3.3.7 \(\int \frac {f^{c (a+b x)^3}}{x^3} \, dx\) [207]

Optimal. Leaf size=263 \[ -\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {3 a^2 b^2 c^2 (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}-\frac {3 a^3 b^2 c^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+3 a b^2 c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right )+\frac {9}{2} a^4 b^2 c^2 \log ^2(f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right ) \]

[Out]

-1/2*f^(c*(b*x+a)^3)/x^2-3/2*a^2*b*c*f^(c*(b*x+a)^3)*ln(f)/x-3/2*a^2*b^2*c^2*(b*x+a)^2*GAMMA(2/3,-c*(b*x+a)^3*
ln(f))*ln(f)^2/(-c*(b*x+a)^3*ln(f))^(2/3)-1/2*b^2*c*(b*x+a)*GAMMA(1/3,-c*(b*x+a)^3*ln(f))*ln(f)/(-c*(b*x+a)^3*
ln(f))^(1/3)-3/2*a^3*b^2*c^2*(b*x+a)*GAMMA(1/3,-c*(b*x+a)^3*ln(f))*ln(f)^2/(-c*(b*x+a)^3*ln(f))^(1/3)+3*a*b^2*
c*ln(f)*Unintegrable(f^(c*(b*x+a)^3)/x,x)+9/2*a^4*b^2*c^2*ln(f)^2*Unintegrable(f^(c*(b*x+a)^3)/x,x)

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Rubi [A]
time = 0.33, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {f^{c (a+b x)^3}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[f^(c*(a + b*x)^3)/x^3,x]

[Out]

-1/2*f^(c*(a + b*x)^3)/x^2 - (3*a^2*b*c*f^(c*(a + b*x)^3)*Log[f])/(2*x) - (3*a^2*b^2*c^2*(a + b*x)^2*Gamma[2/3
, -(c*(a + b*x)^3*Log[f])]*Log[f]^2)/(2*(-(c*(a + b*x)^3*Log[f]))^(2/3)) - (b^2*c*(a + b*x)*Gamma[1/3, -(c*(a
+ b*x)^3*Log[f])]*Log[f])/(2*(-(c*(a + b*x)^3*Log[f]))^(1/3)) - (3*a^3*b^2*c^2*(a + b*x)*Gamma[1/3, -(c*(a + b
*x)^3*Log[f])]*Log[f]^2)/(2*(-(c*(a + b*x)^3*Log[f]))^(1/3)) + 3*a*b^2*c*Log[f]*Defer[Int][f^(c*(a + b*x)^3)/x
, x] + (9*a^4*b^2*c^2*Log[f]^2*Defer[Int][f^(c*(a + b*x)^3)/x, x])/2

Rubi steps

\begin {align*} \int \frac {f^{c (a+b x)^3}}{x^3} \, dx &=-\frac {f^{c (a+b x)^3}}{2 x^2}+\frac {1}{2} (3 b c \log (f)) \int \frac {f^{c (a+b x)^3} (a+b x)^2}{x^2} \, dx\\ &=-\frac {f^{c (a+b x)^3}}{2 x^2}+\frac {1}{2} (3 b c \log (f)) \int \left (b^2 f^{c (a+b x)^3}+\frac {a^2 f^{c (a+b x)^3}}{x^2}+\frac {2 a b f^{c (a+b x)^3}}{x}\right ) \, dx\\ &=-\frac {f^{c (a+b x)^3}}{2 x^2}+\frac {1}{2} \left (3 a^2 b c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x^2} \, dx+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (3 b^3 c \log (f)\right ) \int f^{c (a+b x)^3} \, dx\\ &=-\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^2 b^2 c^2 \log ^2(f)\right ) \int \frac {f^{c (a+b x)^3} (a+b x)^2}{x} \, dx\\ &=-\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^2 b^2 c^2 \log ^2(f)\right ) \int \left (a b f^{c (a+b x)^3}+\frac {a^2 f^{c (a+b x)^3}}{x}+b f^{c (a+b x)^3} (a+b x)\right ) \, dx\\ &=-\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^4 b^2 c^2 \log ^2(f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^2 b^3 c^2 \log ^2(f)\right ) \int f^{c (a+b x)^3} (a+b x) \, dx+\frac {1}{2} \left (9 a^3 b^3 c^2 \log ^2(f)\right ) \int f^{c (a+b x)^3} \, dx\\ &=-\frac {f^{c (a+b x)^3}}{2 x^2}-\frac {3 a^2 b c f^{c (a+b x)^3} \log (f)}{2 x}-\frac {3 a^2 b^2 c^2 (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {b^2 c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}-\frac {3 a^3 b^2 c^2 (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log ^2(f)}{2 \sqrt [3]{-c (a+b x)^3 \log (f)}}+\left (3 a b^2 c \log (f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx+\frac {1}{2} \left (9 a^4 b^2 c^2 \log ^2(f)\right ) \int \frac {f^{c (a+b x)^3}}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{c (a+b x)^3}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[f^(c*(a + b*x)^3)/x^3,x]

[Out]

Integrate[f^(c*(a + b*x)^3)/x^3, x]

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Maple [A]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {f^{c \left (b x +a \right )^{3}}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^3*c)/x^3, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)/x^3, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{c \left (a + b x\right )^{3}}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**3)/x**3, x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^3*c)/x^3, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {f^{c\,{\left (a+b\,x\right )}^3}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(a + b*x)^3)/x^3,x)

[Out]

int(f^(c*(a + b*x)^3)/x^3, x)

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