3.9.36 \(\int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx\)
Optimal. Leaf size=368 \[ \frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-117 a^2 d^2+10 a b c d+11 b^2 c^2\right )}{384 a^2 c^3 x^2}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-585 a^3 d^3+63 a^2 b c d^2+61 a b^2 c^2 d+77 b^3 c^3\right )}{1536 a^3 c^4 x}+\frac {(b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}+\frac {(b c-a d) \left (195 a^3 d^3+135 a^2 b c d^2+105 a b^2 c^2 d+77 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{1024 a^{15/4} c^{17/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{48 a c^2 x^3}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4} \]
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Rubi [A] time = 0.32, antiderivative size = 368, normalized size of antiderivative = 1.00,
number of steps used = 9, 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} \begin {gather*} \frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (-117 a^2 d^2+10 a b c d+11 b^2 c^2\right )}{384 a^2 c^3 x^2}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} \left (63 a^2 b c d^2-585 a^3 d^3+61 a b^2 c^2 d+77 b^3 c^3\right )}{1536 a^3 c^4 x}+\frac {(b c-a d) \left (135 a^2 b c d^2+195 a^3 d^3+105 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 )}{1024 a^{15/4} c^{17/4}}+\frac {(b c-a d) \left (135 a^2 b c d^2+195 a^3 d^3+105 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 )}{1024 a^{15/4} c^{17/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (b c-13 a d)}{48 a c^2 x^3}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
[Out]
-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(4*c*x^4) - ((b*c - 13*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(48*a*c^2*x^3)
+ ((11*b^2*c^2 + 10*a*b*c*d - 117*a^2*d^2)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(384*a^2*c^3*x^2) - ((77*b^3*c^3
+ 61*a*b^2*c^2*d + 63*a^2*b*c*d^2 - 585*a^3*d^3)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(1536*a^3*c^4*x) + ((b*c - a
*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 195*a^3*d^3)*ArcTan[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(
c + d*x)^(1/4))])/(1024*a^(15/4)*c^(17/4)) + ((b*c - a*d)*(77*b^3*c^3 + 105*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 19
5*a^3*d^3)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(1024*a^(15/4)*c^(17/4))
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 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 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]
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 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]
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}+\frac {\int \frac {\frac {1}{4} (b c-13 a d)-3 b d x}{x^4 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 c}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}-\frac {\int \frac {\frac {1}{16} \left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right )+\frac {1}{2} b d (b c-13 a d) x}{x^3 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{12 a c^2}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}+\frac {\int \frac {\frac {1}{64} \left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right )+\frac {1}{16} b d \left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) x}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{24 a^2 c^3}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}-\frac {\left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{1536 a^3 c^4 x}-\frac {\int \frac {3 (b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\right )}{256 x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{24 a^3 c^4}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}-\frac {\left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{1536 a^3 c^4 x}-\frac {\left ((b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\right )\right ) \int \frac {1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2048 a^3 c^4}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}-\frac {\left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{1536 a^3 c^4 x}-\frac {\left ((b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\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 )}{512 a^3 c^4}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}-\frac {\left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{1536 a^3 c^4 x}+\frac {\left ((b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\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 )}{1024 a^{7/2} c^4}+\frac {\left ((b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 a^3 d^3\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 )}{1024 a^{7/2} c^4}\\ &=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{4 c x^4}-\frac {(b c-13 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{48 a c^2 x^3}+\frac {\left (11 b^2 c^2+10 a b c d-117 a^2 d^2\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{384 a^2 c^3 x^2}-\frac {\left (77 b^3 c^3+61 a b^2 c^2 d+63 a^2 b c d^2-585 a^3 d^3\right ) \sqrt [4]{a+b x} (c+d x)^{3/4}}{1536 a^3 c^4 x}+\frac {(b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 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 )}{1024 a^{15/4} c^{17/4}}+\frac {(b c-a d) \left (77 b^3 c^3+105 a b^2 c^2 d+135 a^2 b c d^2+195 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 )}{1024 a^{15/4} c^{17/4}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 212, normalized size = 0.58 \begin {gather*} \frac {\sqrt [4]{a+b x} \left (3 x^4 \left (-195 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 c^2 d^2+28 a b^3 c^3 d+77 b^4 c^4\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^3 \left (384 c^3-416 c^2 d x+468 c d^2 x^2-585 d^3 x^3\right )+a^2 b c x \left (32 c^2-40 c d x+63 d^2 x^2\right )+a b^2 c^2 x^2 (61 d x-44 c)+77 b^3 c^3 x^3\right )\right )}{1536 a^4 c^4 x^4 \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
[Out]
((a + b*x)^(1/4)*(-(a*(c + d*x)*(77*b^3*c^3*x^3 + a*b^2*c^2*x^2*(-44*c + 61*d*x) + a^2*b*c*x*(32*c^2 - 40*c*d*
x + 63*d^2*x^2) + a^3*(384*c^3 - 416*c^2*d*x + 468*c*d^2*x^2 - 585*d^3*x^3))) + 3*(77*b^4*c^4 + 28*a*b^3*c^3*d
+ 30*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 195*a^4*d^4)*x^4*Hypergeometric2F1[1/4, 1, 5/4, (c*(a + b*x))/(a*(c +
d*x))]))/(1536*a^4*c^4*x^4*(c + d*x)^(1/4))
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IntegrateAlgebraic [F] time = 130.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{a+b x}}{x^5 \sqrt [4]{c+d x}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x]
[Out]
Defer[IntegrateAlgebraic][(a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)), x]
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fricas [B] time = 3.42, size = 2877, normalized size = 7.82
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x, algorithm="fricas")
[Out]
1/6144*(12*a^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*
a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 -
989262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^1
0 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*
b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4)*arctan(((77*a^11*b^4*c^17 +
28*a^12*b^3*c^16*d + 30*a^13*b^2*c^15*d^2 + 60*a^14*b*c^14*d^3 - 195*a^15*c^13*d^4)*(b*x + a)^(1/4)*(d*x + c)
^(3/4)*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^
3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^
9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a
^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1
779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(3/4) + (a^11*c^13*d*x + a^11*c^14)*sqrt(((5929*b
^8*c^8 + 4312*a*b^7*c^7*d + 5404*a^2*b^6*c^6*d^2 + 10920*a^3*b^5*c^5*d^3 - 25770*a^4*b^4*c^4*d^4 - 7320*a^5*b^
3*c^3*d^5 - 8100*a^6*b^2*c^2*d^6 - 23400*a^7*b*c*d^7 + 38025*a^8*d^8)*sqrt(b*x + a)*sqrt(d*x + c) + (a^8*c^8*d
*x + a^8*c^9)*sqrt((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b
^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 98926
2960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2
155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c
^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17)))/(d*x + c))*((35153041*b^16*c^16 + 511
31696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 -
191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8
*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^1
