Integrand size = 19, antiderivative size = 106 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {a+b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{a+b x}}\right )}{a^{3/4} b}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )}{a^{3/4} b} \] Output:
-2^(1/2)*arctan(1/2*(a^(1/2)-(b*x+a)^(1/2))*2^(1/2)/a^(1/4)/(b*x+a)^(1/4)) /a^(3/4)/b+2^(1/2)*arctanh(2^(1/2)*a^(1/4)*(b*x+a)^(1/4)/(a^(1/2)+(b*x+a)^ (1/2)))/a^(3/4)/b
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=\frac {\sqrt {2} \left (-\arctan \left (\frac {\sqrt {a}-\sqrt {a+b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{a+b x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )\right )}{a^{3/4} b} \] Input:
Integrate[1/((a + b*x)^(3/4)*(2*a + b*x)),x]
Output:
(Sqrt[2]*(-ArcTan[(Sqrt[a] - Sqrt[a + b*x])/(Sqrt[2]*a^(1/4)*(a + b*x)^(1/ 4))] + ArcTanh[(Sqrt[2]*a^(1/4)*(a + b*x)^(1/4))/(Sqrt[a] + Sqrt[a + b*x]) ]))/(a^(3/4)*b)
Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {73, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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 |
| \(\displaystyle \frac {4 \int \frac {1}{2 a+b x}d\sqrt [4]{a+b x}}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {4 \left (\frac {\int \frac {\sqrt {a}-\sqrt {a+b x}}{2 a+b x}d\sqrt [4]{a+b x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}+\sqrt {a+b x}}{2 a+b x}d\sqrt [4]{a+b x}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {4 \left (\frac {\int \frac {\sqrt {a}-\sqrt {a+b x}}{2 a+b x}d\sqrt [4]{a+b x}}{2 \sqrt {a}}+\frac {\frac {1}{2} \int \frac {1}{\sqrt {a}-\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}+\frac {1}{2} \int \frac {1}{\sqrt {a}+\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {4 \left (\frac {\frac {\int \frac {1}{-\sqrt {a+b x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\int \frac {1}{-\sqrt {a+b x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {a+b x}}{2 a+b x}d\sqrt [4]{a+b x}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \left (\frac {\int \frac {\sqrt {a}-\sqrt {a+b x}}{2 a+b x}d\sqrt [4]{a+b x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {4 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{a+b x}}{\sqrt {a}-\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a}+\sqrt {2} \sqrt [4]{a+b x}\right )}{\sqrt {a}+\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{a+b x}}{\sqrt {a}-\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {2} \sqrt [4]{a}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a}+\sqrt {2} \sqrt [4]{a+b x}\right )}{\sqrt {a}+\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{a+b x}}{\sqrt {a}-\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt {2} \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{a}+\sqrt {2} \sqrt [4]{a+b x}}{\sqrt {a}+\sqrt {2} \sqrt [4]{a+b x} \sqrt [4]{a}+\sqrt {a+b x}}d\sqrt [4]{a+b x}}{2 \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{a+b x}+\sqrt {a+b x}+\sqrt {a}\right )}{2 \sqrt {2} \sqrt [4]{a}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{a+b x}+\sqrt {a+b x}+\sqrt {a}\right )}{2 \sqrt {2} \sqrt [4]{a}}}{2 \sqrt {a}}\right )}{b}\) |
Input:
Int[1/((a + b*x)^(3/4)*(2*a + b*x)),x]
Output:
(4*((-(ArcTan[1 - (Sqrt[2]*(a + b*x)^(1/4))/a^(1/4)]/(Sqrt[2]*a^(1/4))) + ArcTan[1 + (Sqrt[2]*(a + b*x)^(1/4))/a^(1/4)]/(Sqrt[2]*a^(1/4)))/(2*Sqrt[a ]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*(a + b*x)^(1/4) + Sqrt[a + b*x]]/ (Sqrt[2]*a^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*(a + b*x)^(1/4) + Sqrt[a + b*x]]/(2*Sqrt[2]*a^(1/4)))/(2*Sqrt[a])))/b
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, 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_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
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]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {b x +a}+a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}{\sqrt {b x +a}-a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )\right )}{2 b \,a^{\frac {3}{4}}}\) | \(107\) |
