3.31.12 \(\int \frac {1+x}{(1-a x) \sqrt [4]{\frac {1-b x}{c+x}}} \, dx\) [3012]

Optimal. Leaf size=411 \[ \frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}-\frac {(a+4 b+4 a b+a b c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}}{-\sqrt {b}+\sqrt {\frac {1-b x}{c+x}}}\right )}{2 \sqrt {2} a^2 b^{5/4}}-\frac {\sqrt {2} (1+a) \sqrt [4]{1+a c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{-a+b} \sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt {-a+b}-\sqrt {1+a c} \sqrt {\frac {1-b x}{c+x}}}\right )}{a^2 \sqrt [4]{-a+b}}-\frac {(a+4 b+4 a b+a b c) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt {b}+\sqrt {\frac {1-b x}{c+x}}}\right )}{2 \sqrt {2} a^2 b^{5/4}}+\frac {\sqrt {2} (1+a) \sqrt [4]{1+a c} \tanh ^{-1}\left (\frac {\sqrt {-a+b}+\sqrt {1+a c} \sqrt {\frac {1-b x}{c+x}}}{\sqrt {2} \sqrt [4]{-a+b} \sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}\right )}{a^2 \sqrt [4]{-a+b}} \]

[Out]

(c+x)*((-b*x+1)/(c+x))^(3/4)/a/b-1/4*(a*b*c+4*a*b+a+4*b)*arctan(2^(1/2)*b^(1/4)*((-b*x+1)/(c+x))^(1/4)/(-b^(1/
2)+((-b*x+1)/(c+x))^(1/2)))*2^(1/2)/a^2/b^(5/4)-2^(1/2)*(1+a)*(a*c+1)^(1/4)*arctan(2^(1/2)*(-a+b)^(1/4)*(a*c+1
)^(1/4)*((-b*x+1)/(c+x))^(1/4)/((-a+b)^(1/2)-(a*c+1)^(1/2)*((-b*x+1)/(c+x))^(1/2)))/a^2/(-a+b)^(1/4)-1/4*(a*b*
c+4*a*b+a+4*b)*arctanh(2^(1/2)*b^(1/4)*((-b*x+1)/(c+x))^(1/4)/(b^(1/2)+((-b*x+1)/(c+x))^(1/2)))*2^(1/2)/a^2/b^
(5/4)+2^(1/2)*(1+a)*(a*c+1)^(1/4)*arctanh(1/2*((-a+b)^(1/2)+(a*c+1)^(1/2)*((-b*x+1)/(c+x))^(1/2))*2^(1/2)/(-a+
b)^(1/4)/(a*c+1)^(1/4)/((-b*x+1)/(c+x))^(1/4))/a^2/(-a+b)^(1/4)

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Rubi [A]
time = 0.99, antiderivative size = 425, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2002, 593, 598, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} -\frac {(a b (c+4)+a+4 b) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^2 b^{5/4}}+\frac {(a b (c+4)+a+4 b) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{b}}+1\right )}{2 \sqrt {2} a^2 b^{5/4}}-\frac {2 (a+1) \sqrt [4]{a c+1} \text {ArcTan}\left (\frac {\sqrt [4]{a c+1} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {(a b (c+4)+a+4 b) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}+\sqrt {\frac {1-b x}{c+x}}+\sqrt {b}\right )}{4 \sqrt {2} a^2 b^{5/4}}-\frac {(a b (c+4)+a+4 b) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}+\sqrt {\frac {1-b x}{c+x}}+\sqrt {b}\right )}{4 \sqrt {2} a^2 b^{5/4}}+\frac {2 (a+1) \sqrt [4]{a c+1} \tanh ^{-1}\left (\frac {\sqrt [4]{a c+1} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c + x)*((1 - b*x)/(c + x))^(3/4))/(a*b) - (2*(1 + a)*(1 + a*c)^(1/4)*ArcTan[((1 + a*c)^(1/4)*((1 - b*x)/(c +
x))^(1/4))/(a - b)^(1/4)])/(a^2*(a - b)^(1/4)) - ((a + 4*b + a*b*(4 + c))*ArcTan[1 - (Sqrt[2]*((1 - b*x)/(c +
x))^(1/4))/b^(1/4)])/(2*Sqrt[2]*a^2*b^(5/4)) + ((a + 4*b + a*b*(4 + c))*ArcTan[1 + (Sqrt[2]*((1 - b*x)/(c + x
))^(1/4))/b^(1/4)])/(2*Sqrt[2]*a^2*b^(5/4)) + (2*(1 + a)*(1 + a*c)^(1/4)*ArcTanh[((1 + a*c)^(1/4)*((1 - b*x)/(
c + x))^(1/4))/(a - b)^(1/4)])/(a^2*(a - b)^(1/4)) + ((a + 4*b + a*b*(4 + c))*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*((
1 - b*x)/(c + x))^(1/4) + Sqrt[(1 - b*x)/(c + x)]])/(4*Sqrt[2]*a^2*b^(5/4)) - ((a + 4*b + a*b*(4 + c))*Log[Sqr
t[b] + Sqrt[2]*b^(1/4)*((1 - b*x)/(c + x))^(1/4) + Sqrt[(1 - b*x)/(c + x)]])/(4*Sqrt[2]*a^2*b^(5/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2002

Int[(u_)^(p_.)*(v_)^(q_.)*((g_.)*(x_))^(m_.)*(z_)^(r_.), x_Symbol] :> Int[(g*x)^m*ExpandToSum[u, x]^p*ExpandTo
Sum[v, x]^q*ExpandToSum[z, x]^r, x] /; FreeQ[{g, m, p, q, r}, x] && BinomialQ[{u, v, z}, x] && EqQ[BinomialDeg
ree[u, x] - BinomialDegree[v, x], 0] && EqQ[BinomialDegree[u, x] - BinomialDegree[z, x], 0] &&  !BinomialMatch
Q[{u, v, z}, x]

Rubi steps

