\(\int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx\) [1589]

Optimal result

Integrand size = 19, antiderivative size = 126 \[ \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 b^{2/3} \sqrt [3]{d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {61} \[ \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}} \]

[In]

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

[Out]

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

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 {\sqrt {3} \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 )}{b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 b^{2/3} \sqrt [3]{d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-2 \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 b^{2/3} \sqrt [3]{d}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (90) = 180\).

Time = 0.24 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.12 \[ \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx=\left [\frac {\sqrt {3} b d \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (3 \, b^{2} d x + b^{2} c + 2 \, a b d + 3 \, \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b + \sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}\right ) + \left (-b^{2} d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right )}{2 \, b^{2} d}, \frac {2 \, \sqrt {3} b d \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}}{3 \, {\left (b^{2} d x + b^{2} c\right )}}\right ) + \left (-b^{2} d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right )}{2 \, b^{2} d}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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