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

Optimal. Leaf size=195 \[ \frac {1463 b^6 d \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 d \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{32768 a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (-262144 a^7 c x^2+65536 a^6 b c x+327680 a^5 b^2 c+122880 a^5 b^2 d x^8+6144 a^4 b^3 d x^7-7296 a^3 b^4 d x^6+9120 a^2 b^5 d x^5-12540 a b^6 d x^4+21945 b^7 d x^3\right )}{737280 a^5 b^2 x^3} \]

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Rubi [A]  time = 0.75, antiderivative size = 341, normalized size of antiderivative = 1.75, number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2052, 2016, 2014, 2021, 2024, 2032, 63, 331, 298, 203, 206} \begin {gather*} \frac {1463 b^6 d 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 d 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 d \sqrt [4]{a x^4+b x^3}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{a x^4+b x^3}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{a x^4+b x^3}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{a x^4+b x^3}}{1920 a^2}-\frac {16 a c \left (a x^4+b x^3\right )^{5/4}}{45 b^2 x^5}+\frac {4 c \left (a x^4+b x^3\right )^{5/4}}{9 b x^6}+\frac {b d x^4 \sqrt [4]{a x^4+b x^3}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{a x^4+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1463*b^5*d*(b*x^3 + a*x^4)^(1/4))/(49152*a^5) - (209*b^4*d*x*(b*x^3 + a*x^4)^(1/4))/(12288*a^4) + (19*b^3*d*x
^2*(b*x^3 + a*x^4)^(1/4))/(1536*a^3) - (19*b^2*d*x^3*(b*x^3 + a*x^4)^(1/4))/(1920*a^2) + (b*d*x^4*(b*x^3 + a*x
^4)^(1/4))/(120*a) + (d*x^5*(b*x^3 + a*x^4)^(1/4))/6 + (4*c*(b*x^3 + a*x^4)^(5/4))/(9*b*x^6) - (16*a*c*(b*x^3
+ a*x^4)^(5/4))/(45*b^2*x^5) + (1463*b^6*d*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*d*x^(9/4)*(b + a*x)^(3/4)*ArcTanh[(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 (-c+d x^8\right )}{x^4} \, dx &=\int \left (-\frac {c \sqrt [4]{b x^3+a x^4}}{x^4}+d x^4 \sqrt [4]{b x^3+a x^4}\right ) \, dx\\ &=-\left (c \int \frac {\sqrt [4]{b x^3+a x^4}}{x^4} \, dx\right )+d \int x^4 \sqrt [4]{b x^3+a x^4} \, dx\\ &=\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}+\frac {(4 a c) \int \frac {\sqrt [4]{b x^3+a x^4}}{x^3} \, dx}{9 b}+\frac {1}{24} (b d) \int \frac {x^7}{\left (b x^3+a x^4\right )^{3/4}} \, dx\\ &=\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (19 b^2 d\right ) \int \frac {x^6}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{480 a}\\ &=-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {\left (19 b^3 d\right ) \int \frac {x^5}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{512 a^2}\\ &=\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (209 b^4 d\right ) \int \frac {x^4}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{6144 a^3}\\ &=-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {\left (1463 b^5 d\right ) \int \frac {x^3}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{49152 a^4}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d\right ) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{65536 a^5}\\ &=\frac {1463 b^5 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d 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 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d 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 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d 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 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}-\frac {\left (1463 b^6 d 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 d 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 d \sqrt [4]{b x^3+a x^4}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{b x^3+a x^4}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{b x^3+a x^4}}{1920 a^2}+\frac {b d x^4 \sqrt [4]{b x^3+a x^4}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{b x^3+a x^4}+\frac {4 c \left (b x^3+a x^4\right )^{5/4}}{9 b x^6}-\frac {16 a c \left (b x^3+a x^4\right )^{5/4}}{45 b^2 x^5}+\frac {1463 b^6 d 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 d 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 = 0.36, size = 323, normalized size = 1.66 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \left (-4 a^{10} c x^2 \sqrt [4]{\frac {a x}{b}+1}+a^9 b c x \sqrt [4]{\frac {a x}{b}+1}+5 a^8 b^2 c \sqrt [4]{\frac {a x}{b}+1}+44 a^2 b^8 d x^2 \sqrt [4]{\frac {a x}{b}+1}-5 b^{10} d \, _2F_1\left (-\frac {33}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+40 b^{10} d \, _2F_1\left (-\frac {29}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )-140 b^{10} d \, _2F_1\left (-\frac {25}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+280 b^{10} d \, _2F_1\left (-\frac {21}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )-350 b^{10} d \, _2F_1\left (-\frac {17}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+280 b^{10} d \, _2F_1\left (-\frac {13}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )-140 b^{10} d \, _2F_1\left (-\frac {9}{4},-\frac {9}{4};-\frac {5}{4};-\frac {a x}{b}\right )+35 b^{10} d \sqrt [4]{\frac {a x}{b}+1}+79 a b^9 d x \sqrt [4]{\frac {a x}{b}+1}\right )}{45 a^8 b^2 x^3 \sqrt [4]{\frac {a x}{b}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3*(b + a*x))^(1/4)*(5*a^8*b^2*c*(1 + (a*x)/b)^(1/4) + 35*b^10*d*(1 + (a*x)/b)^(1/4) + a^9*b*c*x*(1 + (a*
x)/b)^(1/4) + 79*a*b^9*d*x*(1 + (a*x)/b)^(1/4) - 4*a^10*c*x^2*(1 + (a*x)/b)^(1/4) + 44*a^2*b^8*d*x^2*(1 + (a*x
)/b)^(1/4) - 5*b^10*d*Hypergeometric2F1[-33/4, -9/4, -5/4, -((a*x)/b)] + 40*b^10*d*Hypergeometric2F1[-29/4, -9
/4, -5/4, -((a*x)/b)] - 140*b^10*d*Hypergeometric2F1[-25/4, -9/4, -5/4, -((a*x)/b)] + 280*b^10*d*Hypergeometri
c2F1[-21/4, -9/4, -5/4, -((a*x)/b)] - 350*b^10*d*Hypergeometric2F1[-17/4, -9/4, -5/4, -((a*x)/b)] + 280*b^10*d
*Hypergeometric2F1[-13/4, -9/4, -5/4, -((a*x)/b)] - 140*b^10*d*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 = 1.97, size = 195, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{b x^3+a x^4} \left (327680 a^5 b^2 c+65536 a^6 b c x-262144 a^7 c x^2+21945 b^7 d x^3-12540 a b^6 d x^4+9120 a^2 b^5 d x^5-7296 a^3 b^4 d x^6+6144 a^4 b^3 d x^7+122880 a^5 b^2 d x^8\right )}{737280 a^5 b^2 x^3}+\frac {1463 b^6 d \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 d \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)*(-c + d*x^8))/x^4,x]

