Integrand size = 17, antiderivative size = 140 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}} \] Output:
-3^(1/2)*arctan(1/3*(1+2*b^(1/3)*(d*x+c)^(1/3)/(-a*d+b*c)^(1/3))*3^(1/2))/ b^(1/3)/(-a*d+b*c)^(2/3)-1/2*ln(b*x+a)/b^(1/3)/(-a*d+b*c)^(2/3)+3/2*ln((-a *d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/b^(1/3)/(-a*d+b*c)^(2/3)
Time = 0.15 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b} (-b c+a d)^{2/3}} \] Input:
Integrate[1/((a + b*x)*(c + d*x)^(2/3)),x]
Output:
-1/2*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x)^(1/3))/(-(b*c) + a*d)^(1/ 3))/Sqrt[3]] - 2*Log[(-(b*c) + a*d)^(1/3) + b^(1/3)*(c + d*x)^(1/3)] + Log [(-(b*c) + a*d)^(2/3) - b^(1/3)*(-(b*c) + a*d)^(1/3)*(c + d*x)^(1/3) + b^( 2/3)*(c + d*x)^(2/3)])/(b^(1/3)*(-(b*c) + a*d)^(2/3))
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {69, 16, 1082, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle -\frac {3 \int \frac {1}{\frac {(b c-a d)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{c+d x} \sqrt [3]{b c-a d}}{\sqrt [3]{b}}+(c+d x)^{2/3}}d\sqrt [3]{c+d x}}{2 b^{2/3} \sqrt [3]{b c-a d}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{b}}-\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3 \int \frac {1}{\frac {(b c-a d)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{c+d x} \sqrt [3]{b c-a d}}{\sqrt [3]{b}}+(c+d x)^{2/3}}d\sqrt [3]{c+d x}}{2 b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \int \frac {1}{-(c+d x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}\) |
Input:
Int[1/((a + b*x)*(c + d*x)^(2/3)),x]
Output:
-((Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*(c + d*x)^(1/3))/(b*c - a*d)^(1/3))/Sqrt [3]])/(b^(1/3)*(b*c - a*d)^(2/3))) - Log[a + b*x]/(2*b^(1/3)*(b*c - a*d)^( 2/3)) + (3*Log[(b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)])/(2*b^(1/3)*(b *c - a*d)^(2/3))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\right )+2 \ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )-\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\) | \(143\) |
| derivativedivides | \(\frac {\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\) | \(160\) |
| default | \(\frac {\ln \left (\left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (x d +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (x d +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\) | \(160\) |
Input:
int(1/(b*x+a)/(d*x+c)^(2/3),x,method=_RETURNVERBOSE)
Output:
1/2*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(-2*(d*x+c)^(1/3)+((a*d-b*c)/b)^(1/3))/ ((a*d-b*c)/b)^(1/3))+2*ln((d*x+c)^(1/3)+((a*d-b*c)/b)^(1/3))-ln((d*x+c)^(2 /3)-((a*d-b*c)/b)^(1/3)*(d*x+c)^(1/3)+((a*d-b*c)/b)^(2/3)))/b/((a*d-b*c)/b )^(2/3)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (109) = 218\).
