Integrand size = 17, antiderivative size = 95 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{-b c+a d}}\right )}{d^{3/4} \sqrt [4]{-b c+a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{-b c+a d}}\right )}{d^{3/4} \sqrt [4]{-b c+a d}} \] Output:
2*arctan(d^(1/4)*(b*x+a)^(1/4)/(a*d-b*c)^(1/4))/d^(3/4)/(a*d-b*c)^(1/4)-2* arctanh(d^(1/4)*(b*x+a)^(1/4)/(a*d-b*c)^(1/4))/d^(3/4)/(a*d-b*c)^(1/4)
Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=-\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {b c-a d}-\sqrt {d} \sqrt {a+b x}}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{a+b x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{b c-a d} \sqrt [4]{a+b x}}{\sqrt {b c-a d}+\sqrt {d} \sqrt {a+b x}}\right )\right )}{d^{3/4} \sqrt [4]{b c-a d}} \] Input:
Integrate[1/((a + b*x)^(1/4)*(c + d*x)),x]
Output:
-((Sqrt[2]*(ArcTan[(Sqrt[b*c - a*d] - Sqrt[d]*Sqrt[a + b*x])/(Sqrt[2]*d^(1 /4)*(b*c - a*d)^(1/4)*(a + b*x)^(1/4))] + ArcTanh[(Sqrt[2]*d^(1/4)*(b*c - a*d)^(1/4)*(a + b*x)^(1/4))/(Sqrt[b*c - a*d] + Sqrt[d]*Sqrt[a + b*x])]))/( d^(3/4)*(b*c - a*d)^(1/4)))
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {73, 27, 827, 218, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {4 \int \frac {b \sqrt {a+b x}}{b \left (c-\frac {a d}{b}\right )+d (a+b x)}d\sqrt [4]{a+b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {\sqrt {a+b x}}{b c-a d+d (a+b x)}d\sqrt [4]{a+b x}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle 4 \left (\frac {\int \frac {1}{\sqrt {a d-b c}+\sqrt {d} \sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {d}}-\frac {\int \frac {1}{\sqrt {a d-b c}-\sqrt {d} \sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {d}}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle 4 \left (\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{a d-b c}}\right )}{2 d^{3/4} \sqrt [4]{a d-b c}}-\frac {\int \frac {1}{\sqrt {a d-b c}-\sqrt {d} \sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {d}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 4 \left (\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{a d-b c}}\right )}{2 d^{3/4} \sqrt [4]{a d-b c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{a d-b c}}\right )}{2 d^{3/4} \sqrt [4]{a d-b c}}\right )\) |
Input:
Int[1/((a + b*x)^(1/4)*(c + d*x)),x]
Output:
4*(ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(-(b*c) + a*d)^(1/4)]/(2*d^(3/4)*(-(b* c) + a*d)^(1/4)) - ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(-(b*c) + a*d)^(1/4)] /(2*d^(3/4)*(-(b*c) + a*d)^(1/4)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Time = 0.48 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(-\frac {-2 \arctan \left (\frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\left (b x +a \right )^{\frac {1}{4}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {1}{4}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right )}{\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}} d}\) | \(96\) |
derivativedivides | \(\frac {2 \arctan \left (\frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (b x +a \right )^{\frac {1}{4}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {1}{4}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right )}{d \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\) | \(97\) |
default | \(\frac {2 \arctan \left (\frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (b x +a \right )^{\frac {1}{4}}+\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {1}{4}}-\left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\right )}{d \left (\frac {a d -b c}{d}\right )^{\frac {1}{4}}}\) | \(97\) |
Input:
int(1/(b*x+a)^(1/4)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/((a*d-b*c)/d)^(1/4)*(-2*arctan((b*x+a)^(1/4)/((a*d-b*c)/d)^(1/4))+ln((( b*x+a)^(1/4)+((a*d-b*c)/d)^(1/4))/((b*x+a)^(1/4)-((a*d-b*c)/d)^(1/4))))/d
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {1}{4}} \log \left ({\left (b c d^{2} - a d^{3}\right )} \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {3}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {1}{4}} \log \left (-{\left (b c d^{2} - a d^{3}\right )} \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {3}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) - i \, \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {1}{4}} \log \left ({\left (i \, b c d^{2} - i \, a d^{3}\right )} \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {3}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) + i \, \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {1}{4}} \log \left ({\left (-i \, b c d^{2} + i \, a d^{3}\right )} \left (-\frac {1}{b c d^{3} - a d^{4}}\right )^{\frac {3}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(1/(b*x+a)^(1/4)/(d*x+c),x, algorithm="fricas")
Output:
(-1/(b*c*d^3 - a*d^4))^(1/4)*log((b*c*d^2 - a*d^3)*(-1/(b*c*d^3 - a*d^4))^ (3/4) + (b*x + a)^(1/4)) - (-1/(b*c*d^3 - a*d^4))^(1/4)*log(-(b*c*d^2 - a* d^3)*(-1/(b*c*d^3 - a*d^4))^(3/4) + (b*x + a)^(1/4)) - I*(-1/(b*c*d^3 - a* d^4))^(1/4)*log((I*b*c*d^2 - I*a*d^3)*(-1/(b*c*d^3 - a*d^4))^(3/4) + (b*x + a)^(1/4)) + I*(-1/(b*c*d^3 - a*d^4))^(1/4)*log((-I*b*c*d^2 + I*a*d^3)*(- 1/(b*c*d^3 - a*d^4))^(3/4) + (b*x + a)^(1/4))
\[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\int \frac {1}{\sqrt [4]{a + b x} \left (c + d x\right )}\, dx \] Input:
integrate(1/(b*x+a)**(1/4)/(d*x+c),x)
Output:
Integral(1/((a + b*x)**(1/4)*(c + d*x)), x)
Exception generated. \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a)^(1/4)/(d*x+c),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
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (75) = 150\).
