3.22.81 \(\int \frac {\sqrt [4]{-b x^3+a x^4} (-d+c x^4)}{x^2} \, dx\) [2181]

Optimal. Leaf size=162 \[ \frac {\sqrt [4]{-b x^3+a x^4} \left (6144 a^3 d-77 b^3 c x-44 a b^2 c x^2-32 a^2 b c x^3+384 a^3 c x^4\right )}{1536 a^3 x}+\frac {\left (77 b^4 c+2048 a^4 d\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{1024 a^{15/4}}+\frac {\left (-77 b^4 c-2048 a^4 d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{1024 a^{15/4}} \]

[Out]

1/1536*(a*x^4-b*x^3)^(1/4)*(384*a^3*c*x^4-32*a^2*b*c*x^3-44*a*b^2*c*x^2-77*b^3*c*x+6144*a^3*d)/a^3/x+1/1024*(2
048*a^4*d+77*b^4*c)*arctan(a^(1/4)*x/(a*x^4-b*x^3)^(1/4))/a^(15/4)+1/1024*(-2048*a^4*d-77*b^4*c)*arctanh(a^(1/
4)*x/(a*x^4-b*x^3)^(1/4))/a^(15/4)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(392\) vs. \(2(162)=324\).
time = 0.46, antiderivative size = 392, normalized size of antiderivative = 2.42, number of steps used = 19, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2077, 2045, 2057, 65, 338, 304, 209, 212, 2046, 2049} \begin {gather*} \frac {77 b^4 c x^{9/4} (a x-b)^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{1024 a^{15/4} \left (a x^4-b x^3\right )^{3/4}}-\frac {77 b^4 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 )}{1024 a^{15/4} \left (a x^4-b x^3\right )^{3/4}}-\frac {77 b^3 c \sqrt [4]{a x^4-b x^3}}{1536 a^3}-\frac {11 b^2 c x \sqrt [4]{a x^4-b x^3}}{384 a^2}+\frac {2 \sqrt [4]{a} d x^{9/4} (a x-b)^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{\left (a x^4-b x^3\right )^{3/4}}+\frac {1}{4} c x^3 \sqrt [4]{a x^4-b x^3}-\frac {b c x^2 \sqrt [4]{a x^4-b x^3}}{48 a}+\frac {4 d \sqrt [4]{a x^4-b x^3}}{x}-\frac {2 \sqrt [4]{a} 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 )}{\left (a x^4-b x^3\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^4))/x^2,x]

[Out]

(-77*b^3*c*(-(b*x^3) + a*x^4)^(1/4))/(1536*a^3) + (4*d*(-(b*x^3) + a*x^4)^(1/4))/x - (11*b^2*c*x*(-(b*x^3) + a
*x^4)^(1/4))/(384*a^2) - (b*c*x^2*(-(b*x^3) + a*x^4)^(1/4))/(48*a) + (c*x^3*(-(b*x^3) + a*x^4)^(1/4))/4 + (77*
b^4*c*x^(9/4)*(-b + a*x)^(3/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(1024*a^(15/4)*(-(b*x^3) + a*x^4)^(
3/4)) + (2*a^(1/4)*d*x^(9/4)*(-b + a*x)^(3/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(-(b*x^3) + a*x^4)^(
3/4) - (77*b^4*c*x^(9/4)*(-b + a*x)^(3/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(1024*a^(15/4)*(-(b*x^3
) + a*x^4)^(3/4)) - (2*a^(1/4)*d*x^(9/4)*(-b + a*x)^(3/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(-(b*x^
3) + a*x^4)^(3/4)

Rule 65

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 209

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

Rule 212

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

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 338

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 2045

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 + j*p + 1))), x] - Dist[b*p*((n - j)/(c^n*(m + j*p + 1))), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2046

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 2049

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 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[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 2077