Time = 0.13 (sec) , antiderivative size = 900, normalized size of antiderivative = 6.43 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx =\text {Too large to display} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(2/3),x, algorithm="fricas")
Output:
[-1/2*(sqrt(3)*(b^2*c - a*b*d)*sqrt(-(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^( 1/3)/b)*log(-(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2 + 2*(b^2*c*d - a*b*d^2)*x + sqrt(3)*(2*(b^2*c - a*b*d)*(d*x + c)^(2/3) - (b^3*c^2 - 2*a*b^2*c*d + a^2* b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3))*sqrt(-(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b) - 3*(b^3*c^ 2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d)*(d*x + c)^(1/3))/(b*x + a)) + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c )^(2/3) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3)) - 2*(b^3*c^2 - 2*a*b^2* c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c)^(1/3) + (b^3*c^2 - 2 *a*b^2*c*d + a^2*b*d^2)^(2/3)))/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2), -1/2* (2*sqrt(3)*(b^2*c - a*b*d)*sqrt((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/ b)*arctan(1/3*sqrt(3)*((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a* d) + 2*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3))*sqrt((b^ 3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)/b)/(b^2*c^2 - 2*a*b*c*d + a^2*d^2)) + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c )^(2/3) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(1/3)*(b*c - a*d) - (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)^(2/3)*(d*x + c)^(1/3)) - 2*(b^3*c^2 - 2*a*b^2* c*d + a^2*b*d^2)^(2/3)*log(-(b^2*c - a*b*d)*(d*x + c)^(1/3) + (b^3*c^2 - 2 *a*b^2*c*d + a^2*b*d^2)^(2/3)))/(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)]
\[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {2}{3}}}\, dx \] Input:
integrate(1/(b*x+a)/(d*x+c)**(2/3),x)
Output:
Integral(1/((a + b*x)*(c + d*x)**(2/3)), x)
Exception generated. \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(2/3),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c - \sqrt {3} a b d} - \frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{2} c - a b d\right )}} + \frac {\left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c - a d} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(2/3),x, algorithm="giac")
Output:
-3*(b^3*c - a*b^2*d)^(1/3)*arctan(1/3*sqrt(3)*(2*(d*x + c)^(1/3) + ((b*c - a*d)/b)^(1/3))/((b*c - a*d)/b)^(1/3))/(sqrt(3)*b^2*c - sqrt(3)*a*b*d) - 1 /2*(b^3*c - a*b^2*d)^(1/3)*log((d*x + c)^(2/3) + (d*x + c)^(1/3)*((b*c - a *d)/b)^(1/3) + ((b*c - a*d)/b)^(2/3))/(b^2*c - a*b*d) + ((b*c - a*d)/b)^(1 /3)*log(abs((d*x + c)^(1/3) - ((b*c - a*d)/b)^(1/3)))/(b*c - a*d)
Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}-\frac {9\,b^3\,c-9\,a\,b^2\,d}{b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )}{b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}-\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}-\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}+\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \] Input:
int(1/((a + b*x)*(c + d*x)^(2/3)),x)
Output:
log(9*b^2*(c + d*x)^(1/3) - (9*b^3*c - 9*a*b^2*d)/(b^(1/3)*(a*d - b*c)^(2/ 3)))/(b^(1/3)*(a*d - b*c)^(2/3)) + (log(9*b^2*(c + d*x)^(1/3) - ((3^(1/2)* 1i - 1)*(9*b^3*c - 9*a*b^2*d))/(2*b^(1/3)*(a*d - b*c)^(2/3)))*(3^(1/2)*1i - 1))/(2*b^(1/3)*(a*d - b*c)^(2/3)) - (log(9*b^2*(c + d*x)^(1/3) + ((3^(1/ 2)*1i + 1)*(9*b^3*c - 9*a*b^2*d))/(2*b^(1/3)*(a*d - b*c)^(2/3)))*(3^(1/2)* 1i + 1))/(2*b^(1/3)*(a*d - b*c)^(2/3))
Time = 0.18 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.67 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{6}}}{b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{6}}}{b^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}}}\right )+2 \,\mathrm {log}\left (\left (a d -b c \right )^{\frac {1}{3}}+b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}\right )-\mathrm {log}\left (-b^{\frac {1}{6}} \left (d x +c \right )^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}+\left (a d -b c \right )^{\frac {1}{3}}+b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}\right )-\mathrm {log}\left (b^{\frac {1}{6}} \left (d x +c \right )^{\frac {1}{6}} \left (a d -b c \right )^{\frac {1}{6}} \sqrt {3}+\left (a d -b c \right )^{\frac {1}{3}}+b^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}\right )}{2 b^{\frac {1}{3}} \left (a d -b c \right )^{\frac {2}{3}}} \] Input:
int(1/(b*x+a)/(d*x+c)^(2/3),x)
Output:
( - 2*sqrt(3)*atan((b**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) - 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b*c)**(1/6))) - 2*sqrt(3)*atan((b**(1/6)*(a* d - b*c)**(1/6)*sqrt(3) + 2*b**(1/3)*(c + d*x)**(1/6))/(b**(1/6)*(a*d - b* c)**(1/6))) + 2*log((a*d - b*c)**(1/3) + b**(1/3)*(c + d*x)**(1/3)) - log( - b**(1/6)*(c + d*x)**(1/6)*(a*d - b*c)**(1/6)*sqrt(3) + (a*d - b*c)**(1/ 3) + b**(1/3)*(c + d*x)**(1/3)) - log(b**(1/6)*(c + d*x)**(1/6)*(a*d - b*c )**(1/6)*sqrt(3) + (a*d - b*c)**(1/3) + b**(1/3)*(c + d*x)**(1/3)))/(2*b** (1/3)*(a*d - b*c)**(2/3))