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

Optimal. Leaf size=266 \[ -\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 b c-15 a d) (3 a d+b c)}{96 a^2 c^3 x}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-9 a d)}{24 a c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3} \]

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*c*x^3) - ((b*c - 9*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(24*a*c^2*x^2)
+ ((7*b*c - 15*a*d)*(b*c + 3*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(96*a^2*c^3*x) - ((b*c - a*d)*(7*b^2*c^2 +
10*a*b*c*d + 15*a^2*d^2)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(11/4)*c^(13/4)) -
((b*c - a*d)*(7*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4)
)])/(64*a^(11/4)*c^(13/4))

________________________________________________________________________________________

Rubi [A]  time = 0.205714, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {99, 151, 12, 93, 212, 208, 205} \[ -\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{(b c-a d) \left (15 a^2 d^2+10 a b c d+7 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 b c-15 a d) (3 a d+b c)}{96 a^2 c^3 x}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-9 a d)}{24 a c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*c*x^3) - ((b*c - 9*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(24*a*c^2*x^2)
+ ((7*b*c - 15*a*d)*(b*c + 3*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(96*a^2*c^3*x) - ((b*c - a*d)*(7*b^2*c^2 +
10*a*b*c*d + 15*a^2*d^2)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(11/4)*c^(13/4)) -
((b*c - a*d)*(7*b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4)
)])/(64*a^(11/4)*c^(13/4))

Rule 99

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

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 212

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] &&
!GtQ[a/b, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{a+b x}}{x^4 \sqrt [4]{c+d x}} \, dx &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}+\frac{\int \frac{\frac{1}{4} (b c-9 a d)-2 b d x}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{3 c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}-\frac{(b c-9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a c^2 x^2}-\frac{\int \frac{\frac{1}{16} (7 b c-15 a d) (b c+3 a d)+\frac{1}{4} b d (b c-9 a d) x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{6 a c^2}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}-\frac{(b c-9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a c^2 x^2}+\frac{(7 b c-15 a d) (b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^2 c^3 x}+\frac{\int \frac{3 (b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right )}{64 x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{6 a^2 c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}-\frac{(b c-9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a c^2 x^2}+\frac{(7 b c-15 a d) (b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^2 c^3 x}+\frac{\left ((b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right )\right ) \int \frac{1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{128 a^2 c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}-\frac{(b c-9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a c^2 x^2}+\frac{(7 b c-15 a d) (b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^2 c^3 x}+\frac{\left ((b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^4} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{32 a^2 c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}-\frac{(b c-9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a c^2 x^2}+\frac{(7 b c-15 a d) (b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^2 c^3 x}-\frac{\left ((b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right )\right ) \operatorname{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 )}{64 a^{5/2} c^3}-\frac{\left ((b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right )\right ) \operatorname{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 )}{64 a^{5/2} c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 c x^3}-\frac{(b c-9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a c^2 x^2}+\frac{(7 b c-15 a d) (b c+3 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^2 c^3 x}-\frac{(b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}-\frac{(b c-a d) \left (7 b^2 c^2+10 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{11/4} c^{13/4}}\\ \end{align*}

Mathematica [C]  time = 0.0731166, size = 155, normalized size = 0.58 \[ \frac{\sqrt [4]{a+b x} \left (a (c+d x) \left (a^2 \left (-32 c^2+36 c d x-45 d^2 x^2\right )+2 a b c x (3 d x-2 c)+7 b^2 c^2 x^2\right )-3 x^3 \left (5 a^2 b c d^2-15 a^3 d^3+3 a b^2 c^2 d+7 b^3 c^3\right ) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{c (a+b x)}{a (c+d x)}\right )\right )}{96 a^3 c^3 x^3 \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1/4)*(a*(c + d*x)*(7*b^2*c^2*x^2 + 2*a*b*c*x*(-2*c + 3*d*x) + a^2*(-32*c^2 + 36*c*d*x - 45*d^2*x^2
)) - 3*(7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 - 15*a^3*d^3)*x^3*Hypergeometric2F1[1/4, 1, 5/4, (c*(a + b*x
))/(a*(c + d*x))]))/(96*a^3*c^3*x^3*(c + d*x)^(1/4))

