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

Optimal. Leaf size=261 \[ \frac{9}{2} a^4 b^2 c^2 \log ^2(f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^3}}{x},x\right )-\frac{3 a^3 b^2 c^2 \log ^2(f) (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{3 a^2 b^2 c^2 \log ^2(f) (a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{2 \left (-c \log (f) (a+b x)^3\right )^{2/3}}+3 a b^2 c \log (f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^3}}{x},x\right )-\frac{b^2 c \log (f) (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{3 a^2 b c \log (f) f^{c (a+b x)^3}}{2 x}-\frac{f^{c (a+b x)^3}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.448682, 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 = {} \[ \int \frac{f^{c (a+b x)^3}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[f^(c*(a + b*x)^3)/x^3,x]

[Out]

-f^(c*(a + b*x)^3)/(2*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*}

Mathematica [A]  time = 1.38929, size = 0, normalized size = 0. \[ \int \frac{f^{c (a+b x)^3}}{x^3} \, dx \]

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 = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c \left ( bx+a \right ) ^{3}}}{{x}^{3}}}\, dx \end{align*}

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

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., 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^{3}}, x\right ) \end{align*}

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

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

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)