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

Optimal. Leaf size=288 \[ -\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 a^2 d^2+54 a b c d+77 b^2 c^2\right )}{96 a^3 c^3 x}+\frac{\left (15 a^2 b c d^2+15 a^3 d^3+21 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\left (15 a^2 b c d^2+15 a^3 d^3+21 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{24 a^2 c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3} \]

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*a*c*x^3) + ((11*b*c + 9*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(24*a^2*c^
2*x^2) - ((77*b^2*c^2 + 54*a*b*c*d + 45*a^2*d^2)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(96*a^3*c^3*x) + ((77*b^3*c^
3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])
/(64*a^(15/4)*c^(13/4)) + ((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTanh[(c^(1/4)*(a + b
*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(15/4)*c^(13/4))

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Rubi [A]  time = 0.205308, antiderivative size = 288, 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 = {129, 151, 12, 93, 212, 208, 205} \[ -\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 a^2 d^2+54 a b c d+77 b^2 c^2\right )}{96 a^3 c^3 x}+\frac{\left (15 a^2 b c d^2+15 a^3 d^3+21 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\left (15 a^2 b c d^2+15 a^3 d^3+21 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (9 a d+11 b c)}{24 a^2 c^2 x^2}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*a*c*x^3) + ((11*b*c + 9*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(24*a^2*c^
2*x^2) - ((77*b^2*c^2 + 54*a*b*c*d + 45*a^2*d^2)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(96*a^3*c^3*x) + ((77*b^3*c^
3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])
/(64*a^(15/4)*c^(13/4)) + ((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*ArcTanh[(c^(1/4)*(a + b
*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(64*a^(15/4)*c^(13/4))

Rule 129

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[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

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{1}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}-\frac{\int \frac{\frac{1}{4} (11 b c+9 a d)+2 b d x}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{3 a c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac{(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}+\frac{\int \frac{\frac{1}{16} \left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right )+\frac{1}{4} b d (11 b c+9 a d) x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{6 a^2 c^2}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac{(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}-\frac{\left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^3 c^3 x}-\frac{\int \frac{3 \left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right )}{64 x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{6 a^3 c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac{(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}-\frac{\left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^3 c^3 x}-\frac{\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \int \frac{1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{128 a^3 c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac{(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}-\frac{\left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^3 c^3 x}-\frac{\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\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^3 c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac{(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}-\frac{\left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^3 c^3 x}+\frac{\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\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^{7/2} c^3}+\frac{\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\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^{7/2} c^3}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{3 a c x^3}+\frac{(11 b c+9 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{24 a^2 c^2 x^2}-\frac{\left (77 b^2 c^2+54 a b c d+45 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{96 a^3 c^3 x}+\frac{\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}+\frac{\left (77 b^3 c^3+21 a b^2 c^2 d+15 a^2 b c d^2+15 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{64 a^{15/4} c^{13/4}}\\ \end{align*}

Mathematica [C]  time = 0.0751567, size = 156, normalized size = 0.54 \[ \frac{\sqrt [4]{a+b x} \left (3 x^3 \left (15 a^2 b c d^2+15 a^3 d^3+21 a b^2 c^2 d+77 b^3 c^3\right ) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{c (a+b x)}{a (c+d x)}\right )-a (c+d x) \left (a^2 \left (32 c^2-36 c d x+45 d^2 x^2\right )+2 a b c x (27 d x-22 c)+77 b^2 c^2 x^2\right )\right )}{96 a^4 c^3 x^3 \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1/4)*(-(a*(c + d*x)*(77*b^2*c^2*x^2 + 2*a*b*c*x*(-22*c + 27*d*x) + a^2*(32*c^2 - 36*c*d*x + 45*d^2
*x^2))) + 3*(77*b^3*c^3 + 21*a*b^2*c^2*d + 15*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^4*c^3*x^3*(c + d*x)^(1/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{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(1/x^4/(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^4), x)

