\(\int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx\) [2985]

Optimal result

Integrand size = 26, antiderivative size = 388 \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {\left (\frac {b+a x}{d+c x}\right )^{3/4} \left (-32 b^2 d^3-36 b^2 c d^2 x+36 a b d^3 x-48 b d^4 x+3 b^2 c^2 d x^2+42 a b c d^2 x^2-45 a^2 d^3 x^2-60 b c d^3 x^2+60 a d^4 x^2+7 b^2 c^3 x^3+6 a b c^2 d x^3-45 a^2 c d^2 x^3-12 b c^2 d^2 x^3+60 a c d^3 x^3\right )}{96 b^2 d^2 x^3}+\frac {\left (7 b^3 c^3+3 a b^2 c^2 d+5 a^2 b c d^2-12 b^2 c^2 d^2-15 a^3 d^3-8 a b c d^3+20 a^2 d^4\right ) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}+\frac {\left (-7 b^3 c^3-3 a b^2 c^2 d-5 a^2 b c d^2+12 b^2 c^2 d^2+15 a^3 d^3+8 a b c d^3-20 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}} \]

[Out]

1/96*((a*x+b)/(c*x+d))^(3/4)*(-45*a^2*c*d^2*x^3+6*a*b*c^2*d*x^3+60*a*c*d^3*x^3+7*b^2*c^3*x^3-12*b*c^2*d^2*x^3-
45*a^2*d^3*x^2+42*a*b*c*d^2*x^2+60*a*d^4*x^2+3*b^2*c^2*d*x^2-60*b*c*d^3*x^2+36*a*b*d^3*x-36*b^2*c*d^2*x-48*b*d
^4*x-32*b^2*d^3)/b^2/d^2/x^3+1/64*(-15*a^3*d^3+5*a^2*b*c*d^2+20*a^2*d^4+3*a*b^2*c^2*d-8*a*b*c*d^3+7*b^3*c^3-12
*b^2*c^2*d^2)*arctan(d^(1/4)*((a*x+b)/(c*x+d))^(1/4)/b^(1/4))/b^(9/4)/d^(11/4)+1/64*(15*a^3*d^3-5*a^2*b*c*d^2-
20*a^2*d^4-3*a*b^2*c^2*d+8*a*b*c*d^3-7*b^3*c^3+12*b^2*c^2*d^2)*arctanh(d^(1/4)*((a*x+b)/(c*x+d))^(1/4)/b^(1/4)
)/b^(9/4)/d^(11/4)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1984, 591, 468, 296, 304, 211, 214} \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {\left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) (b c-a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}-\frac {\left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) (b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}-\frac {\left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) (b c-a d) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (a x+b)}{c x+d}\right )}+\frac {(d (9 a-4 d)+7 b c) (b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (a x+b)}{c x+d}\right )^2}-\frac {(b+d x) (b c-a d)^3 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{3 b d (c x+d) \left (b-\frac {d (a x+b)}{c x+d}\right )^3} \]

[In]

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

[Out]

-1/3*((b*c - a*d)^3*((b + a*x)/(d + c*x))^(3/4)*(b + d*x))/(b*d*(d + c*x)*(b - (d*(b + a*x))/(d + c*x))^3) + (
(b*c - a*d)^2*(7*b*c + (9*a - 4*d)*d)*((b + a*x)/(d + c*x))^(3/4))/(24*b*d^2*(b - (d*(b + a*x))/(d + c*x))^2)
- ((b*c - a*d)*(7*b^2*c^2 + 2*b*c*(5*a - 6*d)*d + 5*a*(3*a - 4*d)*d^2)*((b + a*x)/(d + c*x))^(3/4))/(32*b^2*d^
2*(b - (d*(b + a*x))/(d + c*x))) + ((b*c - a*d)*(7*b^2*c^2 + 2*b*c*(5*a - 6*d)*d + 5*a*(3*a - 4*d)*d^2)*ArcTan
[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4)) - ((b*c - a*d)*(7*b^2*c^2 + 2*b*c*(5*a
- 6*d)*d + 5*a*(3*a - 4*d)*d^2)*ArcTanh[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4))

