/d*b^2*ln(
F)^2*F^(b/(d*x+c)^2)*c^5+8/945*F^a/d*b^3*ln(F)^3*F^(b/(d*x+c)^2)*c^3+16/945*F^a/
d*b^4*ln(F)^4*F^(b/(d*x+c)^2)*c+2/63*F^a*d^6*b*ln(F)*F^(b/(d*x+c)^2)*x^7+4/315*F
^a*d^4*b^2*ln(F)^2*F^(b/(d*x+c)^2)*x^5+8/945*F^a*d^2*b^3*ln(F)^3*F^(b/(d*x+c)^2)
*x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{945} \,{\left (105 \, F^{a} d^{8} x^{9} + 945 \, F^{a} c d^{7} x^{8} + 30 \,{\left (126 \, F^{a} c^{2} d^{6} + F^{a} b d^{6} \log \left (F\right )\right )} x^{7} + 210 \,{\left (42 \, F^{a} c^{3} d^{5} + F^{a} b c d^{5} \log \left (F\right )\right )} x^{6} + 6 \,{\left (2205 \, F^{a} c^{4} d^{4} + 105 \, F^{a} b c^{2} d^{4} \log \left (F\right ) + 2 \, F^{a} b^{2} d^{4} \log \left (F\right )^{2}\right )} x^{5} + 30 \,{\left (441 \, F^{a} c^{5} d^{3} + 35 \, F^{a} b c^{3} d^{3} \log \left (F\right ) + 2 \, F^{a} b^{2} c d^{3} \log \left (F\right )^{2}\right )} x^{4} + 2 \,{\left (4410 \, F^{a} c^{6} d^{2} + 525 \, F^{a} b c^{4} d^{2} \log \left (F\right ) + 60 \, F^{a} b^{2} c^{2} d^{2} \log \left (F\right )^{2} + 4 \, F^{a} b^{3} d^{2} \log \left (F\right )^{3}\right )} x^{3} + 6 \,{\left (630 \, F^{a} c^{7} d + 105 \, F^{a} b c^{5} d \log \left (F\right ) + 20 \, F^{a} b^{2} c^{3} d \log \left (F\right )^{2} + 4 \, F^{a} b^{3} c d \log \left (F\right )^{3}\right )} x^{2} +{\left (945 \, F^{a} c^{8} + 210 \, F^{a} b c^{6} \log \left (F\right ) + 60 \, F^{a} b^{2} c^{4} \log \left (F\right )^{2} + 24 \, F^{a} b^{3} c^{2} \log \left (F\right )^{3} + 16 \, F^{a} b^{4} \log \left (F\right )^{4}\right )} x\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \int \frac{2 \,{\left (16 \, F^{a} b^{5} d x \log \left (F\right )^{5} - 105 \, F^{a} b c^{9} \log \left (F\right ) - 30 \, F^{a} b^{2} c^{7} \log \left (F\right )^{2} - 12 \, F^{a} b^{3} c^{5} \log \left (F\right )^{3} - 8 \, F^{a} b^{4} c^{3} \log \left (F\right )^{4}\right )} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{945 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^8*F^(a + b/(d*x + c)^2),x, algorithm="maxima")

[Out]