\begin {align*} \int \frac {1+x}{(1-a x) \sqrt [4]{\frac {1-b x}{c+x}}} \, dx &=-\left ((4 (1+b c)) \text {Subst}\left (\int \frac {x^2 \left (1+b-(-1+c) x^4\right )}{\left (b+x^4\right )^2 \left (b+x^4+a \left (-1+c x^4\right )\right )} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )\right )\\ &=-\left ((4 (1+b c)) \text {Subst}\left (\int \frac {x^2 \left (1+b+(1-c) x^4\right )}{\left (b+x^4\right )^2 \left (-a+b+(1+a c) x^4\right )} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )\right )\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}-\frac {\text {Subst}\left (\int \frac {x^2 \left ((a+3 b+4 a b) (1+b c)-(1+a c) (1+b c) x^4\right )}{\left (b+x^4\right ) \left (-a+b+(1+a c) x^4\right )} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{a b}\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}-\frac {\text {Subst}\left (\int \left (\frac {(-a-4 b-a b (4+c)) x^2}{a \left (b+x^4\right )}+\frac {4 (1+a) b (-1-a c) x^2}{a \left (a-b-(1+a c) x^4\right )}\right ) \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{a b}\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}+\frac {(4 (1+a) (1+a c)) \text {Subst}\left (\int \frac {x^2}{a-b-(1+a c) x^4} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{a^2}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {x^2}{b+x^4} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{a^2 b}\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}+\frac {\left (2 (1+a) \sqrt {1+a c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b}-\sqrt {1+a c} x^2} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{a^2}-\frac {\left (2 (1+a) \sqrt {1+a c}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b}+\sqrt {1+a c} x^2} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{a^2}-\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{2 a^2 b}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{2 a^2 b}\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}-\frac {2 (1+a) \sqrt [4]{1+a c} \tan ^{-1}\left (\frac {\sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {2 (1+a) \sqrt [4]{1+a c} \tanh ^{-1}\left (\frac {\sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {2} a^2 b^{5/4}}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {2} a^2 b^{5/4}}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{4 a^2 b}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{\frac {1-b x}{c+x}}\right )}{4 a^2 b}\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}-\frac {2 (1+a) \sqrt [4]{1+a c} \tan ^{-1}\left (\frac {\sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {2 (1+a) \sqrt [4]{1+a c} \tanh ^{-1}\left (\frac {\sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {(a+4 b+a b (4+c)) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}+\sqrt {\frac {1-b x}{c+x}}\right )}{4 \sqrt {2} a^2 b^{5/4}}-\frac {(a+4 b+a b (4+c)) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}+\sqrt {\frac {1-b x}{c+x}}\right )}{4 \sqrt {2} a^2 b^{5/4}}+\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^2 b^{5/4}}-\frac {(a+4 b+a b (4+c)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^2 b^{5/4}}\\ &=\frac {(c+x) \left (\frac {1-b x}{c+x}\right )^{3/4}}{a b}-\frac {2 (1+a) \sqrt [4]{1+a c} \tan ^{-1}\left (\frac {\sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}-\frac {(a+4 b+a b (4+c)) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^2 b^{5/4}}+\frac {(a+4 b+a b (4+c)) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^2 b^{5/4}}+\frac {2 (1+a) \sqrt [4]{1+a c} \tanh ^{-1}\left (\frac {\sqrt [4]{1+a c} \sqrt [4]{\frac {1-b x}{c+x}}}{\sqrt [4]{a-b}}\right )}{a^2 \sqrt [4]{a-b}}+\frac {(a+4 b+a b (4+c)) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}+\sqrt {\frac {1-b x}{c+x}}\right )}{4 \sqrt {2} a^2 b^{5/4}}-\frac {(a+4 b+a b (4+c)) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{\frac {1-b x}{c+x}}+\sqrt {\frac {1-b x}{c+x}}\right )}{4 \sqrt {2} a^2 b^{5/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.15, size = 151, normalized size = 0.37 \begin {gather*} -\frac {4 \left (5 (1+a) \sqrt [4]{\frac {1-b x}{1+b c}} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {b (c+x)}{1+b c}\right )-5 (1+a) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {(a-b) (c+x)}{(1+a c) (1-b x)}\right )+a (c+x) \sqrt [4]{\frac {1-b x}{1+b c}} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\frac {b (c+x)}{1+b c}\right )\right )}{5 a^2 \sqrt [4]{\frac {1-b x}{c+x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(5*(1 + a)*((1 - b*x)/(1 + b*c))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (b*(c + x))/(1 + b*c)] - 5*(1 + a)
*Hypergeometric2F1[1/4, 1, 5/4, ((a - b)*(c + x))/((1 + a*c)*(1 - b*x))] + a*(c + x)*((1 - b*x)/(1 + b*c))^(1/
4)*Hypergeometric2F1[1/4, 5/4, 9/4, (b*(c + x))/(1 + b*c)]))/(5*a^2*((1 - b*x)/(c + x))^(1/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1+x}{\left (-a x +1\right ) \left (\frac {-b x +1}{c +x}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4062 vs. \(2 (341) = 682\).
