3.142 \(\int \frac{(a+b x^3)^m}{c+d x^3} \, dx\)

Optimal. Leaf size=57 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} F_1\left (\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c} \]

[Out]

(x*(a + b*x^3)^m*AppellF1[1/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(1 + (b*x^3)/a)^m)

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Rubi [A]  time = 0.0251674, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {430, 429} \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} F_1\left (\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^m/(c + d*x^3),x]

[Out]

(x*(a + b*x^3)^m*AppellF1[1/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*(1 + (b*x^3)/a)^m)

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^m}{c+d x^3} \, dx &=\left (\left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m}\right ) \int \frac{\left (1+\frac{b x^3}{a}\right )^m}{c+d x^3} \, dx\\ &=\frac{x \left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m} F_1\left (\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c}\\ \end{align*}

Mathematica [B]  time = 0.184583, size = 162, normalized size = 2.84 \[ -\frac{4 a c x \left (a+b x^3\right )^m F_1\left (\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (c+d x^3\right ) \left (3 x^3 \left (a d F_1\left (\frac{4}{3};-m,2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-b c m F_1\left (\frac{4}{3};1-m,1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};-m,1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*a*c*x*(a + b*x^3)^m*AppellF1[1/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(-4*a*c*AppellF1[1
/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(-(b*c*m*AppellF1[4/3, 1 - m, 1, 7/3, -((b*x^3)/a), -((d*x
^3)/c)]) + a*d*AppellF1[4/3, -m, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))

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Maple [F]  time = 0.431, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{3}+a \right ) ^{m}}{d{x}^{3}+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^m/(d*x^3+c),x)

[Out]

int((b*x^3+a)^m/(d*x^3+c),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^m/(d*x^3 + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^3 + a)^m/(d*x^3 + c), 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((b*x**3+a)**m/(d*x**3+c),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^3 + a)^m/(d*x^3 + c), x)