3.3026 \(\int \frac{1}{(a+b x) \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\)

Optimal. Leaf size=178 \[ -\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{a d+b c+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{a d+b c+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0715279, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {123} \[ -\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{a d+b c+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{a d+b c+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 123

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3
] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rubi steps

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

Mathematica [C]  time = 0.148075, size = 140, normalized size = 0.79 \[ -\frac{3 \sqrt [3]{\frac{b c-a d}{2 d (a+b x)}+1} \sqrt [3]{\frac{b c-a d}{d (a+b x)}+1} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{b c-a d}{d (a+b x)},-\frac{b c-a d}{2 d (a+b x)}\right )}{2 b \sqrt [3]{c+d x} \sqrt [3]{a d+b (c+2 d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(1 + (b*c - a*d)/(2*d*(a + b*x)))^(1/3)*(1 + (b*c - a*d)/(d*(a + b*x)))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3,
-((b*c - a*d)/(d*(a + b*x))), -(b*c - a*d)/(2*d*(a + b*x))])/(2*b*(c + d*x)^(1/3)*(a*d + b*(c + 2*d*x))^(1/3)
)

________________________________________________________________________________________

Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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