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

Optimal. Leaf size=112 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right )}{c e (m+1) (b c-a d)} \]

[Out]

(b*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a*(b*c - a*d)*e*(1 + m)) - (d*(e*x
)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((d*x^3)/c)])/(c*(b*c - a*d)*e*(1 + m))

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Rubi [A]  time = 0.0525424, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {482, 364} \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right )}{c e (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(a*(b*c - a*d)*e*(1 + m)) - (d*(e*x
)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((d*x^3)/c)])/(c*(b*c - a*d)*e*(1 + m))

Rule 482

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

Rule 364

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

Rubi steps

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

Mathematica [A]  time = 0.0625725, size = 86, normalized size = 0.77 \[ \frac{x (e x)^m \left (a d \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right )-b c \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(-(b*c*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)]) + a*d*Hypergeometric2F1[1, (1 + m)
/3, (4 + m)/3, -((d*x^3)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( b{x}^{3}+a \right ) \left ( d{x}^{3}+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m/((b*x^3 + a)*(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 (e x\right )^{m}}{b d x^{6} +{\left (b c + a d\right )} x^{3} + a c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x)^m/(b*d*x^6 + (b*c + a*d)*x^3 + a*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((e*x)**m/(b*x**3+a)/(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 (e x\right )^{m}}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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