________________________________________________________________________________________

Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 3.38579, size = 5308, normalized size = 19.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(12*a^2*c^3*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3
- 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^
4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13
))^(1/4)*arctan(((7*a^8*b^3*c^13 + 3*a^9*b^2*c^12*d + 5*a^10*b*c^11*d^2 - 15*a^11*c^10*d^3)*(b*x + a)^(1/4)*(d
*x + c)^(3/4)*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a
^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8
- 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(3/4)
+ (a^8*c^10*d*x + a^8*c^11)*sqrt(((49*b^6*c^6 + 42*a*b^5*c^5*d + 79*a^2*b^4*c^4*d^2 - 180*a^3*b^3*c^3*d^3 - 65
*a^4*b^2*c^2*d^4 - 150*a^5*b*c*d^5 + 225*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (a^6*c^6*d*x + a^6*c^7)*sqrt((
2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 -
48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3
*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13)))/(d*x + c))*((2401*b^1
2*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a
^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9
- 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(3/4))/(2401*b^12*c^13 + 4116*
a*b^11*c^12*d + 9506*a^2*b^10*c^11*d^2 - 11004*a^3*b^9*c^10*d^3 - 15249*a^4*b^8*c^9*d^4 - 48600*a^5*b^7*c^8*d^
5 + 31580*a^6*b^6*c^7*d^6 + 18600*a^7*b^5*c^6*d^7 + 93775*a^8*b^4*c^5*d^8 - 61500*a^9*b^3*c^4*d^9 - 6750*a^10*
b^2*c^3*d^10 - 67500*a^11*b*c^2*d^11 + 50625*a^12*c*d^12 + (2401*b^12*c^12*d + 4116*a*b^11*c^11*d^2 + 9506*a^2
*b^10*c^10*d^3 - 11004*a^3*b^9*c^9*d^4 - 15249*a^4*b^8*c^8*d^5 - 48600*a^5*b^7*c^7*d^6 + 31580*a^6*b^6*c^6*d^7
+ 18600*a^7*b^5*c^5*d^8 + 93775*a^8*b^4*c^4*d^9 - 61500*a^9*b^3*c^3*d^10 - 6750*a^10*b^2*c^2*d^11 - 67500*a^1
1*b*c*d^12 + 50625*a^12*d^13)*x)) + 3*a^2*c^3*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d
^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7
*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 +
50625*a^12*d^12)/(a^11*c^13))^(1/4)*log(-((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 - 15*a^3*d^3)*(b*x + a)^(
1/4)*(d*x + c)^(3/4) + (a^3*c^3*d*x + a^3*c^4)*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^10*c^10*d^2
- 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 18600*a^7*b^
5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c*d^11 + 506
25*a^12*d^12)/(a^11*c^13))^(1/4))/(d*x + c)) - 3*a^2*c^3*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*
b^10*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6
+ 18600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*
b*c*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(1/4)*log(-((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 - 15*a^3*d^3)*
(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^3*c^3*d*x + a^3*c^4)*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 9506*a^2*b^1
0*c^10*d^2 - 11004*a^3*b^9*c^9*d^3 - 15249*a^4*b^8*c^8*d^4 - 48600*a^5*b^7*c^7*d^5 + 31580*a^6*b^6*c^6*d^6 + 1
8600*a^7*b^5*c^5*d^7 + 93775*a^8*b^4*c^4*d^8 - 61500*a^9*b^3*c^3*d^9 - 6750*a^10*b^2*c^2*d^10 - 67500*a^11*b*c
*d^11 + 50625*a^12*d^12)/(a^11*c^13))^(1/4))/(d*x + c)) + 4*(32*a^2*c^2 - (7*b^2*c^2 + 6*a*b*c*d - 45*a^2*d^2)
*x^2 + 4*(a*b*c^2 - 9*a^2*c*d)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a^2*c^3*x^3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{a + b x}}{x^{4} \sqrt [4]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out