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Fricas [B]  time = 3.40331, size = 5787, normalized size = 20.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(12*a^3*c^3*x^3*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a
^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*
b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d
^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4)*arctan(-((77*a^11*b^3*c^13 + 21*a^12*b^2*c^12*d + 15*a^13*b*c^11*d^2
+ 15*a^14*c^10*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4)*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*
a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*
a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^
2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(3/4) - (a^11*c^10*d*x + a^11*c^11)*sqrt(((5
929*b^6*c^6 + 3234*a*b^5*c^5*d + 2751*a^2*b^4*c^4*d^2 + 2940*a^3*b^3*c^3*d^3 + 855*a^4*b^2*c^2*d^4 + 450*a^5*b
*c*d^5 + 225*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (a^8*c^6*d*x + a^8*c^7)*sqrt((35153041*b^12*c^12 + 3834877
2*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*
a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b
^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13)))/(d*x + c))*((351
53041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^
4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^
4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15
*c^13))^(3/4))/(35153041*b^12*c^13 + 38348772*a*b^11*c^12*d + 43080114*a^2*b^10*c^11*d^2 + 52655988*a^3*b^9*c^
10*d^3 + 36722511*a^4*b^8*c^9*d^4 + 27042120*a^5*b^7*c^8*d^5 + 18926460*a^6*b^6*c^7*d^6 + 8958600*a^7*b^5*c^6*
d^7 + 4614975*a^8*b^4*c^5*d^8 + 2092500*a^9*b^3*c^4*d^9 + 587250*a^10*b^2*c^3*d^10 + 202500*a^11*b*c^2*d^11 +
50625*a^12*c*d^12 + (35153041*b^12*c^12*d + 38348772*a*b^11*c^11*d^2 + 43080114*a^2*b^10*c^10*d^3 + 52655988*a
^3*b^9*c^9*d^4 + 36722511*a^4*b^8*c^8*d^5 + 27042120*a^5*b^7*c^7*d^6 + 18926460*a^6*b^6*c^6*d^7 + 8958600*a^7*
b^5*c^5*d^8 + 4614975*a^8*b^4*c^4*d^9 + 2092500*a^9*b^3*c^3*d^10 + 587250*a^10*b^2*c^2*d^11 + 202500*a^11*b*c*
d^12 + 50625*a^12*d^13)*x)) - 3*a^3*c^3*x^3*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*
c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c
^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^1
0 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4)*log(((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*
d^2 + 15*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a^4*c^3*d*x + a^4*c^4)*((35153041*b^12*c^12 + 38348772*a*
b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*
b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c
^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4))/(d*x + c)) + 3
*a^3*c^3*x^3*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10*c^10*d^2 + 52655988*a^3*b^9*c^9
*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 8958600*a^7*b^5*c^5*d^
7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^10 + 202500*a^11*b*c*d^11 + 5062
5*a^12*d^12)/(a^15*c^13))^(1/4)*log(((77*b^3*c^3 + 21*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 15*a^3*d^3)*(b*x + a)^(1/
4)*(d*x + c)^(3/4) - (a^4*c^3*d*x + a^4*c^4)*((35153041*b^12*c^12 + 38348772*a*b^11*c^11*d + 43080114*a^2*b^10
*c^10*d^2 + 52655988*a^3*b^9*c^9*d^3 + 36722511*a^4*b^8*c^8*d^4 + 27042120*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*
c^6*d^6 + 8958600*a^7*b^5*c^5*d^7 + 4614975*a^8*b^4*c^4*d^8 + 2092500*a^9*b^3*c^3*d^9 + 587250*a^10*b^2*c^2*d^
10 + 202500*a^11*b*c*d^11 + 50625*a^12*d^12)/(a^15*c^13))^(1/4))/(d*x + c)) + 4*(32*a^2*c^2 + (77*b^2*c^2 + 54
*a*b*c*d + 45*a^2*d^2)*x^2 - 4*(11*a*b*c^2 + 9*a^2*c*d)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a^3*c^3*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out