3.3.6 \(\int \frac {f^{c (a+b x)^3}}{x^2} \, dx\) [206]

Optimal. Leaf size=133 \[ -\frac {f^{c (a+b x)^3}}{x}-\frac {b c (a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{\left (-c (a+b x)^3 \log (f)\right )^{2/3}}-\frac {a b c (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \log (f)}{\sqrt [3]{-c (a+b x)^3 \log (f)}}+3 a^2 b c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^3}}{x},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.21, 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^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 1.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{c (a+b x)^3}}{x^2} \, dx \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {f^{c \left (b x +a \right )^{3}}}{x^{2}}\, dx\]

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 = 0.00, 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^2,x, algorithm="maxima")

[Out]

integrate(f^((b*x + a)^3*c)/x^2, 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^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{c \left (a + b x\right )^{3}}}{x^{2}}\, dx \end {gather*}

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 = 0.00, 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^2,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)^3*c)/x^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f^{c\,{\left (a+b\,x\right )}^3}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(a + b*x)^3)/x^2,x)

[Out]

int(f^(c*(a + b*x)^3)/x^2, x)

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