\(\int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) [904]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \]

[Out]

-2*arctan(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(3/4)/c^(1/4)-2*arctanh(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)
/(d*x+c)^(1/4))/a^(3/4)/c^(1/4)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {95, 218, 214, 211} \[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \]

[In]

Int[1/(x*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

(-2*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(a^(3/4)*c^(1/4)) - (2*ArcTanh[(c^(1/4)*(a +
b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(a^(3/4)*c^(1/4))

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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]

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rubi steps \begin{align*} \text {integral}& = 4 \text {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right ) \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {c} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{\sqrt {a}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {c} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{\sqrt {a}} \\ & = -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{a^{3/4} \sqrt [4]{c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )\right )}{a^{3/4} \sqrt [4]{c}} \]

[In]

Integrate[1/(x*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

(-2*(ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))] + ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*
(c + d*x)^(1/4))]))/(a^(3/4)*c^(1/4))

Maple [F]

\[\int \frac {1}{x \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]

[In]

int(1/x/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(1/x/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.61 \[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a d x + a c\right )} \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a d x + a c\right )} \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - i \, \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a d x + i \, a c\right )} \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + i \, \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a d x - i \, a c\right )} \left (\frac {1}{a^{3} c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \]

[In]

integrate(1/x/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

-(1/(a^3*c))^(1/4)*log(((a*d*x + a*c)*(1/(a^3*c))^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + (1/(a^
3*c))^(1/4)*log(-((a*d*x + a*c)*(1/(a^3*c))^(1/4) - (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) - I*(1/(a^3*c)
)^(1/4)*log(((I*a*d*x + I*a*c)*(1/(a^3*c))^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + I*(1/(a^3*c))
^(1/4)*log(((-I*a*d*x - I*a*c)*(1/(a^3*c))^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c))

Sympy [F]

\[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]

[In]

integrate(1/x/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(1/(x*(a + b*x)**(3/4)*(c + d*x)**(1/4)), x)

Maxima [F]

\[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x} \,d x } \]

[In]

integrate(1/x/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x), x)

Giac [F]

\[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x} \,d x } \]

[In]

integrate(1/x/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(3/4)*(d*x + c)^(1/4)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]

[In]

int(1/(x*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x)

[Out]

int(1/(x*(a + b*x)^(3/4)*(c + d*x)^(1/4)), x)