\(\int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx\) [887]

Optimal result

Integrand size = 22, antiderivative size = 169 \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {132, 65, 246, 218, 214, 211, 12, 95} \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 132

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[b*d^(m
+ n)*f^p, Int[(a + b*x)^(m - 1)/(c + d*x)^m, x], x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandTo
Sum[(a + b*x)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n,
-1]))

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]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=\frac {4 \sqrt [4]{a+b x} \left (\sqrt [4]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {d (a+b x)}{-b c+a d}\right )-\operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {c (a+b x)}{a (c+d x)}\right )\right )}{\sqrt [4]{c+d x}} \]

[In]

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

[Out]

(4*(a + b*x)^(1/4)*(((b*(c + d*x))/(b*c - a*d))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (d*(a + b*x))/(-(b*c) +
a*d)] - Hypergeometric2F1[1/4, 1, 5/4, (c*(a + b*x))/(a*(c + d*x))]))/(c + d*x)^(1/4)

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=-\left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - i \, \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, d x + i \, c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + i \, \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, d x - i \, c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + i \, \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, d x + i \, c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - i \, \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, d x - i \, c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=\int \frac {\sqrt [4]{a + b x}}{x \sqrt [4]{c + d x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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