3.2.43 \(\int \frac {(a+b x^3)^m}{(c+d x^3)^2} \, dx\) [143]

Optimal. Leaf size=57 \[ \frac {x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} F_1\left (\frac {1}{3};-m,2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {441, 440} \begin {gather*} \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,2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^m/(c + d*x^3)^2,x]

[Out]

(x*(a + b*x^3)^m*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c^2*(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*} \int \frac {\left (a+b x^3\right )^m}{\left (c+d x^3\right )^2} \, 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}{\left (c+d x^3\right )^2} \, 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,2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(57)=114\).
time = 0.34, size = 162, normalized size = 2.84 \begin {gather*} -\frac {4 a c x \left (a+b x^3\right )^m F_1\left (\frac {1}{3};-m,2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\left (c+d x^3\right )^2 \left (-4 a c F_1\left (\frac {1}{3};-m,2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (b c m F_1\left (\frac {4}{3};1-m,2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 a d F_1\left (\frac {4}{3};-m,3;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{m}}{\left (d \,x^{3}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^m}{{\left (d\,x^3+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^m/(c + d*x^3)^2,x)

[Out]

int((a + b*x^3)^m/(c + d*x^3)^2, x)

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