Integrand size = 22, antiderivative size = 59 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b} \] Output:
2*arctan((-b*x-a)^(1/4)/a^(1/4))/a^(1/4)/b-2*arctanh((-b*x-a)^(1/4)/a^(1/4 ))/a^(1/4)/b
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )\right )}{\sqrt [4]{a} b} \] Input:
Integrate[1/((-a - b*x)^(1/4)*(2*a + b*x)),x]
Output:
(2*(ArcTan[(-a - b*x)^(1/4)/a^(1/4)] - ArcTanh[(-a - b*x)^(1/4)/a^(1/4)])) /(a^(1/4)*b)
Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {73, 827, 216, 219}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {4 \int \frac {\sqrt {-a-b x}}{2 a+b x}d\sqrt [4]{-a-b x}}{b}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {1}{\sqrt {a}-\sqrt {-a-b x}}d\sqrt [4]{-a-b x}-\frac {1}{2} \int \frac {1}{\sqrt {a}+\sqrt {-a-b x}}d\sqrt [4]{-a-b x}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {1}{\sqrt {a}-\sqrt {-a-b x}}d\sqrt [4]{-a-b x}-\frac {\arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {4 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a}}\right )}{b}\) |
Input:
Int[1/((-a - b*x)^(1/4)*(2*a + b*x)),x]
Output:
(-4*(-1/2*ArcTan[(-a - b*x)^(1/4)/a^(1/4)]/a^(1/4) + ArcTanh[(-a - b*x)^(1 /4)/a^(1/4)]/(2*a^(1/4))))/b
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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 = 0.42 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}{\left (-b x -a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}\right )}{b \,a^{\frac {1}{4}}}\) | \(62\) |
default | \(\frac {2 \arctan \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}{\left (-b x -a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}\right )}{b \,a^{\frac {1}{4}}}\) | \(62\) |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )-\ln \left (\frac {-\left (-b x -a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (-b x -a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right )}{b \,a^{\frac {1}{4}}}\) | \(66\) |
Input:
int(1/(-b*x-a)^(1/4)/(b*x+2*a),x,method=_RETURNVERBOSE)
Output:
1/b/a^(1/4)*(2*arctan((-b*x-a)^(1/4)/a^(1/4))-ln(((-b*x-a)^(1/4)+a^(1/4))/ ((-b*x-a)^(1/4)-a^(1/4))))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=-\left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (a b^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (i \, a b^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a b^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{a b^{4}}\right )^{\frac {1}{4}} \log \left (-a b^{3} \left (\frac {1}{a b^{4}}\right )^{\frac {3}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(1/(-b*x-a)^(1/4)/(b*x+2*a),x, algorithm="fricas")
Output:
-(1/(a*b^4))^(1/4)*log(a*b^3*(1/(a*b^4))^(3/4) + (-b*x - a)^(1/4)) + I*(1/ (a*b^4))^(1/4)*log(I*a*b^3*(1/(a*b^4))^(3/4) + (-b*x - a)^(1/4)) - I*(1/(a *b^4))^(1/4)*log(-I*a*b^3*(1/(a*b^4))^(3/4) + (-b*x - a)^(1/4)) + (1/(a*b^ 4))^(1/4)*log(-a*b^3*(1/(a*b^4))^(3/4) + (-b*x - a)^(1/4))
\[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=\int \frac {1}{\sqrt [4]{- a - b x} \left (2 a + b x\right )}\, dx \] Input:
integrate(1/(-b*x-a)**(1/4)/(b*x+2*a),x)
Output:
Integral(1/((-a - b*x)**(1/4)*(2*a + b*x)), x)
Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=\frac {\frac {2 \, \arctan \left (\frac {{\left (-b x - a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (-b x - a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x - a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}}{b} \] Input:
integrate(1/(-b*x-a)^(1/4)/(b*x+2*a),x, algorithm="maxima")
Output:
(2*arctan((-b*x - a)^(1/4)/a^(1/4))/a^(1/4) + log(((-b*x - a)^(1/4) - a^(1 /4))/((-b*x - a)^(1/4) + a^(1/4)))/a^(1/4))/b
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (47) = 94\).
Time = 0.13 (sec) , antiderivative size = 201, normalized size of antiderivative = 3.41 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=-\frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x - a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x - a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (-b x - a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x - a} + \sqrt {-a}\right )}{2 \, \left (-a\right )^{\frac {1}{4}} b} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (-b x - a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x - a} + \sqrt {-a}\right )}{2 \, a b} \] Input:
integrate(1/(-b*x-a)^(1/4)/(b*x+2*a),x, algorithm="giac")
Output:
-sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x - a)^ (1/4))/(-a)^(1/4))/(a*b) - sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2) *(-a)^(1/4) - 2*(-b*x - a)^(1/4))/(-a)^(1/4))/(a*b) - 1/2*sqrt(2)*log(sqrt (2)*(-b*x - a)^(1/4)*(-a)^(1/4) + sqrt(-b*x - a) + sqrt(-a))/((-a)^(1/4)*b ) - 1/2*sqrt(2)*(-a)^(3/4)*log(-sqrt(2)*(-b*x - a)^(1/4)*(-a)^(1/4) + sqrt (-b*x - a) + sqrt(-a))/(a*b)
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {{\left (-a-b\,x\right )}^{1/4}}{a^{1/4}}\right )}{a^{1/4}\,b}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (-a-b\,x\right )}^{1/4}}{a^{1/4}}\right )}{a^{1/4}\,b} \] Input:
int(1/((2*a + b*x)*(- a - b*x)^(1/4)),x)
Output:
(2*atan((- a - b*x)^(1/4)/a^(1/4)))/(a^(1/4)*b) - (2*atanh((- a - b*x)^(1/ 4)/a^(1/4)))/(a^(1/4)*b)
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt [4]{-a-b x} (2 a+b x)} \, dx=\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{4}}-a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )-2 \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{4}}+a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )-\mathrm {log}\left (-a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b x +a}+\sqrt {a}\right )+\mathrm {log}\left (a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b x +a}+\sqrt {a}\right )\right )}{2 a^{\frac {1}{4}} b} \] Input:
int(1/(-b*x-a)^(1/4)/(b*x+2*a),x)
Output:
(a**(3/4)*( - 1)**(3/4)*sqrt(2)*( - 2*atan((2*(a + b*x)**(1/4) - a**(1/4)* sqrt(2))/(a**(1/4)*sqrt(2))) - 2*atan((2*(a + b*x)**(1/4) + a**(1/4)*sqrt( 2))/(a**(1/4)*sqrt(2))) - log( - a**(1/4)*(a + b*x)**(1/4)*sqrt(2) + sqrt( a + b*x) + sqrt(a)) + log(a**(1/4)*(a + b*x)**(1/4)*sqrt(2) + sqrt(a + b*x ) + sqrt(a))))/(2*a*b)