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^4\right )}{x^2} \, dx &=\int \left (-\frac {d \sqrt [4]{-b x^3+a x^4}}{x^2}+c x^2 \sqrt [4]{-b x^3+a x^4}\right ) \, dx\\ &=c \int x^2 \sqrt [4]{-b x^3+a x^4} \, dx-d \int \frac {\sqrt [4]{-b x^3+a x^4}}{x^2} \, dx\\ &=\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}-\frac {1}{16} (b c) \int \frac {x^5}{\left (-b x^3+a x^4\right )^{3/4}} \, dx-(a d) \int \frac {x^2}{\left (-b x^3+a x^4\right )^{3/4}} \, dx\\ &=\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}-\frac {\left (11 b^2 c\right ) \int \frac {x^4}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{192 a}-\frac {\left (a d x^{9/4} (-b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{\left (-b x^3+a x^4\right )^{3/4}}\\ &=\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}-\frac {\left (77 b^3 c\right ) \int \frac {x^3}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{1536 a^2}-\frac {\left (4 a d x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {77 b^3 c \sqrt [4]{-b x^3+a x^4}}{1536 a^3}+\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}-\frac {\left (77 b^4 c\right ) \int \frac {x^2}{\left (-b x^3+a x^4\right )^{3/4}} \, dx}{2048 a^3}-\frac {\left (4 a d x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{\left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {77 b^3 c \sqrt [4]{-b x^3+a x^4}}{1536 a^3}+\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}-\frac {\left (77 b^4 c x^{9/4} (-b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{2048 a^3 \left (-b x^3+a x^4\right )^{3/4}}-\frac {\left (2 \sqrt {a} d x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{\left (-b x^3+a x^4\right )^{3/4}}+\frac {\left (2 \sqrt {a} d x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{\left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {77 b^3 c \sqrt [4]{-b x^3+a x^4}}{1536 a^3}+\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}+\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {\left (77 b^4 c x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 a^3 \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {77 b^3 c \sqrt [4]{-b x^3+a x^4}}{1536 a^3}+\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}+\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {\left (77 b^4 c x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{512 a^3 \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {77 b^3 c \sqrt [4]{-b x^3+a x^4}}{1536 a^3}+\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}+\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {\left (77 b^4 c x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{1024 a^{7/2} \left (-b x^3+a x^4\right )^{3/4}}+\frac {\left (77 b^4 c x^{9/4} (-b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{1024 a^{7/2} \left (-b x^3+a x^4\right )^{3/4}}\\ &=-\frac {77 b^3 c \sqrt [4]{-b x^3+a x^4}}{1536 a^3}+\frac {4 d \sqrt [4]{-b x^3+a x^4}}{x}-\frac {11 b^2 c x \sqrt [4]{-b x^3+a x^4}}{384 a^2}-\frac {b c x^2 \sqrt [4]{-b x^3+a x^4}}{48 a}+\frac {1}{4} c x^3 \sqrt [4]{-b x^3+a x^4}+\frac {77 b^4 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 )}{1024 a^{15/4} \left (-b x^3+a x^4\right )^{3/4}}+\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}-\frac {77 b^4 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 )}{1024 a^{15/4} \left (-b x^3+a x^4\right )^{3/4}}-\frac {2 \sqrt [4]{a} 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 )}{\left (-b x^3+a x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 185, normalized size = 1.14 \begin {gather*} \frac {x^2 (-b+a x)^{3/4} \left (2 a^{3/4} \sqrt [4]{-b+a x} \left (-77 b^3 c x-44 a b^2 c x^2-32 a^2 b c x^3+384 a^3 \left (16 d+c x^4\right )\right )+3 \left (77 b^4 c+2048 a^4 d\right ) \sqrt [4]{x} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-3 \left (77 b^4 c+2048 a^4 d\right ) \sqrt [4]{x} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )\right )}{3072 a^{15/4} \left (x^3 (-b+a x)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-(b*x^3) + a*x^4)^(1/4)*(-d + c*x^4))/x^2,x]

[Out]

(x^2*(-b + a*x)^(3/4)*(2*a^(3/4)*(-b + a*x)^(1/4)*(-77*b^3*c*x - 44*a*b^2*c*x^2 - 32*a^2*b*c*x^3 + 384*a^3*(16
*d + c*x^4)) + 3*(77*b^4*c + 2048*a^4*d)*x^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)] - 3*(77*b^4*c + 20
48*a^4*d)*x^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)]))/(3072*a^(15/4)*(x^3*(-b + a*x))^(3/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}} \left (c \,x^{4}-d \right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4-b*x^3)^(1/4)*(c*x^4-d)/x^2,x)

[Out]

int((a*x^4-b*x^3)^(1/4)*(c*x^4-d)/x^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4-b*x^3)^(1/4)*(c*x^4-d)/x^2,x, algorithm="maxima")

