3.25.45 \(\int \frac {\sqrt [4]{-b x^3+a x^4} (-d+c x^8)}{x^4} \, dx\)

Optimal. Leaf size=198 \[ \frac {1463 b^6 c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{32768 a^{23/4}}+\frac {\sqrt [4]{a x^4-b x^3} \left (-262144 a^7 d x^2-65536 a^6 b d x+122880 a^5 b^2 c x^8+327680 a^5 b^2 d-6144 a^4 b^3 c x^7-7296 a^3 b^4 c x^6-9120 a^2 b^5 c x^5-12540 a b^6 c x^4-21945 b^7 c x^3\right )}{737280 a^5 b^2 x^3} \]

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Rubi [A]  time = 0.80, antiderivative size = 359, normalized size of antiderivative = 1.81, number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2052, 2016, 2014, 2021, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {1463 b^6 c x^{9/4} (a x-b)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{32768 a^{23/4} \left (a x^4-b x^3\right )^{3/4}}-\frac {1463 b^6 c x^{9/4} (a x-b)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{32768 a^{23/4} \left (a x^4-b x^3\right )^{3/4}}-\frac {1463 b^5 c \sqrt [4]{a x^4-b x^3}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{a x^4-b x^3}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{a x^4-b x^3}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{a x^4-b x^3}}{1920 a^2}-\frac {16 a d \left (a x^4-b x^3\right )^{5/4}}{45 b^2 x^5}-\frac {b c x^4 \sqrt [4]{a x^4-b x^3}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{a x^4-b x^3}-\frac {4 d \left (a x^4-b x^3\right )^{5/4}}{9 b x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^8))/x^4,x]

[Out]

(-1463*b^5*c*(-(b*x^3) + a*x^4)^(1/4))/(49152*a^5) - (209*b^4*c*x*(-(b*x^3) + a*x^4)^(1/4))/(12288*a^4) - (19*
b^3*c*x^2*(-(b*x^3) + a*x^4)^(1/4))/(1536*a^3) - (19*b^2*c*x^3*(-(b*x^3) + a*x^4)^(1/4))/(1920*a^2) - (b*c*x^4
*(-(b*x^3) + a*x^4)^(1/4))/(120*a) + (c*x^5*(-(b*x^3) + a*x^4)^(1/4))/6 - (4*d*(-(b*x^3) + a*x^4)^(5/4))/(9*b*
x^6) - (16*a*d*(-(b*x^3) + a*x^4)^(5/4))/(45*b^2*x^5) + (1463*b^6*c*x^(9/4)*(-b + a*x)^(3/4)*ArcTan[(a^(1/4)*x
^(1/4))/(-b + a*x)^(1/4)])/(32768*a^(23/4)*(-(b*x^3) + a*x^4)^(3/4)) - (1463*b^6*c*x^(9/4)*(-b + a*x)^(3/4)*Ar
cTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(32768*a^(23/4)*(-(b*x^3) + a*x^4)^(3/4))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rule 206

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

Rule 298

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 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
 1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
 0])

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
 1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 2052

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^8\right )}{x^4} \, dx &=\int \left (-\frac {d \sqrt [4]{-b x^3+a x^4}}{x^4}+c x^4 \sqrt [4]{-b x^3+a x^4}\right ) \, dx\\ &=c \int x^4 \sqrt [4]{-b x^3+a x^4} \, dx-d \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^4} \, dx\\ &=\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {1}{24} (b c) \int \frac {x^7}{\left (-b x^3+a x^4\right )^{3/4}} \, dx-\frac {(4 a d) \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^3} \, dx}{9 b}\\ &=-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (19 b^2 c\right ) \int \frac {x^6}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{480 a}\\ &=-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (19 b^3 c\right ) \int \frac {x^5}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{512 a^2}\\ &=-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (209 b^4 c\right ) \int \frac {x^4}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{6144 a^3}\\ &=-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^5 c\right ) \int \frac {x^3}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{49152 a^4}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c\right ) \int \frac {x^2}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{65536 a^5}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c x^{9/4} (-b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{65536 a^5 \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c x^{9/4} (-b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{16384 a^5 \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c x^{9/4} (-b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{16384 a^5 \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 c x^{9/4} (-b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{32768 a^{11/2} \left (-b x^3+a x^4\right )^{3/4}}+\frac {\left (1463 b^6 c x^{9/4} (-b+a x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{32768 a^{11/2} \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {1463 b^5 c \sqrt [4]{-b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 c x \sqrt [4]{-b x^3+a x^4}}{12288 a^4}-\frac {19 b^3 c x^2 \sqrt [4]{-b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 c x^3 \sqrt [4]{-b x^3+a x^4}}{1920 a^2}-\frac {b c x^4 \sqrt [4]{-b x^3+a x^4}}{120 a}+\frac {1}{6} c x^5 \sqrt [4]{-b x^3+a x^4}-\frac {4 d \left (-b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a d \left (-b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {1463 b^6 c x^{9/4} (-b+a x)^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{32768 a^{23/4} \left (-b x^3+a x^4\right )^{3/4}}-\frac {1463 b^6 c x^{9/4} (-b+a x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{32768 a^{23/4} \left (-b x^3+a x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.39, size = 325, normalized size = 1.64 \begin {gather*} -\frac {4 \sqrt [4]{x^3 (a x-b)} \left (4 a^{10} d x^2 \sqrt [4]{1-\frac {a x}{b}}+a^9 b d x \sqrt [4]{1-\frac {a x}{b}}-5 a^8 b^2 d \sqrt [4]{1-\frac {a x}{b}}-44 a^2 b^8 c x^2 \sqrt [4]{1-\frac {a x}{b}}+5 b^{10} c \, _2F_1\left (-\frac {33}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-40 b^{10} c \, _2F_1\left (-\frac {29}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )+140 b^{10} c \, _2F_1\left (-\frac {25}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-280 b^{10} c \, _2F_1\left (-\frac {21}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )+350 b^{10} c \, _2F_1\left (-\frac {17}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-280 b^{10} c \, _2F_1\left (-\frac {13}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )+140 b^{10} c \, _2F_1\left (-\frac {9}{4},-\frac {9}{4};-\frac {5}{4};\frac {a x}{b}\right )-35 b^{10} c \sqrt [4]{1-\frac {a x}{b}}+79 a b^9 c x \sqrt [4]{1-\frac {a x}{b}}\right )}{45 a^8 b^2 x^3 \sqrt [4]{1-\frac {a x}{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^8))/x^4,x]

