Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{b^{3/4}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{b^{3/4}} \] Output:
-2*arctan(b^(1/4)*x^(1/4)/(b*x+a)^(1/4))/b^(3/4)+2*arctanh(b^(1/4)*x^(1/4) /(b*x+a)^(1/4))/b^(3/4)
Time = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )\right )}{b^{3/4}} \] Input:
Integrate[1/(x^(1/4)*(a + b*x)^(3/4)),x]
Output:
(-2*(ArcTan[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)] - ArcTanh[(b^(1/4)*x^(1/4)) /(a + b*x)^(1/4)]))/b^(3/4)
Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {73, 854, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 4 \int \frac {\sqrt {x}}{(a+b x)^{3/4}}d\sqrt [4]{x}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle 4 \int \frac {\sqrt {x}}{1-b x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle 4 \left (\frac {\int \frac {1}{1-\sqrt {b} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}-\frac {\int \frac {1}{\sqrt {b} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 4 \left (\frac {\int \frac {1}{1-\sqrt {b} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 4 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )\) |
Input:
Int[1/(x^(1/4)*(a + b*x)^(3/4)),x]
Output:
4*(-1/2*ArcTan[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)]/b^(3/4) + ArcTanh[(b^(1/ 4)*x^(1/4))/(a + b*x)^(1/4)]/(2*b^(3/4)))
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
\[\int \frac {1}{x^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}}}d x\]
Input:
int(1/x^(1/4)/(b*x+a)^(3/4),x)
Output:
int(1/x^(1/4)/(b*x+a)^(3/4),x)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.32 \[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=\frac {1}{b^{3}}^{\frac {1}{4}} \log \left (\frac {b \frac {1}{b^{3}}^{\frac {1}{4}} x + {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{x}\right ) - \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (-\frac {b \frac {1}{b^{3}}^{\frac {1}{4}} x - {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{x}\right ) + i \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (\frac {i \, b \frac {1}{b^{3}}^{\frac {1}{4}} x + {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{x}\right ) - i \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (\frac {-i \, b \frac {1}{b^{3}}^{\frac {1}{4}} x + {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{x}\right ) \] Input:
integrate(1/x^(1/4)/(b*x+a)^(3/4),x, algorithm="fricas")
Output:
(b^(-3))^(1/4)*log((b*(b^(-3))^(1/4)*x + (b*x + a)^(1/4)*x^(3/4))/x) - (b^ (-3))^(1/4)*log(-(b*(b^(-3))^(1/4)*x - (b*x + a)^(1/4)*x^(3/4))/x) + I*(b^ (-3))^(1/4)*log((I*b*(b^(-3))^(1/4)*x + (b*x + a)^(1/4)*x^(3/4))/x) - I*(b ^(-3))^(1/4)*log((-I*b*(b^(-3))^(1/4)*x + (b*x + a)^(1/4)*x^(3/4))/x)
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=\frac {x^{\frac {3}{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate(1/x**(1/4)/(b*x+a)**(3/4),x)
Output:
x**(3/4)*gamma(3/4)*hyper((3/4, 3/4), (7/4,), b*x*exp_polar(I*pi)/a)/(a**( 3/4)*gamma(7/4))
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=\frac {2 \, \arctan \left (\frac {{\left (b x + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}{b^{\frac {1}{4}} + \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}\right )}{b^{\frac {3}{4}}} \] Input:
integrate(1/x^(1/4)/(b*x+a)^(3/4),x, algorithm="maxima")
Output:
2*arctan((b*x + a)^(1/4)/(b^(1/4)*x^(1/4)))/b^(3/4) - log(-(b^(1/4) - (b*x + a)^(1/4)/x^(1/4))/(b^(1/4) + (b*x + a)^(1/4)/x^(1/4)))/b^(3/4)
\[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} x^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/x^(1/4)/(b*x+a)^(3/4),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(3/4)*x^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=\int \frac {1}{x^{1/4}\,{\left (a+b\,x\right )}^{3/4}} \,d x \] Input:
int(1/(x^(1/4)*(a + b*x)^(3/4)),x)
Output:
int(1/(x^(1/4)*(a + b*x)^(3/4)), x)
\[ \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}} \, dx=\int \frac {1}{x^{\frac {1}{4}} \left (b x +a \right )^{\frac {3}{4}}}d x \] Input:
int(1/x^(1/4)/(b*x+a)^(3/4),x)
Output:
int(1/(x**(1/4)*(a + b*x)**(3/4)),x)