\(\int \frac {(a+b x^3)^m}{(c+d x^3)^3} \, dx\) [144]

Optimal result

Integrand size = 19, antiderivative size = 57 \[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\frac {x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{3},-m,3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440} \[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{3},-m,3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3} \]

[In]

Int[(a + b*x^3)^m/(c + d*x^3)^3,x]

[Out]

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

Rule 440

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])

Rule 441

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])

Rubi steps \begin{align*} \text {integral}& = \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}{\left (c+d x^3\right )^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,3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(57)=114\).

Time = 0.55 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=-\frac {4 a c x \left (a+b x^3\right )^m \operatorname {AppellF1}\left (\frac {1}{3},-m,3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\left (c+d x^3\right )^3 \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},-m,3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (b c m \operatorname {AppellF1}\left (\frac {4}{3},1-m,3,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 a d \operatorname {AppellF1}\left (\frac {4}{3},-m,4,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (b \,x^{3}+a \right )^{m}}{\left (d \,x^{3}+c \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{m}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

integral((b*x^3 + a)^m/(d^3*x^9 + 3*c*d^2*x^6 + 3*c^2*d*x^3 + c^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{m}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{m}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^m}{{\left (d\,x^3+c\right )}^3} \,d x \]

[In]

int((a + b*x^3)^m/(c + d*x^3)^3,x)

[Out]

int((a + b*x^3)^m/(c + d*x^3)^3, x)