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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 304

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

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rule 591

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Dis
t[1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(
m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rule 1984

Int[(u_)^(r_.)*(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Wi
th[{q = Denominator[p]}, Dist[q*e*((b*c - a*d)/n), Subst[Int[SimplifyIntegrand[x^(q*(p + 1) - 1)*(((-a)*e + c*
x^q)^((m + 1)/n - 1)/(b*e - d*x^q)^((m + 1)/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r,
x], x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && Frac
tionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]

Rubi steps \begin{align*} \text {integral}& = -\left ((4 (b c-a d)) \text {Subst}\left (\int \frac {x^2 \left (a-c x^4\right ) \left (b (a-d)+\left (-b c+d^2\right ) x^4\right )}{\left (b-d x^4\right )^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )\right ) \\ & = -\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}-\frac {(b c-a d) \text {Subst}\left (\int \frac {x^2 \left (3 b (a-d) (b c+3 a d)-(7 b c+5 a d) \left (b c-d^2\right ) x^4\right )}{\left (b-d x^4\right )^3} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{3 b d} \\ & = -\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (b-d x^4\right )^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{8 b d^2} \\ & = -\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )}-\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{b-d x^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{32 b^2 d^2} \\ & = -\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )}-\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{64 b^2 d^{5/2}}+\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{64 b^2 d^{5/2}} \\ & = -\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )}+\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.40 \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {\left (\frac {b+a x}{d+c x}\right )^{3/4} \left (-32 b^3 d (d+c x)^2+x \left (4 b^2 (7 b c+3 (3 a-4 d) d) (d+c x)^2-\left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) x \left (3 b (d+c x)+(b c-a d) x \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {d (b+a x)}{b (d+c x)}\right )\right )\right )\right )}{96 b^3 d^2 x^3} \]

[In]

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

[Out]

(((b + a*x)/(d + c*x))^(3/4)*(-32*b^3*d*(d + c*x)^2 + x*(4*b^2*(7*b*c + 3*(3*a - 4*d)*d)*(d + c*x)^2 - (7*b^2*
c^2 + 2*b*c*(5*a - 6*d)*d + 5*a*(3*a - 4*d)*d^2)*x*(3*b*(d + c*x) + (b*c - a*d)*x*Hypergeometric2F1[3/4, 1, 7/
4, (d*(b + a*x))/(b*(d + c*x))]))))/(96*b^3*d^2*x^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 6825, normalized size of antiderivative = 17.59 \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.38 \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=-\frac {3 \, {\left (7 \, b^{3} c^{3} d^{2} + 3 \, a b^{2} c^{2} d^{3} + 20 \, a^{2} d^{6} - {\left (15 \, a^{3} + 8 \, a b c\right )} d^{5} + {\left (5 \, a^{2} b c - 12 \, b^{2} c^{2}\right )} d^{4}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {11}{4}} + 6 \, {\left (3 \, b^{4} c^{3} d - 17 \, a b^{3} c^{2} d^{2} - 28 \, a^{2} b d^{5} + 3 \, {\left (7 \, a^{3} b + 8 \, a b^{2} c\right )} d^{4} - {\left (7 \, a^{2} b^{2} c - 4 \, b^{3} c^{2}\right )} d^{3}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {7}{4}} - {\left (7 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 108 \, a^{2} b^{2} d^{4} + {\left (113 \, a^{3} b^{2} + 120 \, a b^{3} c\right )} d^{3} - 3 \, {\left (41 \, a^{2} b^{3} c + 4 \, b^{4} c^{2}\right )} d^{2}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {3}{4}}}{96 \, {\left (b^{5} d^{2} - \frac {3 \, {\left (a x + b\right )} b^{4} d^{3}}{c x + d} + \frac {3 \, {\left (a x + b\right )}^{2} b^{3} d^{4}}{{\left (c x + d\right )}^{2}} - \frac {{\left (a x + b\right )}^{3} b^{2} d^{5}}{{\left (c x + d\right )}^{3}}\right )}} + \frac {{\left (7 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 20 \, a^{2} d^{4} - {\left (15 \, a^{3} + 8 \, a b c\right )} d^{3} + {\left (5 \, a^{2} b c - 12 \, b^{2} c^{2}\right )} d^{2}\right )} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b} \sqrt {d}}}\right )}{\sqrt {\sqrt {b} \sqrt {d}} \sqrt {d}} + \frac {\log \left (\frac {\sqrt {d} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} - \sqrt {\sqrt {b} \sqrt {d}}}{\sqrt {d} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} + \sqrt {\sqrt {b} \sqrt {d}}}\right )}{\sqrt {\sqrt {b} \sqrt {d}} \sqrt {d}}\right )}}{128 \, b^{2} d^{2}} \]

