3.1617 \(\int \frac{\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0308381, 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} \[ -\frac{3 \sqrt [3]{b} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3}}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{d^{4/3}}-\frac{3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}-\frac{\sqrt [3]{b} \log (a+b x)}{2 d^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.0444046, size = 73, normalized size = 0.49 \[ \frac{3 (a+b x)^{4/3} \left (\frac{b (c+d x)}{b c-a d}\right )^{4/3} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{d (a+b x)}{a d-b c}\right )}{4 b (c+d x)^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(4/3)*((b*(c + d*x))/(b*c - a*d))^(4/3)*Hypergeometric2F1[4/3, 4/3, 7/3, (d*(a + b*x))/(-(b*c) +
a*d)])/(4*b*(c + d*x)^(4/3))

________________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{bx+a} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.30863, size = 603, normalized size = 4.05 \begin{align*} -\frac{2 \, \sqrt{3}{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} d \left (-\frac{b}{d}\right )^{\frac{2}{3}} + \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right ) +{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 2 \,{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (-\frac{b}{d}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) + 6 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (d^{2} x + c d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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