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]

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

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Rubi [F]  time = 0.09, 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 = {} \[ \int \frac {\left (d^3+e^3 x^3\right )^p}{d+e x} \, dx \]

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*}

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Mathematica [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (d^3+e^3 x^3\right )^p}{d+e x} \, dx \]

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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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + d^{3}\right )}^{p}}{e x + d}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e^{3} x^{3} + d^{3}\right )}^{p}}{e x + d}\,{d x} \]

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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (e^{3} x^{3}+d^{3}\right )^{p}}{e x +d}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e^{3} x^{3} + d^{3}\right )}^{p}}{e x + d}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^3+e^3\,x^3\right )}^p}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d^3 + e^3*x^3)^p/(d + e*x),x)

[Out]

int((d^3 + e^3*x^3)^p/(d + e*x), x)

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sympy [B]  time = 30.80, size = 636, normalized size = 4.71 \[ \frac {0^{p} \log {\left (1 + \frac {e^{3} x^{3}}{d^{3}} \right )} \Gamma \left (- \frac {2}{3}\right ) \Gamma \left (- \frac {1}{3}\right ) \Gamma \left (\frac {4}{3}\right ) \Gamma \left (\frac {5}{3}\right )}{4 \pi ^{2} e} + \frac {0^{p} e^{\frac {i \pi }{3}} \log {\left (1 - \frac {e x e^{\frac {i \pi }{3}}}{d} \right )} \Gamma \left (- \frac {1}{3}\right ) \Gamma \left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right ) \Gamma \left (\frac {4}{3}\right )}{6 \pi ^{2} e \Gamma \left (\frac {5}{3}\right )} + \frac {0^{p} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {e x e^{\frac {i \pi }{3}}}{d} \right )} \Gamma ^{3}\left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right )}{12 \pi ^{2} e \Gamma \left (\frac {4}{3}\right )} - \frac {0^{p} \log {\left (1 - \frac {e x e^{i \pi }}{d} \right )} \Gamma \left (- \frac {1}{3}\right ) \Gamma \left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right ) \Gamma \left (\frac {4}{3}\right )}{6 \pi ^{2} e \Gamma \left (\frac {5}{3}\right )} + \frac {0^{p} \log {\left (1 - \frac {e x e^{i \pi }}{d} \right )} \Gamma ^{3}\left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right )}{12 \pi ^{2} e \Gamma \left (\frac {4}{3}\right )} + \frac {0^{p} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {e x e^{\frac {5 i \pi }{3}}}{d} \right )} \Gamma ^{3}\left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right )}{12 \pi ^{2} e \Gamma \left (\frac {4}{3}\right )} + \frac {0^{p} e^{- \frac {i \pi }{3}} \log {\left (1 - \frac {e x e^{\frac {5 i \pi }{3}}}{d} \right )} \Gamma \left (- \frac {1}{3}\right ) \Gamma \left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right ) \Gamma \left (\frac {4}{3}\right )}{6 \pi ^{2} e \Gamma \left (\frac {5}{3}\right )} - \frac {d^{2} e^{3 p} p x^{3 p} \Gamma \left (- \frac {2}{3}\right ) \Gamma \left (- \frac {1}{3}\right ) \Gamma \left (\frac {4}{3}\right ) \Gamma \left (\frac {5}{3}\right ) \Gamma \relax (p) \Gamma \left (\frac {2}{3} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {2}{3} - p \\ \frac {5}{3} - p \end {matrix}\middle | {\frac {d^{3} e^{i \pi }}{e^{3} x^{3}}} \right )}}{4 \pi ^{2} e^{3} x^{2} \Gamma \left (\frac {5}{3} - p\right ) \Gamma \left (p + 1\right )} - \frac {d e^{3 p} p x^{3 p} \Gamma \left (- \frac {1}{3}\right ) \Gamma \left (\frac {1}{3}\right ) \Gamma \left (\frac {2}{3}\right ) \Gamma \left (\frac {4}{3}\right ) \Gamma \relax (p) \Gamma \left (\frac {1}{3} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{3} - p \\ \frac {4}{3} - p \end {matrix}\middle | {\frac {d^{3} e^{i \pi }}{e^{3} x^{3}}} \right )}}{4 \pi ^{2} e^{2} x \Gamma \left (\frac {4}{3} - p\right ) \Gamma \left (p + 1\right )} - \frac {d^{3 p} e^{2} x^{3} \Gamma ^{2}\left (\frac {1}{3}\right ) \Gamma ^{2}\left (\frac {2}{3}\right ) \Gamma \relax (p) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{3} x^{3} e^{i \pi }}{d^{3}}} \right )}}{4 \pi ^{2} d^{3} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \]

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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