[Out]

(-4*(x^3*(-b + a*x))^(1/4)*(-35*b^10*c*(1 - (a*x)/b)^(1/4) - 5*a^8*b^2*d*(1 - (a*x)/b)^(1/4) + 79*a*b^9*c*x*(1
 - (a*x)/b)^(1/4) + a^9*b*d*x*(1 - (a*x)/b)^(1/4) - 44*a^2*b^8*c*x^2*(1 - (a*x)/b)^(1/4) + 4*a^10*d*x^2*(1 - (
a*x)/b)^(1/4) + 5*b^10*c*Hypergeometric2F1[-33/4, -9/4, -5/4, (a*x)/b] - 40*b^10*c*Hypergeometric2F1[-29/4, -9
/4, -5/4, (a*x)/b] + 140*b^10*c*Hypergeometric2F1[-25/4, -9/4, -5/4, (a*x)/b] - 280*b^10*c*Hypergeometric2F1[-
21/4, -9/4, -5/4, (a*x)/b] + 350*b^10*c*Hypergeometric2F1[-17/4, -9/4, -5/4, (a*x)/b] - 280*b^10*c*Hypergeomet
ric2F1[-13/4, -9/4, -5/4, (a*x)/b] + 140*b^10*c*Hypergeometric2F1[-9/4, -9/4, -5/4, (a*x)/b]))/(45*a^8*b^2*x^3
*(1 - (a*x)/b)^(1/4))

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IntegrateAlgebraic [A]  time = 1.39, size = 198, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{-b x^3+a x^4} \left (327680 a^5 b^2 d-65536 a^6 b d x-262144 a^7 d x^2-21945 b^7 c x^3-12540 a b^6 c x^4-9120 a^2 b^5 c x^5-7296 a^3 b^4 c x^6-6144 a^4 b^3 c x^7+122880 a^5 b^2 c x^8\right )}{737280 a^5 b^2 x^3}+\frac {1463 b^6 c \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 c \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{32768 a^{23/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^8))/x^4,x]

[Out]

((-(b*x^3) + a*x^4)^(1/4)*(327680*a^5*b^2*d - 65536*a^6*b*d*x - 262144*a^7*d*x^2 - 21945*b^7*c*x^3 - 12540*a*b
^6*c*x^4 - 9120*a^2*b^5*c*x^5 - 7296*a^3*b^4*c*x^6 - 6144*a^4*b^3*c*x^7 + 122880*a^5*b^2*c*x^8))/(737280*a^5*b
^2*x^3) + (1463*b^6*c*ArcTan[(a^(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(32768*a^(23/4)) - (1463*b^6*c*ArcTanh[(a^
(1/4)*x)/(-(b*x^3) + a*x^4)^(1/4)])/(32768*a^(23/4))

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fricas [B]  time = 0.52, size = 390, normalized size = 1.97 \begin {gather*} \frac {263340 \, \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \arctan \left (-\frac {\left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {3}{4}} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} a^{17} b^{6} c - \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {3}{4}} a^{17} x \sqrt {\frac {\sqrt {a x^{4} - b x^{3}} b^{12} c^{2} + \sqrt {\frac {b^{24} c^{4}}{a^{23}}} a^{12} x^{2}}{x^{2}}}}{b^{24} c^{4} x}\right ) - 65835 \, \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} b^{6} c + \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 65835 \, \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} b^{6} c - \left (\frac {b^{24} c^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 4 \, {\left (122880 \, a^{5} b^{2} c x^{8} - 6144 \, a^{4} b^{3} c x^{7} - 7296 \, a^{3} b^{4} c x^{6} - 9120 \, a^{2} b^{5} c x^{5} - 12540 \, a b^{6} c x^{4} - 21945 \, b^{7} c x^{3} - 262144 \, a^{7} d x^{2} - 65536 \, a^{6} b d x + 327680 \, a^{5} b^{2} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{2949120 \, a^{5} b^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-b*x^3)^(1/4)*(c*x^8-d)/x^4,x, algorithm="fricas")