Time = 0.13 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.63 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\frac {2 \, {\left (b c d^{3} - a d^{4}\right )}^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + 2 \, {\left (b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}} + \frac {2 \, {\left (b c d^{3} - a d^{4}\right )}^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} - 2 \, {\left (b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}} - \frac {{\left (b c d^{3} - a d^{4}\right )}^{\frac {3}{4}} \log \left (\sqrt {2} {\left (b x + a\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {b x + a} + \sqrt {\frac {b c - a d}{d}}\right )}{\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}} + \frac {{\left (b c d^{3} - a d^{4}\right )}^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (b x + a\right )}^{\frac {1}{4}} \left (\frac {b c - a d}{d}\right )^{\frac {1}{4}} + \sqrt {b x + a} + \sqrt {\frac {b c - a d}{d}}\right )}{\sqrt {2} b c d^{3} - \sqrt {2} a d^{4}} \] Input:
integrate(1/(b*x+a)^(1/4)/(d*x+c),x, algorithm="giac")
Output:
2*(b*c*d^3 - a*d^4)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*((b*c - a*d)/d)^(1/4 ) + 2*(b*x + a)^(1/4))/((b*c - a*d)/d)^(1/4))/(sqrt(2)*b*c*d^3 - sqrt(2)*a *d^4) + 2*(b*c*d^3 - a*d^4)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*((b*c - a*d )/d)^(1/4) - 2*(b*x + a)^(1/4))/((b*c - a*d)/d)^(1/4))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4) - (b*c*d^3 - a*d^4)^(3/4)*log(sqrt(2)*(b*x + a)^(1/4)*((b*c - a*d)/d)^(1/4) + sqrt(b*x + a) + sqrt((b*c - a*d)/d))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4) + (b*c*d^3 - a*d^4)^(3/4)*log(-sqrt(2)*(b*x + a)^(1/4)*((b *c - a*d)/d)^(1/4) + sqrt(b*x + a) + sqrt((b*c - a*d)/d))/(sqrt(2)*b*c*d^3 - sqrt(2)*a*d^4)
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {d^{1/4}\,{\left (a+b\,x\right )}^{1/4}}{{\left (a\,d-b\,c\right )}^{1/4}}\right )}{d^{3/4}\,{\left (a\,d-b\,c\right )}^{1/4}}-\frac {2\,\mathrm {atanh}\left (\frac {d^{1/4}\,{\left (a+b\,x\right )}^{1/4}}{{\left (a\,d-b\,c\right )}^{1/4}}\right )}{d^{3/4}\,{\left (a\,d-b\,c\right )}^{1/4}} \] Input:
int(1/((a + b*x)^(1/4)*(c + d*x)),x)
Output:
(2*atan((d^(1/4)*(a + b*x)^(1/4))/(a*d - b*c)^(1/4)))/(d^(3/4)*(a*d - b*c) ^(1/4)) - (2*atanh((d^(1/4)*(a + b*x)^(1/4))/(a*d - b*c)^(1/4)))/(d^(3/4)* (a*d - b*c)^(1/4))
Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt [4]{a+b x} (c+d x)} \, dx=\frac {2 \mathit {atan} \left (\frac {\sqrt {d}\, \left (b x +a \right )^{\frac {1}{4}}}{d^{\frac {1}{4}} \left (a d -b c \right )^{\frac {1}{4}}}\right )-\mathrm {log}\left (\left (a d -b c \right )^{\frac {1}{4}}+d^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}}\right )+\mathrm {log}\left (-\left (a d -b c \right )^{\frac {1}{4}}+d^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}}\right )}{d^{\frac {3}{4}} \left (a d -b c \right )^{\frac {1}{4}}} \] Input:
int(1/(b*x+a)^(1/4)/(d*x+c),x)
Output:
(d**(1/4)*(a*d - b*c)**(3/4)*(2*atan((sqrt(d)*(a + b*x)**(1/4))/(d**(1/4)* (a*d - b*c)**(1/4))) - log((a*d - b*c)**(1/4) + d**(1/4)*(a + b*x)**(1/4)) + log( - (a*d - b*c)**(1/4) + d**(1/4)*(a + b*x)**(1/4))))/(d*(a*d - b*c) )