2*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900
625*a^16*d^16)/(a^15*c^17))^(3/4))/(35153041*b^16*c^17 + 51131696*a*b^15*c^16*d + 82673976*a^2*b^14*c^15*d^2 +
176093456*a^3*b^13*c^14*d^3 - 182203364*a^4*b^12*c^13*d^4 - 191017680*a^5*b^11*c^12*d^5 - 318453240*a^6*b^10*
c^11*d^6 - 989262960*a^7*b^9*c^10*d^7 + 665778150*a^8*b^8*c^9*d^8 + 275389200*a^9*b^7*c^8*d^9 + 370974600*a^10
*b^6*c^7*d^10 + 2155086000*a^11*b^5*c^6*d^11 - 1551622500*a^12*b^4*c^5*d^12 - 177606000*a^13*b^3*c^4*d^13 - 68
445000*a^14*b^2*c^3*d^14 - 1779570000*a^15*b*c^2*d^15 + 1445900625*a^16*c*d^16 + (35153041*b^16*c^16*d + 51131
696*a*b^15*c^15*d^2 + 82673976*a^2*b^14*c^14*d^3 + 176093456*a^3*b^13*c^13*d^4 - 182203364*a^4*b^12*c^12*d^5 -
191017680*a^5*b^11*c^11*d^6 - 318453240*a^6*b^10*c^10*d^7 - 989262960*a^7*b^9*c^9*d^8 + 665778150*a^8*b^8*c^8
*d^9 + 275389200*a^9*b^7*c^7*d^10 + 370974600*a^10*b^6*c^6*d^11 + 2155086000*a^11*b^5*c^5*d^12 - 1551622500*a^
12*b^4*c^4*d^13 - 177606000*a^13*b^3*c^3*d^14 - 68445000*a^14*b^2*c^2*d^15 - 1779570000*a^15*b*c*d^16 + 144590
0625*a^16*d^17)*x)) + 3*a^3*c^4*x^4*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2
+ 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^1
0*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^1
0*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 6
8445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4)*log(-((77*b^4*
c^4 + 28*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 195*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (a
^4*c^4*d*x + a^4*c^5)*((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a
^3*b^13*c^13*d^3 - 182203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 9
89262960*a^7*b^9*c^9*d^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10
+ 2155086000*a^11*b^5*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b
^2*c^2*d^14 - 1779570000*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4))/(d*x + c)) - 3*a^3*c^4*x^4*
((35153041*b^16*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182
203364*a^4*b^12*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d
^7 + 665778150*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*b^5
*c^5*d^11 - 1551622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 17795700
00*a^15*b*c*d^15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4)*log(-((77*b^4*c^4 + 28*a*b^3*c^3*d + 30*a^2*b^2*c^
2*d^2 + 60*a^3*b*c*d^3 - 195*a^4*d^4)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^4*c^4*d*x + a^4*c^5)*((35153041*b^1
6*c^16 + 51131696*a*b^15*c^15*d + 82673976*a^2*b^14*c^14*d^2 + 176093456*a^3*b^13*c^13*d^3 - 182203364*a^4*b^1
2*c^12*d^4 - 191017680*a^5*b^11*c^11*d^5 - 318453240*a^6*b^10*c^10*d^6 - 989262960*a^7*b^9*c^9*d^7 + 665778150
*a^8*b^8*c^8*d^8 + 275389200*a^9*b^7*c^7*d^9 + 370974600*a^10*b^6*c^6*d^10 + 2155086000*a^11*b^5*c^5*d^11 - 15
51622500*a^12*b^4*c^4*d^12 - 177606000*a^13*b^3*c^3*d^13 - 68445000*a^14*b^2*c^2*d^14 - 1779570000*a^15*b*c*d^
15 + 1445900625*a^16*d^16)/(a^15*c^17))^(1/4))/(d*x + c)) - 4*(384*a^3*c^3 + (77*b^3*c^3 + 61*a*b^2*c^2*d + 63
*a^2*b*c*d^2 - 585*a^3*d^3)*x^3 - 4*(11*a*b^2*c^3 + 10*a^2*b*c^2*d - 117*a^3*c*d^2)*x^2 + 32*(a^2*b*c^3 - 13*a
^3*c^2*d)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a^3*c^4*x^4)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x, algorithm="giac")
[Out]
integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5), x)
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}} x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
[Out]
int((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(1/4)/x^5/(d*x+c)^(1/4),x, algorithm="maxima")
[Out]
integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^5), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{1/4}}{x^5\,{\left (c+d\,x\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)),x)
[Out]
int((a + b*x)^(1/4)/(x^5*(c + d*x)^(1/4)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{a + b x}}{x^{5} \sqrt [4]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**(1/4)/x**5/(d*x+c)**(1/4),x)
[Out]
Integral((a + b*x)**(1/4)/(x**5*(c + d*x)**(1/4)), x)
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