∫ 1__________ 3√____ a + bx (c+dx)2∕3(e+fx) dx [3019]

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

Optimal. Leaf size=197 \[ -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}}+\frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}} \]

[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)

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Rubi [A]
time = 0.05, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {93} \begin {gather*} -\frac {\sqrt {3} \text {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}} \end {gather*}

Antiderivative was successfully veriﬁed.

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

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Mathematica [A]
time = 0.41, size = 225, normalized size = 1.14 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully veriﬁed.

[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))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )}\, dx\]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (172) = 344\).
time = 1.01, size = 1803, normalized size = 9.15 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(3)*(a*c*f^2 + b*d*e^2 - (b*c + a*d)*f*e)*sqrt(-(a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e +

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

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

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

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

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

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

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

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

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

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

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

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

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

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

f*e)*sqrt((a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(1/3)/(a*f - b*e))*arcta

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

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

f*x + a*c*f - (b*d*x + a*d)*e))*sqrt((a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^

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

2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(2/3)*log(((a*c*f^2 + b*d*e^2 - (b*c +

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

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

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

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

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

a)))/(a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}} \left (e + f x\right )}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (e+f\,x\right )\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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