[Out]

integrate((a*x^4 - b*x^3)^(1/4)*(c*x^4 - d)/x^2, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (142) = 284\).
time = 0.35, size = 810, normalized size = 5.00 \begin {gather*} \frac {12 \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{11} x \sqrt {\frac {a^{8} x^{2} \sqrt {\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}} + {\left (5929 \, b^{8} c^{2} + 315392 \, a^{4} b^{4} c d + 4194304 \, a^{8} d^{2}\right )} \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {3}{4}} - {\left (77 \, a^{11} b^{4} c + 2048 \, a^{15} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {3}{4}}}{{\left (35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}\right )} x}\right ) - 3 \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} + {\left (77 \, b^{4} c + 2048 \, a^{4} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 3 \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{4} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} - {\left (77 \, b^{4} c + 2048 \, a^{4} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (384 \, a^{3} c x^{4} - 32 \, a^{2} b c x^{3} - 44 \, a b^{2} c x^{2} - 77 \, b^{3} c x + 6144 \, a^{3} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{6144 \, a^{3} 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^4-d)/x^2,x, algorithm="fricas")

[Out]

1/6144*(12*a^3*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 264569985433
6*a^12*b^4*c*d^3 + 17592186044416*a^16*d^4)/a^15)^(1/4)*arctan((a^11*x*sqrt((a^8*x^2*sqrt((35153041*b^16*c^4 +
3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16
*d^4)/a^15) + (5929*b^8*c^2 + 315392*a^4*b^4*c*d + 4194304*a^8*d^2)*sqrt(a*x^4 - b*x^3))/x^2)*((35153041*b^16*
c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416
*a^16*d^4)/a^15)^(3/4) - (77*a^11*b^4*c + 2048*a^15*d)*(a*x^4 - b*x^3)^(1/4)*((35153041*b^16*c^4 + 3739918336*
a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16*d^4)/a^15)^
(3/4))/((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4
*c*d^3 + 17592186044416*a^16*d^4)*x)) - 3*a^3*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496
*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16*d^4)/a^15)^(1/4)*log((a^4*x*((35153041*b
^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 1759218604
4416*a^16*d^4)/a^15)^(1/4) + (77*b^4*c + 2048*a^4*d)*(a*x^4 - b*x^3)^(1/4))/x) + 3*a^3*x*((35153041*b^16*c^4 +
3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16
*d^4)/a^15)^(1/4)*log(-(a^4*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 +
2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16*d^4)/a^15)^(1/4) - (77*b^4*c + 2048*a^4*d)*(a*x^4 - b*x^3)
^(1/4))/x) + 4*(384*a^3*c*x^4 - 32*a^2*b*c*x^3 - 44*a*b^2*c*x^2 - 77*b^3*c*x + 6144*a^3*d)*(a*x^4 - b*x^3)^(1/
4))/(a^3*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^{4} - d\right )}{x^{2}}\, 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**4-d)/x**2,x)

[Out]

Integral((x**3*(a*x - b))**(1/4)*(c*x**4 - d)/x**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (142) = 284\).
time = 0.42, size = 350, normalized size = 2.16 \begin {gather*} \frac {49152 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} b d + \frac {6 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \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^{3}} + \frac {6 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \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^{3}} + \frac {3 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \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^{3}} - \frac {3 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \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^{3}} + \frac {8 \, {\left (77 \, {\left (a - \frac {b}{x}\right )}^{\frac {13}{4}} b^{5} c - 275 \, {\left (a - \frac {b}{x}\right )}^{\frac {9}{4}} a b^{5} c + 351 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{4}} a^{2} b^{5} c + 231 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} a^{3} b^{5} c\right )} x^{4}}{a^{3} b^{4}}}{12288 \, 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^4-d)/x^2,x, algorithm="giac")

[Out]

1/12288*(49152*(a - b/x)^(1/4)*b*d + 6*sqrt(2)*(77*b^5*c + 2048*a^4*b*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4
) + 2*(a - b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^3) + 6*sqrt(2)*(77*b^5*c + 2048*a^4*b*d)*arctan(-1/2*sqrt(2)*
(sqrt(2)*(-a)^(1/4) - 2*(a - b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^3) + 3*sqrt(2)*(77*b^5*c + 2048*a^4*b*d)*lo
g(sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/((-a)^(3/4)*a^3) - 3*sqrt(2)*(77*b^5*c + 2048
*a^4*b*d)*log(-sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/((-a)^(3/4)*a^3) + 8*(77*(a - b/
x)^(13/4)*b^5*c - 275*(a - b/x)^(9/4)*a*b^5*c + 351*(a - b/x)^(5/4)*a^2*b^5*c + 231*(a - b/x)^(1/4)*a^3*b^5*c)
*x^4/(a^3*b^4))/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (d-c\,x^4\right )\,{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((d - c*x^4)*(a*x^4 - b*x^3)^(1/4))/x^2,x)

[Out]

-int(((d - c*x^4)*(a*x^4 - b*x^3)^(1/4))/x^2, x)

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