Integrand size = 22, antiderivative size = 59 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{a^{3/4} b}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{a^{3/4} b} \] Output:
-2*arctan((-b*x-a)^(1/4)/a^(1/4))/a^(3/4)/b-2*arctanh((-b*x-a)^(1/4)/a^(1/ 4))/a^(3/4)/b
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )\right )}{a^{3/4} b} \] Input:
Integrate[1/((-a - b*x)^(3/4)*(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^(3/4)*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, 756, 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.
\(\displaystyle \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {4 \int \frac {1}{2 a+b x}d\sqrt [4]{-a-b x}}{b}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle -\frac {4 \left (\frac {\int \frac {1}{\sqrt {a}-\sqrt {-a-b x}}d\sqrt [4]{-a-b x}}{2 \sqrt {a}}+\frac {\int \frac {1}{\sqrt {a}+\sqrt {-a-b x}}d\sqrt [4]{-a-b x}}{2 \sqrt {a}}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {4 \left (\frac {\int \frac {1}{\sqrt {a}-\sqrt {-a-b x}}d\sqrt [4]{-a-b x}}{2 \sqrt {a}}+\frac {\arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {4 \left (\frac {\arctan \left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{-a-b x}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{b}\) |
Input:
Int[1/((-a - b*x)^(3/4)*(2*a + b*x)),x]
Output:
(-4*(ArcTan[(-a - b*x)^(1/4)/a^(1/4)]/(2*a^(3/4)) + ArcTanh[(-a - b*x)^(1/ 4)/a^(1/4)]/(2*a^(3/4))))/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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}{\left (-b x -a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{b \,a^{\frac {3}{4}}}\) | \(61\) |
default | \(-\frac {\ln \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}{\left (-b x -a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{b \,a^{\frac {3}{4}}}\) | \(61\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {-\left (-b x -a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (-b x -a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (-b x -a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{b \,a^{\frac {3}{4}}}\) | \(65\) |
Input:
int(1/(-b*x-a)^(3/4)/(b*x+2*a),x,method=_RETURNVERBOSE)
Output:
-1/b/a^(3/4)*(ln(((-b*x-a)^(1/4)+a^(1/4))/((-b*x-a)^(1/4)-a^(1/4)))+2*arct an((-b*x-a)^(1/4)/a^(1/4)))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (a b \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (i \, a b \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a b \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (-a b \left (\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (-b x - a\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(1/(-b*x-a)^(3/4)/(b*x+2*a),x, algorithm="fricas")
Output:
-(1/(a^3*b^4))^(1/4)*log(a*b*(1/(a^3*b^4))^(1/4) + (-b*x - a)^(1/4)) - I*( 1/(a^3*b^4))^(1/4)*log(I*a*b*(1/(a^3*b^4))^(1/4) + (-b*x - a)^(1/4)) + I*( 1/(a^3*b^4))^(1/4)*log(-I*a*b*(1/(a^3*b^4))^(1/4) + (-b*x - a)^(1/4)) + (1 /(a^3*b^4))^(1/4)*log(-a*b*(1/(a^3*b^4))^(1/4) + (-b*x - a)^(1/4))
\[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=\int \frac {1}{\left (- a - b x\right )^{\frac {3}{4}} \cdot \left (2 a + b x\right )}\, dx \] Input:
integrate(1/(-b*x-a)**(3/4)/(b*x+2*a),x)
Output:
Integral(1/((-a - b*x)**(3/4)*(2*a + b*x)), x)
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\frac {\frac {2 \, \arctan \left (\frac {{\left (-b x - a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {\log \left (\frac {{\left (-b x - a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x - a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}}{b} \] Input:
integrate(1/(-b*x-a)^(3/4)/(b*x+2*a),x, algorithm="maxima")
Output:
-(2*arctan((-b*x - a)^(1/4)/a^(1/4))/a^(3/4) - log(((-b*x - a)^(1/4) - a^( 1/4))/((-b*x - a)^(1/4) + a^(1/4)))/a^(3/4))/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 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x - a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x - a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a b} - \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} {\left (-b x - a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x - a} + \sqrt {-a}\right )}{2 \, a b} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (-b x - a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x - a} + \sqrt {-a}\right )}{2 \, \left (-a\right )^{\frac {3}{4}} b} \] Input:
integrate(1/(-b*x-a)^(3/4)/(b*x+2*a),x, algorithm="giac")
Output:
-sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(-b*x - a)^ (1/4))/(-a)^(1/4))/(a*b) - sqrt(2)*(-a)^(1/4)*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)*(-a)^(1/ 4)*log(sqrt(2)*(-b*x - a)^(1/4)*(-a)^(1/4) + sqrt(-b*x - a) + sqrt(-a))/(a *b) - 1/2*sqrt(2)*log(-sqrt(2)*(-b*x - a)^(1/4)*(-a)^(1/4) + sqrt(-b*x - a ) + sqrt(-a))/((-a)^(3/4)*b)
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\left (-a-b\,x\right )}^{1/4}}{a^{1/4}}\right )}{a^{3/4}\,b}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (-a-b\,x\right )}^{1/4}}{a^{1/4}}\right )}{a^{3/4}\,b} \] Input:
int(1/((2*a + b*x)*(- a - b*x)^(3/4)),x)
Output:
- (2*atan((- a - b*x)^(1/4)/a^(1/4)))/(a^(3/4)*b) - (2*atanh((- a - b*x)^( 1/4)/a^(1/4)))/(a^(3/4)*b)
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.00 \[ \int \frac {1}{(-a-b x)^{3/4} (2 a+b x)} \, dx=-\frac {\sqrt {2}\, \left (2 \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{4}}-a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{4}}+a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )-\mathrm {log}\left (-a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b x +a}+\sqrt {a}\right )+\mathrm {log}\left (a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b x +a}+\sqrt {a}\right )\right ) \left (-1\right )^{\frac {1}{4}}}{2 a^{\frac {3}{4}} b} \] Input:
int(1/(-b*x-a)^(3/4)/(b*x+2*a),x)
Output:
(a**(1/4)*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)*sqr t(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 *( - 1)**(3/4)*a*b)