time = 2.22, size = 4062, normalized size = 9.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*a*b*(-(a^4*b^4*c^4 + 256*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a^2 + a)*b^3
+ 4*(a^4*b^3 + 4*(a^4 + a^3)*b^4)*c^3 + 96*(a^4 + 2*a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3 + a^2)*b^4
+ 8*(a^4 + a^3)*b^3)*c^2 + 16*(a^4 + a^3)*b + 4*(a^4*b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4 + 2*a^3 +
a^2)*b^3 + 12*(a^4 + a^3)*b^2)*c)/(a^8*b^5))^(1/4)*arctan((sqrt((a^6*b^6*c^6 + 4096*(a^6 + 6*a^5 + 15*a^4 + 20
*a^3 + 15*a^2 + 6*a + 1)*b^6 + a^6 + 6144*(a^6 + 5*a^5 + 10*a^4 + 10*a^3 + 5*a^2 + a)*b^5 + 6*(a^6*b^5 + 4*(a^
6 + a^5)*b^6)*c^5 + 3840*(a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*b^4 + 15*(a^6*b^4 + 16*(a^6 + 2*a^5 + a^4)*b^6 +
8*(a^6 + a^5)*b^5)*c^4 + 1280*(a^6 + 3*a^5 + 3*a^4 + a^3)*b^3 + 20*(a^6*b^3 + 64*(a^6 + 3*a^5 + 3*a^4 + a^3)*b
^6 + 48*(a^6 + 2*a^5 + a^4)*b^5 + 12*(a^6 + a^5)*b^4)*c^3 + 240*(a^6 + 2*a^5 + a^4)*b^2 + 15*(a^6*b^2 + 256*(a
^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*b^6 + 256*(a^6 + 3*a^5 + 3*a^4 + a^3)*b^5 + 96*(a^6 + 2*a^5 + a^4)*b^4 + 16*
(a^6 + a^5)*b^3)*c^2 + 24*(a^6 + a^5)*b + 6*(a^6*b + 1024*(a^6 + 5*a^5 + 10*a^4 + 10*a^3 + 5*a^2 + a)*b^6 + 12
80*(a^6 + 4*a^5 + 6*a^4 + 4*a^3 + a^2)*b^5 + 640*(a^6 + 3*a^5 + 3*a^4 + a^3)*b^4 + 160*(a^6 + 2*a^5 + a^4)*b^3
+ 20*(a^6 + a^5)*b^2)*c)*sqrt(-(b*x - 1)/(c + x)) - (a^8*b^7*c^4 + a^8*b^3 + 256*(a^8 + 4*a^7 + 6*a^6 + 4*a^5
+ a^4)*b^7 + 256*(a^8 + 3*a^7 + 3*a^6 + a^5)*b^6 + 96*(a^8 + 2*a^7 + a^6)*b^5 + 16*(a^8 + a^7)*b^4 + 4*(a^8*b
^6 + 4*(a^8 + a^7)*b^7)*c^3 + 6*(a^8*b^5 + 16*(a^8 + 2*a^7 + a^6)*b^7 + 8*(a^8 + a^7)*b^6)*c^2 + 4*(a^8*b^4 +
64*(a^8 + 3*a^7 + 3*a^6 + a^5)*b^7 + 48*(a^8 + 2*a^7 + a^6)*b^6 + 12*(a^8 + a^7)*b^5)*c)*sqrt(-(a^4*b^4*c^4 +
256*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a^2 + a)*b^3 + 4*(a^4*b^3 + 4*(a^4 + a^3)
*b^4)*c^3 + 96*(a^4 + 2*a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3 + a^2)*b^4 + 8*(a^4 + a^3)*b^3)*c^2 + 16
*(a^4 + a^3)*b + 4*(a^4*b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4 + 2*a^3 + a^2)*b^3 + 12*(a^4 + a^3)*b^2
)*c)/(a^8*b^5)))*a^2*b*(-(a^4*b^4*c^4 + 256*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a
^2 + a)*b^3 + 4*(a^4*b^3 + 4*(a^4 + a^3)*b^4)*c^3 + 96*(a^4 + 2*a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3
+ a^2)*b^4 + 8*(a^4 + a^3)*b^3)*c^2 + 16*(a^4 + a^3)*b + 4*(a^4*b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4
+ 2*a^3 + a^2)*b^3 + 12*(a^4 + a^3)*b^2)*c)/(a^8*b^5))^(1/4) - (a^5*b^4*c^3 + a^5*b + 64*(a^5 + 3*a^4 + 3*a^3
+ a^2)*b^4 + 48*(a^5 + 2*a^4 + a^3)*b^3 + 12*(a^5 + a^4)*b^2 + 3*(a^5*b^3 + 4*(a^5 + a^4)*b^4)*c^2 + 3*(a^5*b
^2 + 16*(a^5 + 2*a^4 + a^3)*b^4 + 8*(a^5 + a^4)*b^3)*c)*(-(b*x - 1)/(c + x))^(1/4)*(-(a^4*b^4*c^4 + 256*(a^4 +