[Out]

((b*x^3 + a*x^4)^(1/4)*(327680*a^5*b^2*c + 65536*a^6*b*c*x - 262144*a^7*c*x^2 + 21945*b^7*d*x^3 - 12540*a*b^6*
d*x^4 + 9120*a^2*b^5*d*x^5 - 7296*a^3*b^4*d*x^6 + 6144*a^4*b^3*d*x^7 + 122880*a^5*b^2*d*x^8))/(737280*a^5*b^2*
x^3) + (1463*b^6*d*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(32768*a^(23/4)) - (1463*b^6*d*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 = 385, normalized size = 1.97 \begin {gather*} \frac {263340 \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \arctan \left (-\frac {\left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {3}{4}} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{17} b^{6} d - \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {3}{4}} a^{17} x \sqrt {\frac {\sqrt {a x^{4} + b x^{3}} b^{12} d^{2} + \sqrt {\frac {b^{24} d^{4}}{a^{23}}} a^{12} x^{2}}{x^{2}}}}{b^{24} d^{4} x}\right ) - 65835 \, \left (\frac {b^{24} d^{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} d + \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 65835 \, \left (\frac {b^{24} d^{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} d - \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 4 \, {\left (122880 \, a^{5} b^{2} d x^{8} + 6144 \, a^{4} b^{3} d x^{7} - 7296 \, a^{3} b^{4} d x^{6} + 9120 \, a^{2} b^{5} d x^{5} - 12540 \, a b^{6} d x^{4} + 21945 \, b^{7} d x^{3} - 262144 \, a^{7} c x^{2} + 65536 \, a^{6} b c x + 327680 \, a^{5} b^{2} c\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)*(d*x^8-c)/x^4,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.86, size = 359, normalized size = 1.84 \begin {gather*} \frac {\frac {131670 \, \sqrt {2} b^{7} d \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} d \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} d \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} d \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} d - 40755 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} a b^{7} d + 92910 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a^{2} b^{7} d - 109782 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{3} b^{7} d + 69327 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{4} b^{7} d + 21945 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{5} b^{7} d\right )} x^{6}}{a^{5} b^{6}} + \frac {524288 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{8} c - 9 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{8} c\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)*(d*x^8-c)/x^4,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x^{8} - c\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)*(d*x^8-c)/x^4,x, algorithm="maxima")

[Out]

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

[Out]

int(-((c - d*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 + d x^{8}\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)*(d*x**8-c)/x**4,x)

[Out]

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

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