[Out]

1/2949120*(263340*(b^24*c^4/a^23)^(1/4)*a^5*b^2*x^3*arctan(-((b^24*c^4/a^23)^(3/4)*(a*x^4 - b*x^3)^(1/4)*a^17*
b^6*c - (b^24*c^4/a^23)^(3/4)*a^17*x*sqrt((sqrt(a*x^4 - b*x^3)*b^12*c^2 + sqrt(b^24*c^4/a^23)*a^12*x^2)/x^2))/
(b^24*c^4*x)) - 65835*(b^24*c^4/a^23)^(1/4)*a^5*b^2*x^3*log(1463*((a*x^4 - b*x^3)^(1/4)*b^6*c + (b^24*c^4/a^23
)^(1/4)*a^6*x)/x) + 65835*(b^24*c^4/a^23)^(1/4)*a^5*b^2*x^3*log(1463*((a*x^4 - b*x^3)^(1/4)*b^6*c - (b^24*c^4/
a^23)^(1/4)*a^6*x)/x) + 4*(122880*a^5*b^2*c*x^8 - 6144*a^4*b^3*c*x^7 - 7296*a^3*b^4*c*x^6 - 9120*a^2*b^5*c*x^5
 - 12540*a*b^6*c*x^4 - 21945*b^7*c*x^3 - 262144*a^7*d*x^2 - 65536*a^6*b*d*x + 327680*a^5*b^2*d)*(a*x^4 - b*x^3
)^(1/4))/(a^5*b^2*x^3)

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giac [B]  time = 0.67, size = 373, normalized size = 1.88 \begin {gather*} \frac {\frac {131670 \, \sqrt {2} b^{7} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {131670 \, \sqrt {2} b^{7} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {65835 \, \sqrt {2} b^{7} c \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {65835 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{7} c \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{a^{6}} + \frac {24 \, {\left (7315 \, {\left (a - \frac {b}{x}\right )}^{\frac {21}{4}} b^{7} c - 40755 \, {\left (a - \frac {b}{x}\right )}^{\frac {17}{4}} a b^{7} c + 92910 \, {\left (a - \frac {b}{x}\right )}^{\frac {13}{4}} a^{2} b^{7} c - 109782 \, {\left (a - \frac {b}{x}\right )}^{\frac {9}{4}} a^{3} b^{7} c + 69327 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{4}} a^{4} b^{7} c + 21945 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} a^{5} b^{7} c\right )} x^{6}}{a^{5} b^{6}} + \frac {524288 \, {\left (5 \, {\left (a - \frac {b}{x}\right )}^{\frac {9}{4}} b^{8} d - 9 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{4}} a b^{8} d\right )}}{b^{9}}}{5898240 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-b*x^3)^(1/4)*(c*x^8-d)/x^4,x, algorithm="giac")

[Out]

1/5898240*(131670*sqrt(2)*b^7*c*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x)^(1/4))/(-a)^(1/4))/((-a)^
(3/4)*a^5) + 131670*sqrt(2)*b^7*c*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a - b/x)^(1/4))/(-a)^(1/4))/((-
a)^(3/4)*a^5) + 65835*sqrt(2)*b^7*c*log(sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/((-a)^(
3/4)*a^5) + 65835*sqrt(2)*(-a)^(1/4)*b^7*c*log(-sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))
/a^6 + 24*(7315*(a - b/x)^(21/4)*b^7*c - 40755*(a - b/x)^(17/4)*a*b^7*c + 92910*(a - b/x)^(13/4)*a^2*b^7*c - 1
09782*(a - b/x)^(9/4)*a^3*b^7*c + 69327*(a - b/x)^(5/4)*a^4*b^7*c + 21945*(a - b/x)^(1/4)*a^5*b^7*c)*x^6/(a^5*
b^6) + 524288*(5*(a - b/x)^(9/4)*b^8*d - 9*(a - b/x)^(5/4)*a*b^8*d)/b^9)/b

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}} \left (c \,x^{8}-d \right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4-b*x^3)^(1/4)*(c*x^8-d)/x^4,x)

[Out]

int((a*x^4-b*x^3)^(1/4)*(c*x^8-d)/x^4,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{8} - d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-b*x^3)^(1/4)*(c*x^8-d)/x^4,x, algorithm="maxima")

[Out]

integrate((c*x^8 - d)*(a*x^4 - b*x^3)^(1/4)/x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (d-c\,x^8\right )\,{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((d - c*x^8)*(a*x^4 - b*x^3)^(1/4))/x^4,x)

[Out]

-int(((d - c*x^8)*(a*x^4 - b*x^3)^(1/4))/x^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x - b\right )} \left (c x^{8} - d\right )}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4-b*x**3)**(1/4)*(c*x**8-d)/x**4,x)

[Out]

Integral((x**3*(a*x - b))**(1/4)*(c*x**8 - d)/x**4, x)

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