4*a^3 + 6*a^2 + 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a^2 + a)*b^3 + 4*(a^4*b^3 + 4*(a^4 + a^3)*b^4)*c^3
+ 96*(a^4 + 2*a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3 + a^2)*b^4 + 8*(a^4 + a^3)*b^3)*c^2 + 16*(a^4 + a^
3)*b + 4*(a^4*b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4 + 2*a^3 + a^2)*b^3 + 12*(a^4 + a^3)*b^2)*c)/(a^8*
b^5))^(1/4))/(a^4*b^4*c^4 + 256*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a^2 + a)*b^3
+ 4*(a^4*b^3 + 4*(a^4 + a^3)*b^4)*c^3 + 96*(a^4 + 2*a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3 + a^2)*b^4 +
8*(a^4 + a^3)*b^3)*c^2 + 16*(a^4 + a^3)*b + 4*(a^4*b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4 + 2*a^3 + a
^2)*b^3 + 12*(a^4 + a^3)*b^2)*c)) - 16*a*b*((a^4 + 4*a^3 + 6*a^2 + (a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)*c + 4*a +
1)/(a^9 - a^8*b))^(1/4)*arctan((sqrt((a^9 + 4*a^8 + 6*a^7 + 4*a^6 + a^5 - (a^8 + 4*a^7 + 6*a^6 + 4*a^5 + a^4)
*b + (a^10 + 4*a^9 + 6*a^8 + 4*a^7 + a^6 - (a^9 + 4*a^8 + 6*a^7 + 4*a^6 + a^5)*b)*c)*sqrt((a^4 + 4*a^3 + 6*a^2
+ (a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)*c + 4*a + 1)/(a^9 - a^8*b)) + (a^6 + 6*a^5 + 15*a^4 + 20*a^3 + (a^8 + 6*a
^7 + 15*a^6 + 20*a^5 + 15*a^4 + 6*a^3 + a^2)*c^2 + 15*a^2 + 2*(a^7 + 6*a^6 + 15*a^5 + 20*a^4 + 15*a^3 + 6*a^2
+ a)*c + 6*a + 1)*sqrt(-(b*x - 1)/(c + x)))*a^2*((a^4 + 4*a^3 + 6*a^2 + (a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)*c +
4*a + 1)/(a^9 - a^8*b))^(1/4) - (a^5 + 3*a^4 + 3*a^3 + a^2 + (a^6 + 3*a^5 + 3*a^4 + a^3)*c)*((a^4 + 4*a^3 + 6*
a^2 + (a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)*c + 4*a + 1)/(a^9 - a^8*b))^(1/4)*(-(b*x - 1)/(c + x))^(1/4))/(a^4 + 4
*a^3 + 6*a^2 + (a^5 + 4*a^4 + 6*a^3 + 4*a^2 + a)*c + 4*a + 1)) - a*b*(-(a^4*b^4*c^4 + 256*(a^4 + 4*a^3 + 6*a^2
+ 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a^2 + a)*b^3 + 4*(a^4*b^3 + 4*(a^4 + a^3)*b^4)*c^3 + 96*(a^4 + 2*
a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3 + a^2)*b^4 + 8*(a^4 + a^3)*b^3)*c^2 + 16*(a^4 + a^3)*b + 4*(a^4*
b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4 + 2*a^3 + a^2)*b^3 + 12*(a^4 + a^3)*b^2)*c)/(a^8*b^5))^(1/4)*lo
g(a^6*b^4*(-(a^4*b^4*c^4 + 256*(a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*b^4 + a^4 + 256*(a^4 + 3*a^3 + 3*a^2 + a)*b^3 +
4*(a^4*b^3 + 4*(a^4 + a^3)*b^4)*c^3 + 96*(a^4 + 2*a^3 + a^2)*b^2 + 6*(a^4*b^2 + 16*(a^4 + 2*a^3 + a^2)*b^4 +
8*(a^4 + a^3)*b^3)*c^2 + 16*(a^4 + a^3)*b + 4*(a^4*b + 64*(a^4 + 3*a^3 + 3*a^2 + a)*b^4 + 48*(a^4 + 2*a^3 + a^
2)*b^3 + 12*(a^4 + a^3)*b^2)*c)/(a^8*b^5))^(3/4...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{a x \sqrt [4]{- \frac {b x}{c + x} + \frac {1}{c + x}} - \sqrt [4]{- \frac {b x}{c + x} + \frac {1}{c + x}}}\, dx - \int \frac {1}{a x \sqrt [4]{- \frac {b x}{c + x} + \frac {1}{c + x}} - \sqrt [4]{- \frac {b x}{c + x} + \frac {1}{c + x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1194 vs. \(2 (341) = 682\).
