3.1578 \(\int \frac{\sqrt [3]{c+d x}}{(a+b x)^{7/3}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{3 (c+d x)^{4/3}}{4 (a+b x)^{4/3} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0029945, antiderivative size = 32, 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} \[ -\frac{3 (c+d x)^{4/3}}{4 (a+b x)^{4/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(1/3)/(a + b*x)^(7/3),x]

[Out]

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

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{\sqrt [3]{c+d x}}{(a+b x)^{7/3}} \, dx &=-\frac{3 (c+d x)^{4/3}}{4 (b c-a d) (a+b x)^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.0117813, size = 32, normalized size = 1. \[ -\frac{3 (c+d x)^{4/3}}{4 (a+b x)^{4/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(1/3)/(a + b*x)^(7/3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{3}{4\,ad-4\,bc} \left ( dx+c \right ) ^{{\frac{4}{3}}} \left ( bx+a \right ) ^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/3)/(b*x+a)^(7/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/3)/(b*x+a)^(7/3),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(7/3), x)

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Fricas [B]  time = 1.72426, size = 143, normalized size = 4.47 \begin{align*} -\frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{4}{3}}}{4 \,{\left (a^{2} b c - a^{3} d +{\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \,{\left (a b^{2} c - a^{2} b d\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/3)/(b*x+a)^(7/3),x, algorithm="fricas")

[Out]

-3/4*(b*x + a)^(2/3)*(d*x + c)^(4/3)/(a^2*b*c - a^3*d + (b^3*c - a*b^2*d)*x^2 + 2*(a*b^2*c - a^2*b*d)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{c + d x}}{\left (a + b x\right )^{\frac{7}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/3)/(b*x+a)**(7/3),x)

[Out]

Integral((c + d*x)**(1/3)/(a + b*x)**(7/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/3)/(b*x+a)^(7/3),x, algorithm="giac")

[Out]

integrate((d*x + c)^(1/3)/(b*x + a)^(7/3), x)