[In]

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

[Out]

-1/96*(3*(7*b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 + 20*a^2*d^6 - (15*a^3 + 8*a*b*c)*d^5 + (5*a^2*b*c - 12*b^2*c^2)*d^4
)*((a*x + b)/(c*x + d))^(11/4) + 6*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 - 28*a^2*b*d^5 + 3*(7*a^3*b + 8*a*b^2*c)*d^
4 - (7*a^2*b^2*c - 4*b^3*c^2)*d^3)*((a*x + b)/(c*x + d))^(7/4) - (7*b^5*c^3 + 3*a*b^4*c^2*d - 108*a^2*b^2*d^4
+ (113*a^3*b^2 + 120*a*b^3*c)*d^3 - 3*(41*a^2*b^3*c + 4*b^4*c^2)*d^2)*((a*x + b)/(c*x + d))^(3/4))/(b^5*d^2 -
3*(a*x + b)*b^4*d^3/(c*x + d) + 3*(a*x + b)^2*b^3*d^4/(c*x + d)^2 - (a*x + b)^3*b^2*d^5/(c*x + d)^3) + 1/128*(
7*b^3*c^3 + 3*a*b^2*c^2*d + 20*a^2*d^4 - (15*a^3 + 8*a*b*c)*d^3 + (5*a^2*b*c - 12*b^2*c^2)*d^2)*(2*arctan(sqrt
(d)*((a*x + b)/(c*x + d))^(1/4)/sqrt(sqrt(b)*sqrt(d)))/(sqrt(sqrt(b)*sqrt(d))*sqrt(d)) + log((sqrt(d)*((a*x +
b)/(c*x + d))^(1/4) - sqrt(sqrt(b)*sqrt(d)))/(sqrt(d)*((a*x + b)/(c*x + d))^(1/4) + sqrt(sqrt(b)*sqrt(d))))/(s
qrt(sqrt(b)*sqrt(d))*sqrt(d)))/(b^2*d^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1632 vs. \(2 (360) = 720\).

