3.1604 \(\int \frac{1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0030281, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(4/3)*(c + d*x)^(2/3)),x]

[Out]

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

Rule 37

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

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx &=-\frac{3 \sqrt [3]{c+d x}}{(b c-a d) \sqrt [3]{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.008687, size = 30, normalized size = 1. \[ \frac{3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(4/3)*(c + d*x)^(2/3)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.9 \begin{align*} 3\,{\frac{\sqrt [3]{dx+c}}{\sqrt [3]{bx+a} \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(4/3)/(d*x+c)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.47829, size = 97, normalized size = 3.23 \begin{align*} -\frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(4/3)/(d*x+c)^(2/3),x, algorithm="fricas")

[Out]

-3*(b*x + a)^(2/3)*(d*x + c)^(1/3)/(a*b*c - a^2*d + (b^2*c - a*b*d)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x\right )^{\frac{4}{3}} \left (c + d x\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(4/3)/(d*x+c)**(2/3),x)

[Out]

Integral(1/((a + b*x)**(4/3)*(c + d*x)**(2/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(4/3)/(d*x+c)^(2/3),x, algorithm="giac")

[Out]

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