time = 1.97, size = 1194, normalized size = 2.91 \begin {gather*} \frac {1}{8} \, {\left (\frac {8 \, {\left (-a^{4} c^{3} + a^{3} b c^{3} - 3 \, a^{3} c^{2} + 3 \, a^{2} b c^{2} - 3 \, a^{2} c + 3 \, a b c - a + b\right )}^{\frac {3}{4}} {\left (\sqrt {2} a b c + \sqrt {2} b c + \sqrt {2} a + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a - b}{a c + 1}\right )^{\frac {1}{4}} + 2 \, \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a - b}{a c + 1}\right )^{\frac {1}{4}}}\right )}{a^{5} c^{2} - a^{4} b c^{2} + 2 \, a^{4} c - 2 \, a^{3} b c + a^{3} - a^{2} b} + \frac {8 \, {\left (-a^{4} c^{3} + a^{3} b c^{3} - 3 \, a^{3} c^{2} + 3 \, a^{2} b c^{2} - 3 \, a^{2} c + 3 \, a b c - a + b\right )}^{\frac {3}{4}} {\left (\sqrt {2} a b c + \sqrt {2} b c + \sqrt {2} a + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a - b}{a c + 1}\right )^{\frac {1}{4}} - 2 \, \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a - b}{a c + 1}\right )^{\frac {1}{4}}}\right )}{a^{5} c^{2} - a^{4} b c^{2} + 2 \, a^{4} c - 2 \, a^{3} b c + a^{3} - a^{2} b} - \frac {8 \, {\left (-a^{4} c^{3} + a^{3} b c^{3} - 3 \, a^{3} c^{2} + 3 \, a^{2} b c^{2} - 3 \, a^{2} c + 3 \, a b c - a + b\right )}^{\frac {3}{4}} {\left (a b c + b c + a + 1\right )} \log \left (\sqrt {2} \left (-\frac {a - b}{a c + 1}\right )^{\frac {1}{4}} \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}} + \sqrt {-\frac {a - b}{a c + 1}} + \sqrt {-\frac {b x - 1}{c + x}}\right )}{\sqrt {2} a^{5} c^{2} - \sqrt {2} a^{4} b c^{2} + 2 \, \sqrt {2} a^{4} c - 2 \, \sqrt {2} a^{3} b c + \sqrt {2} a^{3} - \sqrt {2} a^{2} b} + \frac {8 \, {\left (-a^{4} c^{3} + a^{3} b c^{3} - 3 \, a^{3} c^{2} + 3 \, a^{2} b c^{2} - 3 \, a^{2} c + 3 \, a b c - a + b\right )}^{\frac {3}{4}} {\left (a b c + b c + a + 1\right )} \log \left (-\sqrt {2} \left (-\frac {a - b}{a c + 1}\right )^{\frac {1}{4}} \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}} + \sqrt {-\frac {a - b}{a c + 1}} + \sqrt {-\frac {b x - 1}{c + x}}\right )}{\sqrt {2} a^{5} c^{2} - \sqrt {2} a^{4} b c^{2} + 2 \, \sqrt {2} a^{4} c - 2 \, \sqrt {2} a^{3} b c + \sqrt {2} a^{3} - \sqrt {2} a^{2} b} + \frac {2 \, \sqrt {2} {\left (a b^{\frac {11}{4}} c^{2} + 2 \, {\left (2 \, b^{2} + b\right )} a b^{\frac {3}{4}} c + 4 \, b^{\frac {11}{4}} c + a {\left (4 \, b + 1\right )} b^{\frac {3}{4}} + 4 \, b^{\frac {7}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{a^{2} b^{2}} + \frac {2 \, \sqrt {2} {\left (a b^{\frac {11}{4}} c^{2} + 2 \, {\left (2 \, b^{2} + b\right )} a b^{\frac {3}{4}} c + 4 \, b^{\frac {11}{4}} c + a {\left (4 \, b + 1\right )} b^{\frac {3}{4}} + 4 \, b^{\frac {7}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{a^{2} b^{2}} - \frac {\sqrt {2} {\left (a b^{\frac {11}{4}} c^{2} + 2 \, {\left (2 \, b^{2} + b\right )} a b^{\frac {3}{4}} c + 4 \, b^{\frac {11}{4}} c + a {\left (4 \, b + 1\right )} b^{\frac {3}{4}} + 4 \, b^{\frac {7}{4}}\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}} + \sqrt {b} + \sqrt {-\frac {b x - 1}{c + x}}\right )}{a^{2} b^{2}} + \frac {\sqrt {2} {\left (a b^{\frac {11}{4}} c^{2} + 2 \, {\left (2 \, b^{2} + b\right )} a b^{\frac {3}{4}} c + 4 \, b^{\frac {11}{4}} c + a {\left (4 \, b + 1\right )} b^{\frac {3}{4}} + 4 \, b^{\frac {7}{4}}\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} \left (-\frac {b x - 1}{c + x}\right )^{\frac {1}{4}} + \sqrt {b} + \sqrt {-\frac {b x - 1}{c + x}}\right )}{a^{2} b^{2}} + \frac {8 \, {\left (b^{2} c^{2} \left (-\frac {b x - 1}{c + x}\right )^{\frac {3}{4}} + 2 \, b c \left (-\frac {b x - 1}{c + x}\right )^{\frac {3}{4}} + \left (-\frac {b x - 1}{c + x}\right )^{\frac {3}{4}}\right )}}{a {\left (b - \frac {b x - 1}{c + x}\right )} b}\right )} {\left (\frac {b c}{{\left (b c + 1\right )}^{2}} + \frac {1}{{\left (b c + 1\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*(-a^4*c^3 + a^3*b*c^3 - 3*a^3*c^2 + 3*a^2*b*c^2 - 3*a^2*c + 3*a*b*c - a + b)^(3/4)*(sqrt(2)*a*b*c + sqr
t(2)*b*c + sqrt(2)*a + sqrt(2))*arctan(1/2*sqrt(2)*(sqrt(2)*(-(a - b)/(a*c + 1))^(1/4) + 2*(-(b*x - 1)/(c + x)
)^(1/4))/(-(a - b)/(a*c + 1))^(1/4))/(a^5*c^2 - a^4*b*c^2 + 2*a^4*c - 2*a^3*b*c + a^3 - a^2*b) + 8*(-a^4*c^3 +
a^3*b*c^3 - 3*a^3*c^2 + 3*a^2*b*c^2 - 3*a^2*c + 3*a*b*c - a + b)^(3/4)*(sqrt(2)*a*b*c + sqrt(2)*b*c + sqrt(2)
*a + sqrt(2))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-(a - b)/(a*c + 1))^(1/4) - 2*(-(b*x - 1)/(c + x))^(1/4))/(-(a - b
)/(a*c + 1))^(1/4))/(a^5*c^2 - a^4*b*c^2 + 2*a^4*c - 2*a^3*b*c + a^3 - a^2*b) - 8*(-a^4*c^3 + a^3*b*c^3 - 3*a^
3*c^2 + 3*a^2*b*c^2 - 3*a^2*c + 3*a*b*c - a + b)^(3/4)*(a*b*c + b*c + a + 1)*log(sqrt(2)*(-(a - b)/(a*c + 1))^
(1/4)*(-(b*x - 1)/(c + x))^(1/4) + sqrt(-(a - b)/(a*c + 1)) + sqrt(-(b*x - 1)/(c + x)))/(sqrt(2)*a^5*c^2 - sqr
t(2)*a^4*b*c^2 + 2*sqrt(2)*a^4*c - 2*sqrt(2)*a^3*b*c + sqrt(2)*a^3 - sqrt(2)*a^2*b) + 8*(-a^4*c^3 + a^3*b*c^3
- 3*a^3*c^2 + 3*a^2*b*c^2 - 3*a^2*c + 3*a*b*c - a + b)^(3/4)*(a*b*c + b*c + a + 1)*log(-sqrt(2)*(-(a - b)/(a*c
+ 1))^(1/4)*(-(b*x - 1)/(c + x))^(1/4) + sqrt(-(a - b)/(a*c + 1)) + sqrt(-(b*x - 1)/(c + x)))/(sqrt(2)*a^5*c^
2 - sqrt(2)*a^4*b*c^2 + 2*sqrt(2)*a^4*c - 2*sqrt(2)*a^3*b*c + sqrt(2)*a^3 - sqrt(2)*a^2*b) + 2*sqrt(2)*(a*b^(1
1/4)*c^2 + 2*(2*b^2 + b)*a*b^(3/4)*c + 4*b^(11/4)*c + a*(4*b + 1)*b^(3/4) + 4*b^(7/4))*arctan(1/2*sqrt(2)*(sqr
t(2)*b^(1/4) + 2*(-(b*x - 1)/(c + x))^(1/4))/b^(1/4))/(a^2*b^2) + 2*sqrt(2)*(a*b^(11/4)*c^2 + 2*(2*b^2 + b)*a*
b^(3/4)*c + 4*b^(11/4)*c + a*(4*b + 1)*b^(3/4) + 4*b^(7/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(-(b*x -
1)/(c + x))^(1/4))/b^(1/4))/(a^2*b^2) - sqrt(2)*(a*b^(11/4)*c^2 + 2*(2*b^2 + b)*a*b^(3/4)*c + 4*b^(11/4)*c + a
*(4*b + 1)*b^(3/4) + 4*b^(7/4))*log(sqrt(2)*b^(1/4)*(-(b*x - 1)/(c + x))^(1/4) + sqrt(b) + sqrt(-(b*x - 1)/(c
