3.31.11 \(\int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx\) [3011]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {132, 61, 12, 93} \begin {gather*} \frac {\sqrt {3} \sqrt [3]{b e-a f} \text {ArcTan}\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 {\sqrt {3} \sqrt [3]{b} \text {ArcTan}\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 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 {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]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \end {gather*}

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 61

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

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]

Rule 132

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

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*}

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Mathematica [A]
time = 6.08, size = 491, normalized size = 1.45 \begin {gather*} \frac {\frac {2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{d} \sqrt [3]{a+b x}+2 \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt {3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}-2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{-d e+c f}}-\frac {2 \sqrt [3]{b} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{b e-a f} \log \left (\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}+\sqrt [3]{b e-a f} \sqrt [3]{c+d x}\right )}{\sqrt [3]{-d e+c f}}+\frac {\sqrt [3]{b} \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{\sqrt [3]{d}}+\frac {\sqrt [3]{b e-a f} \log \left ((-d e+c f)^{2/3} (a+b x)^{2/3}-\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+(b e-a f)^{2/3} (c+d x)^{2/3}\right )}{\sqrt [3]{-d e+c f}}}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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 = 0.00, 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((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 = 3.33, size = 524, normalized size = 1.55 \begin {gather*} -\frac {2 \, \sqrt {3} \left (\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (c f - d e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} + \sqrt {3} {\left (a d f x + a c f - {\left (b d x + b c\right )} e\right )}}{3 \, {\left (a d f x + a c f - {\left (b d x + b c\right )} e\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 {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {a f - b e}{c f - d e}\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 {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {a f - b e}{c f - d e}\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 {gather*}

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

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 = 0.00, 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((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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