3.206 \(\int \frac{f^{c (a+b x)^3}}{x^2} \, dx\)

Optimal. Leaf size=132 \[ 3 a^2 b c \log (f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^3}}{x},x\right )-\frac{a b c \log (f) (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{\sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{b c \log (f) (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{\left (-c \log (f) (a+b x)^3\right )^{2/3}}-\frac{f^{c (a+b x)^3}}{x} \]

[Out]

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

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Rubi [A]  time = 0.302344, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{f^{c (a+b x)^3}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[f^(c*(a + b*x)^3)/x^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.08802, size = 0, normalized size = 0. \[ \int \frac{f^{c (a+b x)^3}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[f^(c*(a + b*x)^3)/x^2,x]

[Out]

Integrate[f^(c*(a + b*x)^3)/x^2, x]

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Maple [A]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c \left ( bx+a \right ) ^{3}}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{{\left (b x + a\right )}^{3} c}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^3)/x^2,x, algorithm="maxima")

[Out]

integrate(f^((b*x + a)^3*c)/x^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{f^{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^3)/x^2,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^2, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{c \left (a + b x\right )^{3}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**3)/x**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{{\left (b x + a\right )}^{3} c}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^3)/x^2,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)^3*c)/x^2, x)