+ x)))/(a^2*b^2) + sqrt(2)*(a*b^(11/4)*c^2 + 2*(2*b^2 + b)*a*b^(3/4)*c + 4*b^(11/4)*c + a*(4*b + 1)*b^(3/4) +
4*b^(7/4))*log(-sqrt(2)*b^(1/4)*(-(b*x - 1)/(c + x))^(1/4) + sqrt(b) + sqrt(-(b*x - 1)/(c + x)))/(a^2*b^2) + 8
*(b^2*c^2*(-(b*x - 1)/(c + x))^(3/4) + 2*b*c*(-(b*x - 1)/(c + x))^(3/4) + (-(b*x - 1)/(c + x))^(3/4))/(a*(b -
(b*x - 1)/(c + x))*b))*(b*c/(b*c + 1)^2 + 1/(b*c + 1)^2)

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Mupad [B]
time = 44.07, size = 2500, normalized size = 6.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((((256*a*(a - b)*(b*c + 1)^7*(10*a^3*b - 20*a*b^3 - 288*a*b^4 + 12*a^4*b + a^4 - 80*b^4 + 25*a^2*b^2 + 12*
a^2*b^3 + 72*a^3*b^2 - 336*a^2*b^4 + 96*a^3*b^3 + 48*a^4*b^2 - 128*a^3*b^4 + 64*a^4*b^3 - 22*a^2*b^3*c + 8*a^3
*b^2*c - 180*a^2*b^4*c - 24*a^3*b^3*c + 12*a^4*b^2*c - 192*a^3*b^4*c - 64*a^4*b^4*c + a^2*b^4*c^2 - 2*a^3*b^3*
c^2 + a^4*b^2*c^2 - 52*a*b^4*c + 2*a^4*b*c))/((-b)^(51/4)*(a*c + 1)^9) + (1024*a^2*(a - b)*(b*c + 1)^6*(-(b*x
- 1)/(c + x))^(1/4)*(((a*c + 1)*(a + 1)^4)/(a^8*(a - b)))^(1/4)*(6*a^3*b - 24*a*b^3 + 64*a*b^4 + 8*a^4*b + a^4
+ 32*b^4 + a^2*b^2 - 56*a^2*b^3 + 16*a^3*b^2 + 32*a^2*b^4 - 32*a^3*b^3 + 16*a^4*b^2 - 14*a^2*b^3*c + 4*a^3*b^
2*c + 72*a^2*b^4*c - 16*a^3*b^3*c + 8*a^4*b^2*c + 32*a^3*b^4*c + 17*a^2*b^4*c^2 - 2*a^3*b^3*c^2 + a^4*b^2*c^2
+ 32*a^3*b^4*c^2 + 16*a^4*b^4*c^2 + 40*a*b^4*c + 2*a^4*b*c))/((-b)^(47/4)*(a*c + 1)^9))*(((a*c + 1)*(a + 1)^4)
/(a^8*(a - b)))^(3/4) - (64*(a - b)*(b*c + 1)^6*(-(b*x - 1)/(c + x))^(1/4)*(a + 1)^2*(a + 4*b + 4*a*b + a*b*c)
^2*(64*a*b^2 - 8*a*b - 24*a^2*b - 16*a^3*b + a^3*c + a^2 + 32*b^2 + 32*a^2*b^2 + 40*a^2*b^2*c + 2*a^3*b*c^2 +
16*a^3*b^2*c + 9*a^2*b^2*c^2 + 8*a^3*b^2*c^2 + a^3*b^2*c^3 + 24*a*b^2*c + 10*a^2*b*c + 8*a^3*b*c))/(a^6*(-b)^(
51/4)*(a*c + 1)^8))*(((a*c + 1)*(a + 1)^4)/(a^8*(a - b)))^(1/4) - (64*(a - b)*(b*c + 1)^7*(a + 1)^3*(4*a + a*c
+ 5)*(a + 4*b + 4*a*b + a*b*c)^3)/(a^7*(-b)^(51/4)*(a*c + 1)^8))*(-(a^2*(4*b^3*c + 6*b^3) + a^3*(6*b^3*c + 4*
b^3) + b^3 + a*(b^3*c + 4*b^3) + a^4*(4*b^3*c + b^3) + a^5*b^3*c)/(a^8*b^4 - a^9*b^3))^(1/4) - log((((256*a*(a
- b)*(b*c + 1)^7*(10*a^3*b - 20*a*b^3 - 288*a*b^4 + 12*a^4*b + a^4 - 80*b^4 + 25*a^2*b^2 + 12*a^2*b^3 + 72*a^
3*b^2 - 336*a^2*b^4 + 96*a^3*b^3 + 48*a^4*b^2 - 128*a^3*b^4 + 64*a^4*b^3 - 22*a^2*b^3*c + 8*a^3*b^2*c - 180*a^
2*b^4*c - 24*a^3*b^3*c + 12*a^4*b^2*c - 192*a^3*b^4*c - 64*a^4*b^4*c + a^2*b^4*c^2 - 2*a^3*b^3*c^2 + a^4*b^2*c
^2 - 52*a*b^4*c + 2*a^4*b*c))/((-b)^(51/4)*(a*c + 1)^9) - (1024*a^2*(a - b)*(b*c + 1)^6*(-(b*x - 1)/(c + x))^(
1/4)*(((a*c + 1)*(a + 1)^4)/(a^8*(a - b)))^(1/4)*(6*a^3*b - 24*a*b^3 + 64*a*b^4 + 8*a^4*b + a^4 + 32*b^4 + a^2
*b^2 - 56*a^2*b^3 + 16*a^3*b^2 + 32*a^2*b^4 - 32*a^3*b^3 + 16*a^4*b^2 - 14*a^2*b^3*c + 4*a^3*b^2*c + 72*a^2*b^
