\(\int \frac {x \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx\) [885]

Optimal result

Integrand size = 20, antiderivative size = 188 \[ \int \frac {x \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=-\frac {(5 b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {(b c-a d) (5 b c+3 a d) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}+\frac {(b c-a d) (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}} \]

[Out]

-1/8*(3*a*d+5*b*c)*(b*x+a)^(1/4)*(d*x+c)^(3/4)/b/d^2+1/2*(b*x+a)^(5/4)*(d*x+c)^(3/4)/b/d+1/16*(-a*d+b*c)*(3*a*
d+5*b*c)*arctan(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(7/4)/d^(9/4)+1/16*(-a*d+b*c)*(3*a*d+5*b*c)*arc
tanh(d^(1/4)*(b*x+a)^(1/4)/b^(1/4)/(d*x+c)^(1/4))/b^(7/4)/d^(9/4)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {81, 52, 65, 246, 218, 214, 211} \[ \int \frac {x \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\frac {(b c-a d) (3 a d+5 b c) \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}+\frac {(b c-a d) (3 a d+5 b c) \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (3 a d+5 b c)}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d} \]

[In]

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

[Out]

-1/8*((5*b*c + 3*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(b*d^2) + ((a + b*x)^(5/4)*(c + d*x)^(3/4))/(2*b*d) + (
(b*c - a*d)*(5*b*c + 3*a*d)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(7/4)*d^(9/4))
+ ((b*c - a*d)*(5*b*c + 3*a*d)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(7/4)*d^(9/
4))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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 81

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

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}& = \frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {\left (-\frac {5 b c}{4}-\frac {3 a d}{4}\right ) \int \frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx}{2 b d} \\ & = -\frac {(5 b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {((b c-a d) (5 b c+3 a d)) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 b d^2} \\ & = -\frac {(5 b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {((b c-a d) (5 b c+3 a d)) \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 )}{8 b^2 d^2} \\ & = -\frac {(5 b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {((b c-a d) (5 b c+3 a d)) \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 )}{8 b^2 d^2} \\ & = -\frac {(5 b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {((b c-a d) (5 b c+3 a d)) \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 )}{16 b^{3/2} d^2}+\frac {((b c-a d) (5 b c+3 a d)) \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 )}{16 b^{3/2} d^2} \\ & = -\frac {(5 b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b d^2}+\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 b d}+\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}}+\frac {(b c-a d) (5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{7/4} d^{9/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.78 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93 \[ \int \frac {x \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\frac {\sqrt [4]{a+b x} \left (2 b^{3/4} \sqrt [4]{d (a+b x)} (c+d x)^{3/4} (-5 b c+a d+4 b d x)+\left (-5 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d (a+b x)}}\right )+\left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{d (a+b x)}}\right )\right )}{16 b^{7/4} d^2 \sqrt [4]{d (a+b x)}} \]

[In]

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

[Out]

((a + b*x)^(1/4)*(2*b^(3/4)*(d*(a + b*x))^(1/4)*(c + d*x)^(3/4)*(-5*b*c + a*d + 4*b*d*x) + (-5*b^2*c^2 + 2*a*b
*c*d + 3*a^2*d^2)*ArcTan[(b^(1/4)*(c + d*x)^(1/4))/(d*(a + b*x))^(1/4)] + (5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*
ArcTanh[(b^(1/4)*(c + d*x)^(1/4))/(d*(a + b*x))^(1/4)]))/(16*b^(7/4)*d^2*(d*(a + b*x))^(1/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 1262, normalized size of antiderivative = 6.71 \[ \int \frac {x \sqrt [4]{a+b x}}{\sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/32*(b*d^2*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^
4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4)*log(-((5*b^2*c^
2 - 2*a*b*c*d - 3*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^2*d^3*x + b^2*c*d^2)*((625*b^8*c^8 - 1000*a*b^
7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2
*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4))/(d*x + c)) - b*d^2*((625*b^8*c^8 - 1000*a*b^7*c^7*d
- 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^
6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4)*log(-((5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*(b*x + a)^(1/4)*(
d*x + c)^(3/4) - (b^2*d^3*x + b^2*c*d^2)*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5
*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^
7*d^9))^(1/4))/(d*x + c)) + I*b*d^2*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*
d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9
))^(1/4)*log(-((5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (I*b^2*d^3*x + I*b^2*c*d^
2)*((625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a
^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4))/(d*x + c)) - I*b*d^2*((
625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^
3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4)*log(-((5*b^2*c^2 - 2*a*b*c*d
- 3*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (-I*b^2*d^3*x - I*b^2*c*d^2)*((625*b^8*c^8 - 1000*a*b^7*c^7*d -
900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6
+ 216*a^7*b*c*d^7 + 81*a^8*d^8)/(b^7*d^9))^(1/4))/(d*x + c)) + 4*(4*b*d*x - 5*b*c + a*d)*(b*x + a)^(1/4)*(d*x
+ c)^(3/4))/(b*d^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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