3.236 \(\int f^{\frac {c}{(a+b x)^3}} \, dx\)

Optimal. Leaf size=44 \[ \frac {(a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2208} \[ \frac {(a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \text {Gamma}\left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[f^(c/(a + b*x)^3),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.00 \[ \frac {(a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(c/(a + b*x)^3),x]

[Out]

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

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fricas [B]  time = 0.49, size = 94, normalized size = 2.14 \[ -\frac {b \left (-\frac {c \log \relax (f)}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \relax (f)}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b x + a\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c/(b*x + a)^3), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int f^{\frac {c}{\left (b x +a \right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.97, size = 68, normalized size = 1.55 \[ \frac {\left (a+b\,x\right )\,\left (\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {c\,\ln \relax (f)}{{\left (a+b\,x\right )}^3}\right )}^{1/3}-\Gamma \left (\frac {2}{3},-\frac {c\,\ln \relax (f)}{{\left (a+b\,x\right )}^3}\right )\,{\left (-\frac {c\,\ln \relax (f)}{{\left (a+b\,x\right )}^3}\right )}^{1/3}+f^{\frac {c}{{\left (a+b\,x\right )}^3}}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c/(a + b*x)^3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c/(a + b*x)**3), x)

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