4*c - 16*a^3*b^3*c + 8*a^4*b^2*c + 32*a^3*b^4*c + 17*a^2*b^4*c^2 - 2*a^3*b^3*c^2 + a^4*b^2*c^2 + 32*a^3*b^4*c^
2 + 16*a^4*b^4*c^2 + 40*a*b^4*c + 2*a^4*b*c))/((-b)^(47/4)*(a*c + 1)^9))*(((a*c + 1)*(a + 1)^4)/(a^8*(a - b)))
^(3/4) + (64*(a - b)*(b*c + 1)^6*(-(b*x - 1)/(c + x))^(1/4)*(a + 1)^2*(a + 4*b + 4*a*b + a*b*c)^2*(64*a*b^2 -
8*a*b - 24*a^2*b - 16*a^3*b + a^3*c + a^2 + 32*b^2 + 32*a^2*b^2 + 40*a^2*b^2*c + 2*a^3*b*c^2 + 16*a^3*b^2*c +
9*a^2*b^2*c^2 + 8*a^3*b^2*c^2 + a^3*b^2*c^3 + 24*a*b^2*c + 10*a^2*b*c + 8*a^3*b*c))/(a^6*(-b)^(51/4)*(a*c + 1)
^8))*(((a*c + 1)*(a + 1)^4)/(a^8*(a - b)))^(1/4) - (64*(a - b)*(b*c + 1)^7*(a + 1)^3*(4*a + a*c + 5)*(a + 4*b
+ 4*a*b + a*b*c)^3)/(a^7*(-b)^(51/4)*(a*c + 1)^8))*(-(4*a*b^3 + b^3 + 6*a^2*b^3 + 4*a^3*b^3 + a^4*b^3 + 4*a^2*
b^3*c + 6*a^3*b^3*c + 4*a^4*b^3*c + a^5*b^3*c + a*b^3*c)/(a^8*b^4 - a^9*b^3))^(1/4) + 2*atan(((-(4*a*b^3 + b^3
+ 6*a^2*b^3 + 4*a^3*b^3 + a^4*b^3 + 4*a^2*b^3*c + 6*a^3*b^3*c + 4*a^4*b^3*c + a^5*b^3*c + a*b^3*c)/(a^8*b^4 -
a^9*b^3))^(1/4)*((-(4*a*b^3 + b^3 + 6*a^2*b^3 + 4*a^3*b^3 + a^4*b^3 + 4*a^2*b^3*c + 6*a^3*b^3*c + 4*a^4*b^3*c
+ a^5*b^3*c + a*b^3*c)/(a^8*b^4 - a^9*b^3))^(3/4)*((64*(36*a^12*(-b)^(25/4) - 4*a^13*(-b)^(21/4) + 48*a^13*(-
b)^(25/4) - 60*a^11*(-b)^(29/4) - 240*a^12*(-b)^(29/4) - 192*a^13*(-b)^(29/4) - 180*a^10*(-b)^(33/4) - 240*a^1
1*(-b)^(33/4) + 192*a^12*(-b)^(33/4) + 240*a^9*(-b)^(37/4) + 256*a^13*(-b)^(33/4) + 1200*a^10*(-b)^(37/4) + 17
28*a^11*(-b)^(37/4) + 320*a^8*(-b)^(41/4) + 768*a^12*(-b)^(37/4) + 1152*a^9*(-b)^(41/4) + 1344*a^10*(-b)^(41/4
) + 512*a^11*(-b)^(41/4) - 144*a^13*(-b)^(29/4)*c^2 + 912*a^12*(-b)^(33/4)*c^2 + 1344*a^13*(-b)^(33/4)*c^2 + 3
36*a^13*(-b)^(33/4)*c^3 - 432*a^11*(-b)^(37/4)*c^2 - 4032*a^12*(-b)^(37/4)*c^2 - 1680*a^12*(-b)^(37/4)*c^3 - 4
032*a^13*(-b)^(37/4)*c^2 - 4624*a^10*(-b)^(41/4)*c^2 - 2688*a^13*(-b)^(37/4)*c^3 - 9408*a^11*(-b)^(41/4)*c^2 -
504*a^13*(-b)^(37/4)*c^4 - 336*a^11*(-b)^(41/4)*c^3 - 1344*a^12*(-b)^(41/4)*c^2 + 3584*a^9*(-b)^(45/4)*c^2 +
5376*a^12*(-b)^(41/4)*c^3 + 3584*a^13*(-b)^(41/4)*c^2 + 20160*a^10*(-b)^(45/4)*c^2 + 1848*a^12*(-b)^(41/4)*c^4
+ 6720*a^13*(-b)^(41/4)*c^3 + 8848*a^10*(-b)^(45/4)*c^3 + 30912*a^11*(-b)^(45/4)*c^2 + 3360*a^13*(-b)^(41/4)*
c^4 + 6720*a^8*(-b)^(49/4)*c^2 + 21504*a^11*(-b)^(45/4)*c^3 + 14336*a^12*(-b)^(45/4)*c^2 + 504*a^13*(-b)^(41/4
)*c^5 + 24192*a^9*(-b)^(49/4)*c^2 + 1848*a^11*(-b)^(45/4)*c^4 + 9408*a^12*(-b)^(45/4)*c^3 - 4032*a^9*(-b)^(49/
4)*c^3 + 28224*a^10*(-b)^(49/4)*c^2 - 3360*a^12*(-b)^(45/4)*c^4 - 3584*a^13*(-b)^(45/4)*c^3 - 26880*a^10*(-b)^
(49/4)*c^3 + 10752*a^11*(-b)^(49/4)*c^2 - 1176*a^12*(-b)^(45/4)*c^5 - 6720*a^13*(-b)^(45/4)*c^4 - 10584*a^10*(
-b)^(49/4)*c^4 - 44352*a^11*(-b)^(49/4)*c^3 - 2688*a^13*(-b)^(45/4)*c^5 - 11200*a^8*(-b)^(53/4)*c^3 - 30240*a^
11*(-b)^(49/4)*c^4 - 21504*a^12*(-b)^(49/4)*c^3...

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