3.3011 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx\)

Optimal. Leaf size=339 \[ -\frac{\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac{3 \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}+\frac{\sqrt{3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f \sqrt [3]{d e-c f}}-\frac{3 \sqrt [3]{b} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{d} f}-\frac{\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]

[Out]

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

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Rubi [A]  time = 0.101591, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {105, 59, 91} \[ -\frac{\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac{3 \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}+\frac{\sqrt{3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{f \sqrt [3]{d e-c f}}-\frac{3 \sqrt [3]{b} \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac{\sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt [3]{d} f}-\frac{\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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]

Rule 91

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

Mathematica [C]  time = 0.0608696, size = 114, normalized size = 0.34 \[ \frac{3 \sqrt [3]{a+b x} \left (\sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{d (a+b x)}{a d-b c}\right )-\, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{f \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(1/3)*(((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (d*(a + b*x))/(-(b*c) +
a*d)] - Hypergeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/(f*(c + d*x)^(1/3))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{fx+e}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{1}{3}}{\left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.80597, size = 1239, normalized size = 3.65 \begin{align*} -\frac{2 \, \sqrt{3} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (d e - c f\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{2}{3}} + \sqrt{3}{\left (b c e - a c f +{\left (b d e - a d f\right )} x\right )}}{3 \,{\left (b c e - a c f +{\left (b d e - a d f\right )} x\right )}}\right ) + 2 \, \sqrt{3} \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 (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} \log \left (\frac{{\left (d x + c\right )} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} +{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{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 (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} \log \left (-\frac{{\left (d x + c\right )} \left (\frac{b e - a f}{d e - c f}\right )^{\frac{1}{3}} -{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 2 \, \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 )}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{1}{3}}{\left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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