3.194 \(\int \frac{x^2 (c+d x)^{1+n}}{a+b x^3} \, dx\)

Optimal. Leaf size=253 \[ -\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )} \]

[Out]

-((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^(2/3
)*(b^(1/3)*c - a^(1/3)*d)*(2 + n)) - ((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))
/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*b^(2/3)*(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)*(2 + n)) - ((c + d*x)^(2 +
n)*Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*b^(2/3)*(b^
(1/3)*c - (-1)^(2/3)*a^(1/3)*d)*(2 + n))

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Rubi [A]  time = 0.585119, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6725, 68} \[ -\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c\right )}-\frac{(c+d x)^{n+2} \, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2) \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x)^(1 + n))/(a + b*x^3),x]

[Out]

-((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^(2/3
)*(b^(1/3)*c - a^(1/3)*d)*(2 + n)) - ((c + d*x)^(2 + n)*Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))
/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*b^(2/3)*(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)*(2 + n)) - ((c + d*x)^(2 +
n)*Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*b^(2/3)*(b^
(1/3)*c - (-1)^(2/3)*a^(1/3)*d)*(2 + n))

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{x^2 (c+d x)^{1+n}}{a+b x^3} \, dx &=\int \left (\frac{(c+d x)^{1+n}}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{(c+d x)^{1+n}}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{(c+d x)^{1+n}}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx\\ &=\frac{\int \frac{(c+d x)^{1+n}}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\int \frac{(c+d x)^{1+n}}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac{\int \frac{(c+d x)^{1+n}}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}\\ &=-\frac{(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) (2+n)}-\frac{(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d\right ) (2+n)}-\frac{(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} \left (\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d\right ) (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.37598, size = 213, normalized size = 0.84 \[ \frac{(c+d x)^{n+2} \left (-\frac{\, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}-\frac{\, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}-\frac{\, _2F_1\left (1,n+2;n+3;\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^{2/3} (n+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x)^(1 + n))/(a + b*x^3),x]

[Out]

((c + d*x)^(2 + n)*(-(Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)]/(b^(1/3)
*c - a^(1/3)*d)) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]/
(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d) - Hypergeometric2F1[1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^
(2/3)*a^(1/3)*d)]/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)))/(3*b^(2/3)*(2 + n))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( dx+c \right ) ^{1+n}}{b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d*x+c)^(1+n)/(b*x^3+a),x)

[Out]

int(x^2*(d*x+c)^(1+n)/(b*x^3+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^(1+n)/(b*x^3+a),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^(1+n)/(b*x^3+a),x, algorithm="fricas")

[Out]

integral((d*x + c)^(n + 1)*x^2/(b*x^3 + a), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d*x+c)**(1+n)/(b*x**3+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{n + 1} x^{2}}{b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^(1+n)/(b*x^3+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^(n + 1)*x^2/(b*x^3 + a), x)