\(\int \frac {\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx\) [1576]

Optimal result

Integrand size = 19, antiderivative size = 172 \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx=\frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{b}-\frac {(b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} b^{4/3} d^{2/3}}-\frac {(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac {(b c-a d) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{2/3}} \]

[Out]

(b*x+a)^(2/3)*(d*x+c)^(1/3)/b-1/6*(-a*d+b*c)*ln(d*x+c)/b^(4/3)/d^(2/3)-1/2*(-a*d+b*c)*ln(-1+d^(1/3)*(b*x+a)^(1
/3)/b^(1/3)/(d*x+c)^(1/3))/b^(4/3)/d^(2/3)-1/3*(-a*d+b*c)*arctan(1/3*3^(1/2)+2/3*d^(1/3)*(b*x+a)^(1/3)/b^(1/3)
/(d*x+c)^(1/3)*3^(1/2))/b^(4/3)/d^(2/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {52, 61} \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx=-\frac {(b c-a d) \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d^{2/3}}-\frac {(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac {(b c-a d) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{2/3}}+\frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{b} \]

[In]

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

[Out]

((a + b*x)^(2/3)*(c + d*x)^(1/3))/b - ((b*c - a*d)*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*b^(
1/3)*(c + d*x)^(1/3))])/(Sqrt[3]*b^(4/3)*d^(2/3)) - ((b*c - a*d)*Log[c + d*x])/(6*b^(4/3)*d^(2/3)) - ((b*c - a
*d)*Log[-1 + (d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(2*b^(4/3)*d^(2/3))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{b}+\frac {(b c-a d) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{3 b} \\ & = \frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{b}-\frac {(b c-a d) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} b^{4/3} d^{2/3}}-\frac {(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac {(b c-a d) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx=\frac {6 \sqrt [3]{b} d^{2/3} (a+b x)^{2/3} \sqrt [3]{c+d x}+2 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-2 (b c-a d) \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+(b c-a d) \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{6 b^{4/3} d^{2/3}} \]

[In]

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

[Out]

(6*b^(1/3)*d^(2/3)*(a + b*x)^(2/3)*(c + d*x)^(1/3) + 2*Sqrt[3]*(b*c - a*d)*ArcTan[(Sqrt[3]*b^(1/3)*(c + d*x)^(
1/3))/(2*d^(1/3)*(a + b*x)^(1/3) + b^(1/3)*(c + d*x)^(1/3))] - 2*(b*c - a*d)*Log[d^(1/3)*(a + b*x)^(1/3) - b^(
1/3)*(c + d*x)^(1/3)] + (b*c - a*d)*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)])/(6*b^(4/3)*d^(2/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (132) = 264\).

Time = 0.25 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.47 \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx=\left [\frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{2} c d - a b d^{2}\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \log \left (-3 \, b d^{2} x - 2 \, b c d - a d^{2} + 3 \, \left (b d^{2}\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} d + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}\right ) - 2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{6 \, b^{2} d^{2}}, \frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} + 6 \, \sqrt {\frac {1}{3}} {\left (b^{2} c d - a b d^{2}\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}}{b d^{2} x + a d^{2}}\right ) - 2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{6 \, b^{2} d^{2}}\right ] \]

[In]

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

[Out]

[1/6*(6*(b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d^2 - 3*sqrt(1/3)*(b^2*c*d - a*b*d^2)*sqrt(-(b*d^2)^(1/3)/b)*log(-3*
b*d^2*x - 2*b*c*d - a*d^2 + 3*(b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3)*d + 3*sqrt(1/3)*(2*(b*x + a)^(1/3)
*(d*x + c)^(2/3)*b*d - (b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (b*d^2)^(1/3)*(b*d*x + a*d))*sqrt(-(b*d
^2)^(1/3)/b)) - 2*(b*d^2)^(2/3)*(b*c - a*d)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d - (b*d^2)^(2/3)*(b*x + a)
)/(b*x + a)) + (b*d^2)^(2/3)*(b*c - a*d)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d + (b*d^2)^(2/3)*(b*x + a)^(2
/3)*(d*x + c)^(1/3) + (b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a)))/(b^2*d^2), 1/6*(6*(b*x + a)^(2/3)*(d*x + c)^(1/
3)*b*d^2 + 6*sqrt(1/3)*(b^2*c*d - a*b*d^2)*sqrt((b*d^2)^(1/3)/b)*arctan(sqrt(1/3)*(2*(b*d^2)^(2/3)*(b*x + a)^(
2/3)*(d*x + c)^(1/3) + (b*d^2)^(1/3)*(b*d*x + a*d))*sqrt((b*d^2)^(1/3)/b)/(b*d^2*x + a*d^2)) - 2*(b*d^2)^(2/3)
*(b*c - a*d)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d - (b*d^2)^(2/3)*(b*x + a))/(b*x + a)) + (b*d^2)^(2/3)*(b
*c - a*d)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d + (b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b*d^2)^(
1/3)*(b*d*x + a*d))/(b*x + a)))/(b^2*d^2)]

Sympy [F]

\[ \int \frac {\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx=\int \frac {\sqrt [3]{c + d x}}{\sqrt [3]{a + b x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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