\(\int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx\) [889]

Optimal result

Integrand size = 22, antiderivative size = 194 \[ \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx=\frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}+\frac {(b c-a d) (3 b c+5 a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (3 b c+5 a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {98, 96, 95, 218, 214, 211} \[ \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx=\frac {(b c-a d) (5 a d+3 b c) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (5 a d+3 b c) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+3 b c)}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2} \]

[In]

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

[Out]

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

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 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 {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}-\frac {\left (\frac {3 b c}{4}+\frac {5 a d}{4}\right ) \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx}{2 a c} \\ & = \frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}-\frac {((b c-a d) (3 b c+5 a d)) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 a c^2} \\ & = \frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}-\frac {((b c-a d) (3 b c+5 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 )}{8 a c^2} \\ & = \frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}+\frac {((b c-a d) (3 b c+5 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 )}{16 a^{3/2} c^2}+\frac {((b c-a d) (3 b c+5 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 )}{16 a^{3/2} c^2} \\ & = \frac {(3 b c+5 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 a c^2 x}-\frac {(a+b x)^{5/4} (c+d x)^{3/4}}{2 a c x^2}+\frac {(b c-a d) (3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}}+\frac {(b c-a d) (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{7/4} c^{9/4}} \\ \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 = 106, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx=\frac {\sqrt [4]{a+b x} \left (-a (c+d x) (4 a c+b c x-5 a d x)+\left (3 b^2 c^2+2 a b c d-5 a^2 d^2\right ) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {c (a+b x)}{a (c+d x)}\right )\right )}{8 a^2 c^2 x^2 \sqrt [4]{c+d x}} \]

[In]

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

[Out]

((a + b*x)^(1/4)*(-(a*(c + d*x)*(4*a*c + b*c*x - 5*a*d*x)) + (3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*x^2*Hypergeom
etric2F1[1/4, 1, 5/4, (c*(a + b*x))/(a*(c + d*x))]))/(8*a^2*c^2*x^2*(c + d*x)^(1/4))

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 1279, normalized size of antiderivative = 6.59 \[ \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^3 \sqrt [4]{c+d x}} \, dx=\int \frac {\sqrt [4]{a + b x}}{x^{3} \sqrt [4]{c + d x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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