\(\int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx\) [3019]

Optimal result

Integrand size = 26, antiderivative size = 197 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}}+\frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}} \]

[Out]

1/2*ln(f*x+e)/(-a*f+b*e)^(1/3)/(-c*f+d*e)^(2/3)-3/2*ln((-c*f+d*e)^(1/3)*(b*x+a)^(1/3)/(-a*f+b*e)^(1/3)-(d*x+c)
^(1/3))/(-a*f+b*e)^(1/3)/(-c*f+d*e)^(2/3)-arctan(1/3*3^(1/2)+2/3*(-c*f+d*e)^(1/3)*(b*x+a)^(1/3)/(-a*f+b*e)^(1/
3)/(d*x+c)^(1/3)*3^(1/2))*3^(1/2)/(-a*f+b*e)^(1/3)/(-c*f+d*e)^(2/3)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {93} \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}}+\frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}} \]

[In]

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

[Out]

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

Rule 93

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}}+\frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{d e-c f}+\frac {\sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+\log \left ((d e-c f)^{2/3}-\frac {\sqrt [3]{-b e+a f} \sqrt [3]{d e-c f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {(-b e+a f)^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{2 \sqrt [3]{-b e+a f} (d e-c f)^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 836 vs. \(2 (161) = 322\).

Time = 0.31 (sec) , antiderivative size = 1825, normalized size of antiderivative = 9.26 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/2*(sqrt(3)*(b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*sqrt((-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f -
(b*c^2 + 2*a*c*d)*e*f^2)^(1/3)/(b*e - a*f))*log((3*a*c^2*f^2 + (2*b*c*d + a*d^2)*e^2 - 2*(b*c^2 + 2*a*c*d)*e*f
+ 3*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(d*e - c*f)*(b*x + a)^
(2/3)*(d*x + c)^(1/3) + (3*b*d^2*e^2 - 2*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)*x - sqrt(3)*(2*(b*d*e^
2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e
^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a
*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x))*sqrt((-b*d^2*e^3 + a*c^2*f^3
+ (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)/(b*e - a*f)))/(f*x + e)) + 2*(-b*d^2*e^3 + a*c^2*f
^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x
+ a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3
)*(b*x + a))/(b*x + a)) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*l
og(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d
+ a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (
2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x))/(b*x + a)))/(b*d^
2*e^3 - a*c^2*f^3 - (2*b*c*d + a*d^2)*e^2*f + (b*c^2 + 2*a*c*d)*e*f^2), 1/2*(2*sqrt(3)*(b*d*e^2 + a*c*f^2 - (b
*c + a*d)*e*f)*sqrt(-(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)/(b*e -
a*f))*arctan(1/3*sqrt(3)*(2*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3
)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^
2)^(1/3)*(a*d*e - a*c*f + (b*d*e - b*c*f)*x))*sqrt(-(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2
+ 2*a*c*d)*e*f^2)^(1/3)/(b*e - a*f))/(a*d^2*e^2 - 2*a*c*d*e*f + a*c^2*f^2 + (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*
f^2)*x)) - 2*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*log(((b*d*e^2
+ a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2
*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a))/(b*x + a)) + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f
- (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) +
(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(
1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(a*d*e - a*c*f + (b*
d*e - b*c*f)*x))/(b*x + a)))/(b*d^2*e^3 - a*c^2*f^3 - (2*b*c*d + a*d^2)*e^2*f + (b*c^2 + 2*a*c*d)*e*f^2)]

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=\int \frac {1}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}} \left (e + f x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx=\int \frac {1}{\left (e+f\,x\right )\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \]

[In]

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

[Out]

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