| default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {b x +a}+a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}{\sqrt {b x +a}-a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )\right )}{2 b \,a^{\frac {3}{4}}}\) | \(107\) |
| pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {b x +a}+a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}{\sqrt {b x +a}-a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {a}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \left (b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \left (b x +a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}+1\right )\right )}{2 b \,a^{\frac {3}{4}}}\) | \(107\) |
Input:
int(1/(b*x+a)^(3/4)/(b*x+2*a),x,method=_RETURNVERBOSE)
Output:
1/2/b/a^(3/4)*2^(1/2)*(ln(((b*x+a)^(1/2)+a^(1/4)*(b*x+a)^(1/4)*2^(1/2)+a^( 1/2))/((b*x+a)^(1/2)-a^(1/4)*(b*x+a)^(1/4)*2^(1/2)+a^(1/2)))+2*arctan(2^(1 /2)/a^(1/4)*(b*x+a)^(1/4)+1)-2*arctan(-2^(1/2)/a^(1/4)*(b*x+a)^(1/4)+1))
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=\left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (a b \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) + i \, \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (i \, a b \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) - i \, \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a b \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} \log \left (-a b \left (-\frac {1}{a^{3} b^{4}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}}\right ) \] 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 = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {b x + a} + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {b x + a} + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{2 \, b} \] Input:
integrate(1/(b*x+a)^(3/4)/(b*x+2*a),x, algorithm="maxima")
Output:
1/2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) + 2*(b*x + a)^(1/4))/a^ (1/4))/a^(3/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4) - 2*(b*x + a)^(1/4))/a^(1/4))/a^(3/4) + sqrt(2)*log(sqrt(2)*(b*x + a)^(1/4)*a^(1/4) + sqrt(b*x + a) + sqrt(a))/a^(3/4) - sqrt(2)*log(-sqrt(2)*(b*x + a)^(1/4)* a^(1/4) + sqrt(b*x + a) + sqrt(a))/a^(3/4))/b
Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} + 2 \, {\left (b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}} b} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} - 2 \, {\left (b x + a\right )}^{\frac {1}{4}}\right )}}{2 \, a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}} b} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {b x + a} + \sqrt {a}\right )}{2 \, a^{\frac {3}{4}} b} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (b x + a\right )}^{\frac {1}{4}} a^{\frac {1}{4}} + \sqrt {b x + a} + \sqrt {a}\right )}{2 \, a^{\frac {3}{4}} b} \] Input:
integrate(1/(b*x+a)^(3/4)/(b*x+2*a),x, algorithm="giac")
Output:
sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) + 2*(b*x + a)^(1/4))/a^(1/4))/ (a^(3/4)*b) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4) - 2*(b*x + a)^( 1/4))/a^(1/4))/(a^(3/4)*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) - 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 = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\left (a+b\,x\right )}^{1/4}}{{\left (-a\right )}^{1/4}}\right )}{{\left (-a\right )}^{3/4}\,b}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a+b\,x\right )}^{1/4}}{{\left (-a\right )}^{1/4}}\right )}{{\left (-a\right )}^{3/4}\,b} \] 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 = 115, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(a+b x)^{3/4} (2 a+b x)} \, dx=\frac {\sqrt {2}\, \left (2 \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{4}}-a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{4}}+a^{\frac {1}{4}} \sqrt {2}}{a^{\frac {1}{4}} \sqrt {2}}\right )-\mathrm {log}\left (-a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b x +a}+\sqrt {a}\right )+\mathrm {log}\left (a^{\frac {1}{4}} \left (b x +a \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b x +a}+\sqrt {a}\right )\right )}{2 a^{\frac {3}{4}} b} \] 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 *a*b)