3.145 \(\int (a+b x^3)^{-1-\frac{b c}{3 b c-3 a d}} (c+d x^3)^{-1+\frac{a d}{3 b c-3 a d}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x \left (a+b x^3\right )^{-\frac{b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac{a d}{3 b c-3 a d}}}{a c} \]

[Out]

(x*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c*(a + b*x^3)^((b*c)/(3*b*c - 3*a*d)))

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Rubi [A]  time = 0.019047, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02, Rules used = {381} \[ \frac{x \left (a+b x^3\right )^{-\frac{b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac{a d}{3 b c-3 a d}}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c*(a + b*x^3)^((b*c)/(3*b*c - 3*a*d)))

Rule 381

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1)*(c +
d*x^n)^(q + 1))/(a*c), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0
] && EqQ[a*d*(p + 1) + b*c*(q + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.032566, size = 52, normalized size = 0.98 \[ \frac{x \left (a+b x^3\right )^{\frac{b c}{3 a d-3 b c}} \left (c+d x^3\right )^{\frac{a d}{3 b c-3 a d}}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x^3)^((b*c)/(-3*b*c + 3*a*d))*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c)

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Maple [A]  time = 0.003, size = 71, normalized size = 1.3 \begin{align*}{\frac{x}{ac} \left ( b{x}^{3}+a \right ) ^{1-{\frac{3\,ad-4\,bc}{3\,ad-3\,bc}}} \left ( d{x}^{3}+c \right ) ^{1-{\frac{4\,ad-3\,bc}{3\,ad-3\,bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47237, size = 182, normalized size = 3.43 \begin{align*} \frac{b d x^{7} +{\left (b c + a d\right )} x^{4} + a c x}{{\left (b x^{3} + a\right )}^{\frac{4 \, b c - 3 \, a d}{3 \,{\left (b c - a d\right )}}}{\left (d x^{3} + c\right )}^{\frac{3 \, b c - 4 \, a d}{3 \,{\left (b c - a d\right )}}} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*d*x^7 + (b*c + a*d)*x^4 + a*c*x)/((b*x^3 + a)^(1/3*(4*b*c - 3*a*d)/(b*c - a*d))*(d*x^3 + c)^(1/3*(3*b*c - 4
*a*d)/(b*c - a*d))*a*c)

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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)**(-1-b*c/(-3*a*d+3*b*c))*(d*x**3+c)**(-1+a*d/(-3*a*d+3*b*c)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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