\(\int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx\) [888]

Optimal result

Integrand size = 22, antiderivative size = 131 \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac {(b c-a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}} \]

[Out]

-(b*x+a)^(1/4)*(d*x+c)^(3/4)/c/x-1/2*(-a*d+b*c)*arctan(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(3/4)/c^
(5/4)-1/2*(-a*d+b*c)*arctanh(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(3/4)/c^(5/4)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {96, 95, 218, 214, 211} \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=-\frac {(b c-a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]

[In]

Int[(a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x]

[Out]

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(c*x)) - ((b*c - a*d)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^
(1/4))])/(2*a^(3/4)*c^(5/4)) - ((b*c - a*d)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a
^(3/4)*c^(5/4))

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 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}+\frac {(b c-a d) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}+\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac {(b c-a d) \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 )}{2 \sqrt {a} c}-\frac {(b c-a d) \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 )}{2 \sqrt {a} c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.82 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\frac {-2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4}+(b c-a d) x \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{c+d x}}{\sqrt [4]{c} \sqrt [4]{a+b x}}\right )+(-b c x+a d x) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4} x} \]

[In]

Integrate[(a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x]

[Out]

(-2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4) + (b*c - a*d)*x*ArcTan[(a^(1/4)*(c + d*x)^(1/4))/(c^(1/4)*
(a + b*x)^(1/4))] + (-(b*c*x) + a*d*x)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(3/4
)*c^(5/4)*x)

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{2} \left (d x +c \right )^{\frac {1}{4}}}d x\]

[In]

int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)

[Out]

int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 693, normalized size of antiderivative = 5.29 \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=-\frac {c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (a c d x + a c^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (a c d x + a c^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + i \, c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (i \, a c d x + i \, a c^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - i \, c x \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (b c - a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (-i \, a c d x - i \, a c^{2}\right )} \left (\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{3} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + 4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, c x} \]

[In]

integrate((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

-1/4*(c*x*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c
- a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a*c*d*x + a*c^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*
a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) - c*x*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^
3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a*c*d*x + a*c^2)*((
b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) + I*c*x*((
b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c - a*d)*(b*x
+ a)^(1/4)*(d*x + c)^(3/4) + (I*a*c*d*x + I*a*c^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*
d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) - I*c*x*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*
d^3 + a^4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (-I*a*c*d*x - I*a*c^2)*((b
^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) + 4*(b*x +
a)^(1/4)*(d*x + c)^(3/4))/(c*x)

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int \frac {\sqrt [4]{a + b x}}{x^{2} \sqrt [4]{c + d x}}\, dx \]

[In]

integrate((b*x+a)**(1/4)/x**2/(d*x+c)**(1/4),x)

[Out]

Integral((a + b*x)**(1/4)/(x**2*(c + d*x)**(1/4)), x)

Maxima [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \]

[In]

integrate((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2), x)

Giac [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \]

[In]

integrate((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/4}}{x^2\,{\left (c+d\,x\right )}^{1/4}} \,d x \]

[In]

int((a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x)

[Out]

int((a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)), x)