1/945*(105*F^a*d^8*x^9 + 945*F^a*c*d^7*x^8 + 30*(126*F^a*c^2*d^6 + F^a*b*d^6*log
(F))*x^7 + 210*(42*F^a*c^3*d^5 + F^a*b*c*d^5*log(F))*x^6 + 6*(2205*F^a*c^4*d^4 +
105*F^a*b*c^2*d^4*log(F) + 2*F^a*b^2*d^4*log(F)^2)*x^5 + 30*(441*F^a*c^5*d^3 +
35*F^a*b*c^3*d^3*log(F) + 2*F^a*b^2*c*d^3*log(F)^2)*x^4 + 2*(4410*F^a*c^6*d^2 +
525*F^a*b*c^4*d^2*log(F) + 60*F^a*b^2*c^2*d^2*log(F)^2 + 4*F^a*b^3*d^2*log(F)^3)
*x^3 + 6*(630*F^a*c^7*d + 105*F^a*b*c^5*d*log(F) + 20*F^a*b^2*c^3*d*log(F)^2 + 4
*F^a*b^3*c*d*log(F)^3)*x^2 + (945*F^a*c^8 + 210*F^a*b*c^6*log(F) + 60*F^a*b^2*c^
4*log(F)^2 + 24*F^a*b^3*c^2*log(F)^3 + 16*F^a*b^4*log(F)^4)*x)*F^(b/(d^2*x^2 + 2
*c*d*x + c^2)) + integrate(2/945*(16*F^a*b^5*d*x*log(F)^5 - 105*F^a*b*c^9*log(F)
- 30*F^a*b^2*c^7*log(F)^2 - 12*F^a*b^3*c^5*log(F)^3 - 8*F^a*b^4*c^3*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 = 0.264092, size = 591, normalized size = 12.06 \[ -\frac{16 \, \sqrt{\pi } F^{a} b^{5} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) \log \left (F\right )^{5} -{\left (105 \, d^{10} x^{9} + 945 \, c d^{9} x^{8} + 3780 \, c^{2} d^{8} x^{7} + 8820 \, c^{3} d^{7} x^{6} + 13230 \, c^{4} d^{6} x^{5} + 13230 \, c^{5} d^{5} x^{4} + 8820 \, c^{6} d^{4} x^{3} + 3780 \, c^{7} d^{3} x^{2} + 945 \, c^{8} d^{2} x + 105 \, c^{9} d + 16 \,{\left (b^{4} d^{2} x + b^{4} c d\right )} \log \left (F\right )^{4} + 8 \,{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d\right )} \log \left (F\right )^{3} + 12 \,{\left (b^{2} d^{6} x^{5} + 5 \, b^{2} c d^{5} x^{4} + 10 \, b^{2} c^{2} d^{4} x^{3} + 10 \, b^{2} c^{3} d^{3} x^{2} + 5 \, b^{2} c^{4} d^{2} x + b^{2} c^{5} d\right )} \log \left (F\right )^{2} + 30 \,{\left (b d^{8} x^{7} + 7 \, b c d^{7} x^{6} + 21 \, b c^{2} d^{6} x^{5} + 35 \, b c^{3} d^{5} x^{4} + 35 \, b c^{4} d^{4} x^{3} + 21 \, b c^{5} d^{3} x^{2} + 7 \, b c^{6} d^{2} x + b c^{7} d\right )} \log \left (F\right )\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{945 \, d^{2} \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^8*F^(a + b/(d*x + c)^2),x, algorithm="fricas")

[Out]

-1/945*(16*sqrt(pi)*F^a*b^5*erf(d*sqrt(-b*log(F)/d^2)/(d*x + c))*log(F)^5 - (105
*d^10*x^9 + 945*c*d^9*x^8 + 3780*c^2*d^8*x^7 + 8820*c^3*d^7*x^6 + 13230*c^4*d^6*
x^5 + 13230*c^5*d^5*x^4 + 8820*c^6*d^4*x^3 + 3780*c^7*d^3*x^2 + 945*c^8*d^2*x +
105*c^9*d + 16*(b^4*d^2*x + b^4*c*d)*log(F)^4 + 8*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2
+ 3*b^3*c^2*d^2*x + b^3*c^3*d)*log(F)^3 + 12*(b^2*d^6*x^5 + 5*b^2*c*d^5*x^4 + 1
0*b^2*c^2*d^4*x^3 + 10*b^2*c^3*d^3*x^2 + 5*b^2*c^4*d^2*x + b^2*c^5*d)*log(F)^2 +
30*(b*d^8*x^7 + 7*b*c*d^7*x^6 + 21*b*c^2*d^6*x^5 + 35*b*c^3*d^5*x^4 + 35*b*c^4*
d^4*x^3 + 21*b*c^5*d^3*x^2 + 7*b*c^6*d^2*x + b*c^7*d)*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))*sqrt(-b*log(F)/d^2))/(d^2*sqrt(-
b*log(F)/d^2))

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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)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{8} F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^8*F^(a + b/(d*x + c)^2),x, algorithm="giac")

[Out]

integrate((d*x + c)^8*F^(a + b/(d*x + c)^2), x)