Time = 1.35 (sec) , antiderivative size = 1632, normalized size of antiderivative = 4.21 \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/768*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)*(6*sqrt(2)*(7*b^4*c^4 - 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b
^3*c^3*d^2 - 20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 28*a^2*b*c*d^4 - 20*a^3*d^5)*arctan(1/2*sqrt(2)*(
sqrt(2)*(-b/d)^(1/4) + 2*((a*x + b)/(c*x + d))^(1/4))/(-b/d)^(1/4))/((-b*d^3)^(1/4)*b^2*d^2) + 6*sqrt(2)*(7*b^
4*c^4 - 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b^3*c^3*d^2 - 20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 2
8*a^2*b*c*d^4 - 20*a^3*d^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b/d)^(1/4) - 2*((a*x + b)/(c*x + d))^(1/4))/(-b/d)^
(1/4))/((-b*d^3)^(1/4)*b^2*d^2) - 3*sqrt(2)*(7*b^4*c^4 - 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b^3*c^3*d^2 -
20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 28*a^2*b*c*d^4 - 20*a^3*d^5)*log(sqrt(2)*((a*x + b)/(c*x + d))
^(1/4)*(-b/d)^(1/4) + sqrt((a*x + b)/(c*x + d)) + sqrt(-b/d))/((-b*d^3)^(1/4)*b^2*d^2) + 3*sqrt(2)*(7*b^4*c^4
- 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b^3*c^3*d^2 - 20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 28*a^2*
b*c*d^4 - 20*a^3*d^5)*log(-sqrt(2)*((a*x + b)/(c*x + d))^(1/4)*(-b/d)^(1/4) + sqrt((a*x + b)/(c*x + d)) + sqrt
(-b/d))/((-b*d^3)^(1/4)*b^2*d^2) + 8*(7*b^6*c^4*((a*x + b)/(c*x + d))^(3/4) - 4*a*b^5*c^3*d*((a*x + b)/(c*x +
d))^(3/4) - 18*(a*x + b)*b^5*c^4*d*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 126*a^2*b^4*c^2*d^2*((a*x + b)/(c*x
+ d))^(3/4) + 120*(a*x + b)*a*b^4*c^3*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 12*b^5*c^3*d^2*((a*x + b)/(
c*x + d))^(3/4) - 21*(a*x + b)^2*b^4*c^4*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 236*a^3*b^3*c*d^3*((a*x
+ b)/(c*x + d))^(3/4) - 60*(a*x + b)*a^2*b^3*c^2*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 132*a*b^4*c^2*d^
3*((a*x + b)/(c*x + d))^(3/4) + 12*(a*x + b)^2*a*b^3*c^3*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 24*(a*x
+ b)*b^4*c^3*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 113*a^4*b^2*d^4*((a*x + b)/(c*x + d))^(3/4) - 168*(a
*x + b)*a^3*b^2*c*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 228*a^2*b^3*c*d^4*((a*x + b)/(c*x + d))^(3/4) -
6*(a*x + b)^2*a^2*b^2*c^2*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 120*(a*x + b)*a*b^3*c^2*d^4*((a*x + b)
/(c*x + d))^(3/4)/(c*x + d) + 36*(a*x + b)^2*b^3*c^3*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 126*(a*x +
b)*a^4*b*d^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 108*a^3*b^2*d^5*((a*x + b)/(c*x + d))^(3/4) + 60*(a*x + b
)^2*a^3*b*c*d^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 312*(a*x + b)*a^2*b^2*c*d^5*((a*x + b)/(c*x + d))^(3
/4)/(c*x + d) - 12*(a*x + b)^2*a*b^2*c^2*d^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 45*(a*x + b)^2*a^4*d^6*
((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 168*(a*x + b)*a^3*b*d^6*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 84*(
a*x + b)^2*a^2*b*c*d^6*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 60*(a*x + b)^2*a^3*d^7*((a*x + b)/(c*x + d))^
(3/4)/(c*x + d)^2)/((b - (a*x + b)*d/(c*x + d))^3*b^2*d^2))

Mupad [B] (verification not implemented)

