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

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 = 0.0088327, 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, Rules used = {246, 245} \[ x \left (a+d x^3\right )^n \left (\frac{d x^3}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{3},-n;\frac{4}{3};-\frac{d x^3}{a}\right ) \]

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (a+d x^3\right )^n \, dx &=\left (\left (a+d x^3\right )^n \left (1+\frac{d x^3}{a}\right )^{-n}\right ) \int \left (1+\frac{d x^3}{a}\right )^n \, dx\\ &=x \left (a+d x^3\right )^n \left (1+\frac{d x^3}{a}\right )^{-n} \, _2F_1\left (\frac{1}{3},-n;\frac{4}{3};-\frac{d x^3}{a}\right )\\ \end{align*}

Mathematica [C]  time = 0.150824, 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 = 0.029, size = 0, normalized size = 0. \begin{align*} \int \left ( d{x}^{3}+a \right ) ^{n}\, dx \end{align*}

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. \begin{align*} \int{\left (d x^{3} + a\right )}^{n}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (d x^{3} + a\right )}^{n}, x\right ) \end{align*}

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 [C]  time = 16.0086, size = 34, normalized size = 0.97 \begin{align*} \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 )} \end{align*}

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

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)