3.197 \(\int \frac{(d^3+e^3 x^3)^p}{d+e x} \, dx\)

Optimal. Leaf size=135 \[ \frac{\left (d^3+e^3 x^3\right )^p \left (1+\frac{2 (d+e x)}{\left (-3+i \sqrt{3}\right ) d}\right )^{-p} \left (1-\frac{2 (d+e x)}{\left (3+i \sqrt{3}\right ) d}\right )^{-p} F_1\left (p;-p,-p;p+1;-\frac{2 (d+e x)}{\left (-3+i \sqrt{3}\right ) d},\frac{2 (d+e x)}{\left (3+i \sqrt{3}\right ) d}\right )}{e p} \]

[Out]

((d^3 + e^3*x^3)^p*AppellF1[p, -p, -p, 1 + p, (-2*(d + e*x))/((-3 + I*Sqrt[3])*d), (2*(d + e*x))/((3 + I*Sqrt[
3])*d)])/(e*p*(1 + (2*(d + e*x))/((-3 + I*Sqrt[3])*d))^p*(1 - (2*(d + e*x))/((3 + I*Sqrt[3])*d))^p)

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Rubi [F]  time = 0.0856207, 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 = {} \[ \int \frac{\left (d^3+e^3 x^3\right )^p}{d+e x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d^3 + e^3*x^3)^p/(d + e*x),x]

[Out]

Defer[Int][(d^3 + e^3*x^3)^p/(d + e*x), x]

Rubi steps

\begin{align*} \int \frac{\left (d^3+e^3 x^3\right )^p}{d+e x} \, dx &=\int \frac{\left (d^3+e^3 x^3\right )^p}{d+e x} \, dx\\ \end{align*}

Mathematica [F]  time = 0.0492502, size = 0, normalized size = 0. \[ \int \frac{\left (d^3+e^3 x^3\right )^p}{d+e x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d^3 + e^3*x^3)^p/(d + e*x),x]

[Out]

Integrate[(d^3 + e^3*x^3)^p/(d + e*x), x]

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({e}^{3}{x}^{3}+{d}^{3} \right ) ^{p}}{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e^3*x^3+d^3)^p/(e*x+d),x)

[Out]

int((e^3*x^3+d^3)^p/(e*x+d),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e^{3} x^{3} + d^{3}\right )}^{p}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e^3*x^3+d^3)^p/(e*x+d),x, algorithm="maxima")

[Out]

integrate((e^3*x^3 + d^3)^p/(e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{3} x^{3} + d^{3}\right )}^{p}}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e^3*x^3+d^3)^p/(e*x+d),x, algorithm="fricas")

[Out]

integral((e^3*x^3 + d^3)^p/(e*x + d), x)

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Sympy [B]  time = 56.4874, size = 636, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e**3*x**3+d**3)**p/(e*x+d),x)

[Out]

0**p*log(1 + e**3*x**3/d**3)*gamma(-2/3)*gamma(-1/3)*gamma(4/3)*gamma(5/3)/(4*pi**2*e) + 0**p*exp(I*pi/3)*log(
1 - e*x*exp_polar(I*pi/3)/d)*gamma(-1/3)*gamma(1/3)*gamma(2/3)**2*gamma(4/3)/(6*pi**2*e*gamma(5/3)) + 0**p*exp
(2*I*pi/3)*log(1 - e*x*exp_polar(I*pi/3)/d)*gamma(1/3)**3*gamma(2/3)**2/(12*pi**2*e*gamma(4/3)) - 0**p*log(1 -
e*x*exp_polar(I*pi)/d)*gamma(-1/3)*gamma(1/3)*gamma(2/3)**2*gamma(4/3)/(6*pi**2*e*gamma(5/3)) + 0**p*log(1 -
e*x*exp_polar(I*pi)/d)*gamma(1/3)**3*gamma(2/3)**2/(12*pi**2*e*gamma(4/3)) + 0**p*exp(-2*I*pi/3)*log(1 - e*x*e
xp_polar(5*I*pi/3)/d)*gamma(1/3)**3*gamma(2/3)**2/(12*pi**2*e*gamma(4/3)) + 0**p*exp(-I*pi/3)*log(1 - e*x*exp_
polar(5*I*pi/3)/d)*gamma(-1/3)*gamma(1/3)*gamma(2/3)**2*gamma(4/3)/(6*pi**2*e*gamma(5/3)) - d**2*e**(3*p)*p*x*
*(3*p)*gamma(-2/3)*gamma(-1/3)*gamma(4/3)*gamma(5/3)*gamma(p)*gamma(2/3 - p)*hyper((1 - p, 2/3 - p), (5/3 - p,
), d**3*exp_polar(I*pi)/(e**3*x**3))/(4*pi**2*e**3*x**2*gamma(5/3 - p)*gamma(p + 1)) - d*e**(3*p)*p*x**(3*p)*g
amma(-1/3)*gamma(1/3)*gamma(2/3)*gamma(4/3)*gamma(p)*gamma(1/3 - p)*hyper((1 - p, 1/3 - p), (4/3 - p,), d**3*e
xp_polar(I*pi)/(e**3*x**3))/(4*pi**2*e**2*x*gamma(4/3 - p)*gamma(p + 1)) - d**(3*p)*e**2*x**3*gamma(1/3)**2*ga
mma(2/3)**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), e**3*x**3*exp_polar(I*pi)/d**3)/(4*pi**2*d**3*g
amma(-p)*gamma(p + 1))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e^{3} x^{3} + d^{3}\right )}^{p}}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e^3*x^3+d^3)^p/(e*x+d),x, algorithm="giac")

[Out]

integrate((e^3*x^3 + d^3)^p/(e*x + d), x)