3.32 \(\int \left (a+d x^3\right )^n \, dx\)

Optimal. Leaf size=35 \[ \frac{x \left (a+d x^3\right )^{n+1} \, _2F_1\left (1,n+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{a}\right )}{a} \]

[Out]

(x*(a + d*x^3)^(1 + n)*Hypergeometric2F1[1, 4/3 + n, 4/3, -((d*x^3)/a)])/a

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Rubi [A]  time = 0.0244163, antiderivative size = 44, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ x \left (a+d x^3\right )^n \left (\frac{d x^3}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{3},-n;\frac{4}{3};-\frac{d x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + d*x^3)^n,x]

[Out]

(x*(a + d*x^3)^n*Hypergeometric2F1[1/3, -n, 4/3, -((d*x^3)/a)])/(1 + (d*x^3)/a)^
n

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Rubi in Sympy [A]  time = 3.71689, size = 34, normalized size = 0.97 \[ x \left (1 + \frac{d x^{3}}{a}\right )^{- n} \left (a + d x^{3}\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{d x^{3}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+a)**n,x)

[Out]

x*(1 + d*x**3/a)**(-n)*(a + d*x**3)**n*hyper((-n, 1/3), (4/3,), -d*x**3/a)

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Mathematica [C]  time = 0.304042, size = 196, normalized size = 5.6 \[ \frac{2^{-n} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{d} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{d} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-n} \left (\frac{i \left (\frac{\sqrt [3]{d} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-n} \left (a+d x^3\right )^n F_1\left (n+1;-n,-n;n+2;-\frac{i \left (\sqrt [3]{d} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{d} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{d} (n+1)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + d*x^3)^n,x]

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int \left ( d{x}^{3}+a \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+a)^n,x)

[Out]

int((d*x^3+a)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{3} + a\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + a)^n,x, algorithm="maxima")

[Out]

integrate((d*x^3 + a)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d x^{3} + a\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + a)^n,x, algorithm="fricas")

[Out]

integral((d*x^3 + a)^n, x)

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Sympy [A]  time = 68.8391, size = 34, normalized size = 0.97 \[ \frac{a^{n} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - n \\ \frac{4}{3} \end{matrix}\middle |{\frac{d x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+a)**n,x)

[Out]

a**n*x*gamma(1/3)*hyper((1/3, -n), (4/3,), d*x**3*exp_polar(I*pi)/a)/(3*gamma(4/
3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{3} + a\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + a)^n,x, algorithm="giac")

[Out]

integrate((d*x^3 + a)^n, x)