Time = 8.80 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.37 \[ \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx=\frac {\mathrm {atanh}\left (\frac {d^{1/4}\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{1/4}}{b^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d-20\,a\,d^3+7\,b^2\,c^2-12\,b\,c\,d^2\right )}{64\,b^{9/4}\,d^{11/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{1/4}}{b^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d-20\,a\,d^3+7\,b^2\,c^2-12\,b\,c\,d^2\right )}{64\,b^{9/4}\,d^{11/4}}-\frac {\frac {c^2\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{7/4}\,\left (\frac {21\,a^3\,c\,d^3}{16}-\frac {7\,a^2\,b\,c^2\,d^2}{16}-\frac {7\,a^2\,c\,d^4}{4}-\frac {17\,a\,b^2\,c^3\,d}{16}+\frac {3\,a\,b\,c^2\,d^3}{2}+\frac {3\,b^3\,c^4}{16}+\frac {b^2\,c^3\,d^2}{4}\right )}{a^3\,b\,d^4}-\frac {c\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{3/4}\,\left (\frac {113\,a^3\,c^2\,d^3}{96}-\frac {41\,a^2\,b\,c^3\,d^2}{32}-\frac {9\,a^2\,c^2\,d^4}{8}+\frac {a\,b^2\,c^4\,d}{32}+\frac {5\,a\,b\,c^3\,d^3}{4}+\frac {7\,b^3\,c^5}{96}-\frac {b^2\,c^4\,d^2}{8}\right )}{a^3\,d^5}+\frac {c^3\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{11/4}\,\left (-\frac {15\,a^3\,d^3}{32}+\frac {5\,a^2\,b\,c\,d^2}{32}+\frac {5\,a^2\,d^4}{8}+\frac {3\,a\,b^2\,c^2\,d}{32}-\frac {a\,b\,c\,d^3}{4}+\frac {7\,b^3\,c^3}{32}-\frac {3\,b^2\,c^2\,d^2}{8}\right )}{a^3\,b^2\,d^3}}{\frac {b^3\,c^3}{a^3\,d^3}-\frac {c^3\,{\left (b+a\,x\right )}^3}{a^3\,{\left (d+c\,x\right )}^3}+\frac {3\,b\,c^3\,{\left (b+a\,x\right )}^2}{a^3\,d\,{\left (d+c\,x\right )}^2}-\frac {3\,b^2\,c^3\,\left (b+a\,x\right )}{a^3\,d^2\,\left (d+c\,x\right )}} \]

[In]

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

[Out]

(atanh((d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4))*(a*d - b*c)*(15*a^2*d^2 - 20*a*d^3 + 7*b^2*c^2 - 12*b*c*
d^2 + 10*a*b*c*d))/(64*b^(9/4)*d^(11/4)) - (atan((d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4))*(a*d - b*c)*(1
5*a^2*d^2 - 20*a*d^3 + 7*b^2*c^2 - 12*b*c*d^2 + 10*a*b*c*d))/(64*b^(9/4)*d^(11/4)) - ((c^2*((b + a*x)/(d + c*x
))^(7/4)*((3*b^3*c^4)/16 - (7*a^2*c*d^4)/4 + (21*a^3*c*d^3)/16 + (b^2*c^3*d^2)/4 - (7*a^2*b*c^2*d^2)/16 + (3*a
*b*c^2*d^3)/2 - (17*a*b^2*c^3*d)/16))/(a^3*b*d^4) - (c*((b + a*x)/(d + c*x))^(3/4)*((7*b^3*c^5)/96 - (9*a^2*c^
2*d^4)/8 + (113*a^3*c^2*d^3)/96 - (b^2*c^4*d^2)/8 - (41*a^2*b*c^3*d^2)/32 + (5*a*b*c^3*d^3)/4 + (a*b^2*c^4*d)/
32))/(a^3*d^5) + (c^3*((b + a*x)/(d + c*x))^(11/4)*((5*a^2*d^4)/8 - (15*a^3*d^3)/32 + (7*b^3*c^3)/32 - (3*b^2*
c^2*d^2)/8 - (a*b*c*d^3)/4 + (3*a*b^2*c^2*d)/32 + (5*a^2*b*c*d^2)/32))/(a^3*b^2*d^3))/((b^3*c^3)/(a^3*d^3) - (
c^3*(b + a*x)^3)/(a^3*(d + c*x)^3) + (3*b*c^3*(b + a*x)^2)/(a^3*d*(d + c*x)^2) - (3*b^2*c^3*(b + a*x))/(a^3*d^
2*(d + c*x)))