3.1577 \(\int \frac{\sqrt [3]{c+d x}}{(a+b x)^{4/3}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{3 \sqrt [3]{d} \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3}}-\frac{\sqrt{3} \sqrt [3]{d} \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{4/3}}-\frac{3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac{\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}} \]

[Out]

(-3*(c + d*x)^(1/3))/(b*(a + b*x)^(1/3)) - (Sqrt[3]*d^(1/3)*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sq
rt[3]*b^(1/3)*(c + d*x)^(1/3))])/b^(4/3) - (d^(1/3)*Log[c + d*x])/(2*b^(4/3)) - (3*d^(1/3)*Log[-1 + (d^(1/3)*(
a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(2*b^(4/3))

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Rubi [A]  time = 0.032848, antiderivative size = 149, normalized size of antiderivative = 1., 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 = {47, 59} \[ -\frac{3 \sqrt [3]{d} \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3}}-\frac{\sqrt{3} \sqrt [3]{d} \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{b^{4/3}}-\frac{3 \sqrt [3]{c+d x}}{b \sqrt [3]{a+b x}}-\frac{\sqrt [3]{d} \log (c+d x)}{2 b^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(c + d*x)^(1/3))/(b*(a + b*x)^(1/3)) - (Sqrt[3]*d^(1/3)*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sq
rt[3]*b^(1/3)*(c + d*x)^(1/3))])/b^(4/3) - (d^(1/3)*Log[c + d*x])/(2*b^(4/3)) - (3*d^(1/3)*Log[-1 + (d^(1/3)*(
a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(2*b^(4/3))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

Rubi steps

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

Mathematica [C]  time = 0.0272532, size = 71, normalized size = 0.48 \[ -\frac{3 \sqrt [3]{c+d x} \, _2F_1\left (-\frac{1}{3},-\frac{1}{3};\frac{2}{3};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt [3]{a+b x} \sqrt [3]{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(a + b*x)^(1/3)*((b*(
c + d*x))/(b*c - a*d))^(1/3))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{dx+c} \left ( bx+a \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{4}{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)^(4/3),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.8196, size = 603, normalized size = 4.05 \begin{align*} -\frac{2 \, \sqrt{3}{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} b \left (-\frac{d}{b}\right )^{\frac{2}{3}} + \sqrt{3}{\left (b d x + a d\right )}}{3 \,{\left (b d x + a d\right )}}\right ) +{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} \log \left (\frac{{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} \left (-\frac{d}{b}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) - 2 \,{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} \log \left (\frac{{\left (b x + a\right )} \left (-\frac{d}{b}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) + 6 \,{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{2 \,{\left (b^{2} x + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(3)*(b*x + a)*(-d/b)^(1/3)*arctan(1/3*(2*sqrt(3)*(b*x + a)^(2/3)*(d*x + c)^(1/3)*b*(-d/b)^(2/3) +
sqrt(3)*(b*d*x + a*d))/(b*d*x + a*d)) + (b*x + a)*(-d/b)^(1/3)*log(((b*x + a)*(-d/b)^(2/3) - (b*x + a)^(2/3)*(
d*x + c)^(1/3)*(-d/b)^(1/3) + (b*x + a)^(1/3)*(d*x + c)^(2/3))/(b*x + a)) - 2*(b*x + a)*(-d/b)^(1/3)*log(((b*x
+ a)*(-d/b)^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(b*x + a)) + 6*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(b^2*x +
a*b)

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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{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)**(1/3)/(a + b*x)**(4/3), 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{4}{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)^(4/3),x, algorithm="giac")

[Out]

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