3.355 \(\int F^{a+\frac {b}{(c+d x)^3}} \, dx\)

Optimal. Leaf size=47 \[ \frac {F^a (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

1/3*F^a*(d*x+c)*GAMMA(-1/3,-b*ln(F)/(d*x+c)^3)*(-b*ln(F)/(d*x+c)^3)^(1/3)/d

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Rubi [A]  time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2208} \[ \frac {F^a (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3),x]

[Out]

(F^a*(c + d*x)*Gamma[-1/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(1/3))/(3*d)

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin {align*} \int F^{a+\frac {b}{(c+d x)^3}} \, dx &=\frac {F^a (c+d x) \Gamma \left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right ) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}}}{3 d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.00 \[ \frac {F^a (c+d x) \sqrt [3]{-\frac {b \log (F)}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3),x]

[Out]

(F^a*(c + d*x)*Gamma[-1/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^3))^(1/3))/(3*d)

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fricas [B]  time = 0.43, size = 129, normalized size = 2.74 \[ -\frac {F^{a} d \left (-\frac {b \log \relax (F)}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (F)}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (d x + c\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3),x, algorithm="fricas")

[Out]

-(F^a*d*(-b*log(F)/d^3)^(1/3)*gamma(2/3, -b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - (d*x + c)*F^((
a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3), x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int F^{a +\frac {b}{\left (d x +c \right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^3*b),x)

[Out]

int(F^(a+1/(d*x+c)^3*b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 3 \, F^{a} b d \int \frac {F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} x}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\,{d x} \log \relax (F) + F^{a} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3),x, algorithm="maxima")

[Out]

3*F^a*b*d*integrate(F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*x/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 +
4*c^3*d*x + c^4), x)*log(F) + F^a*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*x

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mupad [B]  time = 3.93, size = 71, normalized size = 1.51 \[ \frac {F^a\,\left (c+d\,x\right )\,\left (F^{\frac {b}{{\left (c+d\,x\right )}^3}}-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{1/3}+\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (F)}{{\left (c+d\,x\right )}^3}\right )}^{1/3}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^3),x)

[Out]

(F^a*(c + d*x)*(F^(b/(c + d*x)^3) - igamma(2/3, -(b*log(F))/(c + d*x)^3)*(-(b*log(F))/(c + d*x)^3)^(1/3) + gam
ma(2/3)*(-(b*log(F))/(c + d*x)^3)^(1/3)))/d

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{a + \frac {b}{\left (c + d x\right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3),x)

[Out]

Integral(F**(a + b/(c + d*x)**3), x)

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