\(\int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\) [905]

Optimal result

Integrand size = 22, antiderivative size = 134 \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac {(3 b c+a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac {(3 b c+a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}} \]

[Out]

-(b*x+a)^(1/4)*(d*x+c)^(3/4)/a/c/x+1/2*(a*d+3*b*c)*arctan(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(7/4)
/c^(5/4)+1/2*(a*d+3*b*c)*arctanh(c^(1/4)*(b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(7/4)/c^(5/4)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 134, 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 = {98, 95, 218, 214, 211} \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {(a d+3 b c) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac {(a d+3 b c) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \]

[In]

Int[1/(x^2*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) + ((3*b*c + a*d)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d
*x)^(1/4))])/(2*a^(7/4)*c^(5/4)) + ((3*b*c + a*d)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]
)/(2*a^(7/4)*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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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}}{a c x}-\frac {\left (\frac {3 b c}{4}+\frac {a d}{4}\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{a c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}-\frac {\left (4 \left (\frac {3 b c}{4}+\frac {a d}{4}\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^4} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{a c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac {(3 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 a^{3/2} c}+\frac {(3 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 a^{3/2} c} \\ & = -\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x}+\frac {(3 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^{7/4} c^{5/4}}+\frac {(3 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^{7/4} c^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {-2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4}+(3 b c+a d) x \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )+(3 b c+a d) x \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4} x} \]

[In]

Integrate[1/(x^2*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

(-2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4) + (3*b*c + a*d)*x*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4
)*(c + d*x)^(1/4))] + (3*b*c + a*d)*x*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(7/4)
*c^(5/4)*x)

Maple [F]

\[\int \frac {1}{x^{2} \left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}d x\]

[In]

int(1/x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(1/x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 722, normalized size of antiderivative = 5.39 \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\frac {a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) - i \, a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (i \, a^{2} c d x + i \, a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + i \, a c x \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (3 \, b c + a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (-i \, a^{2} c d x - i \, a^{2} c^{2}\right )} \left (\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} 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 \, a c x} \]

[In]

integrate(1/x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

1/4*(a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*lo
g(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a
^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) - a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d
+ 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c
)^(3/4) - (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4
)/(a^7*c^5))^(1/4))/(d*x + c)) - I*a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3
+ a^4*d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (I*a^2*c*d*x + I*a^2*c^2)*((8
1*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) + I*
a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(((3
*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (-I*a^2*c*d*x - I*a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a
^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) - 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(
a*c*x)

Sympy [F]

\[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \]

[In]

integrate(1/x**2/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(1/(x**2*(a + b*x)**(3/4)*(c + d*x)**(1/4)), x)

Maxima [F]

\[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(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^2), x)

Giac [F]

\[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(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^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{x^2\,{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]

[In]

int(1/(x^2*(a + b*x)^(3/4)*(c + d*x)^(1/4)),x)

[Out]

int(1/(x^2*(a + b*x)^(3/4)*(c + d*x)^(1/4)), x)