3.166 \(\int \frac{(c+d x^4)^5}{(a+b x^4)^2} \, dx\)

Optimal. Leaf size=407 \[ \frac{d^3 x^5 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{5 b^4}+\frac{d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}-\frac{(b c-a d)^4 (17 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (17 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}-\frac{(b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{d^4 x^9 (5 b c-2 a d)}{9 b^3}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^4\right )}+\frac{d^5 x^{13}}{13 b^2} \]

[Out]

(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2
*d^2)*x^5)/(5*b^4) + (d^4*(5*b*c - 2*a*d)*x^9)/(9*b^3) + (d^5*x^13)/(13*b^2) + ((b*c - a*d)^5*x)/(4*a*b^5*(a +
b*x^4)) - ((b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(21/4
)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(21/4)) - (
(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(
21/4)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a
^(7/4)*b^(21/4))

________________________________________________________________________________________

Rubi [A]  time = 0.395549, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {390, 385, 211, 1165, 628, 1162, 617, 204} \[ \frac{d^3 x^5 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{5 b^4}+\frac{d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}-\frac{(b c-a d)^4 (17 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (17 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}-\frac{(b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{d^4 x^9 (5 b c-2 a d)}{9 b^3}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^4\right )}+\frac{d^5 x^{13}}{13 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^4)^5/(a + b*x^4)^2,x]

[Out]

(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2
*d^2)*x^5)/(5*b^4) + (d^4*(5*b*c - 2*a*d)*x^9)/(9*b^3) + (d^5*x^13)/(13*b^2) + ((b*c - a*d)^5*x)/(4*a*b^5*(a +
b*x^4)) - ((b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(21/4
)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(21/4)) - (
(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(
21/4)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a
^(7/4)*b^(21/4))

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 211

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

Rule 1165

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 628

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 1162

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 617

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 204

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

Rubi steps

\begin{align*} \int \frac{\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx &=\int \left (\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{b^4}+\frac{d^4 (5 b c-2 a d) x^8}{b^3}+\frac{d^5 x^{12}}{b^2}+\frac{(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^4}{b^5 \left (a+b x^4\right )^2}\right ) \, dx\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac{d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}+\frac{\int \frac{(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^4}{\left (a+b x^4\right )^2} \, dx}{b^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac{d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^4\right )}+\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{1}{a+b x^4} \, dx}{4 a b^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac{d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^4\right )}+\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^5}+\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac{d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^4\right )}+\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{11/2}}+\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{11/2}}-\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt{2} a^{7/4} b^{21/4}}-\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt{2} a^{7/4} b^{21/4}}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac{d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^4\right )}-\frac{(b c-a d)^4 (3 b c+17 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (3 b c+17 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}+\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}-\frac{\left ((b c-a d)^4 (3 b c+17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac{d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac{d^5 x^{13}}{13 b^2}+\frac{(b c-a d)^5 x}{4 a b^5 \left (a+b x^4\right )}-\frac{(b c-a d)^4 (3 b c+17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (3 b c+17 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} b^{21/4}}-\frac{(b c-a d)^4 (3 b c+17 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}+\frac{(b c-a d)^4 (3 b c+17 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{16 \sqrt{2} a^{7/4} b^{21/4}}\\ \end{align*}

Mathematica [A]  time = 0.36496, size = 391, normalized size = 0.96 \[ \frac{3744 b^{5/4} d^3 x^5 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )+18720 \sqrt [4]{b} d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )-\frac{585 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{585 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}-\frac{1170 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{1170 \sqrt{2} (b c-a d)^4 (17 a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+2080 b^{9/4} d^4 x^9 (5 b c-2 a d)+\frac{4680 \sqrt [4]{b} x (b c-a d)^5}{a \left (a+b x^4\right )}+1440 b^{13/4} d^5 x^{13}}{18720 b^{21/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^4)^5/(a + b*x^4)^2,x]

[Out]

(18720*b^(1/4)*d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x + 3744*b^(5/4)*d^3*(10*b^2*c^2
- 10*a*b*c*d + 3*a^2*d^2)*x^5 + 2080*b^(9/4)*d^4*(5*b*c - 2*a*d)*x^9 + 1440*b^(13/4)*d^5*x^13 + (4680*b^(1/4)
*(b*c - a*d)^5*x)/(a*(a + b*x^4)) - (1170*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x
)/a^(1/4)])/a^(7/4) + (1170*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^
(7/4) - (585*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^
(7/4) + (585*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^
(7/4))/(18720*b^(21/4))

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 1118, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^4+c)^5/(b*x^4+a)^2,x)

[Out]

-20*d^3/b^3*a*c^2*x+3/16/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c^5+1/13*d^5*x^13/b^2+45/
8/b^3*a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^2*d^3+5/16/b/a*(1/b*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/b*a)^(1/4)*x-1)*c^4*d-65/32/b^4*a^2*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(
1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*c*d^4+45/16/b^3*a*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/
4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*c^2*d^3+5/32/b/a*(1/b*a)^(1/4)*2^(1/2
)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*c^4*d-65/16/b^4*
a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c*d^4+45/8/b^3*a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1
/2)/(1/b*a)^(1/4)*x+1)*c^2*d^3+5/16/b/a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c^4*d-65/16/b^
4*a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c*d^4+17/16/b^5*a^3*(1/b*a)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(1/b*a)^(1/4)*x+1)*d^5-25/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)*c^3*d^2+17/16
/b^5*a^3*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*d^5+5/4/b^4*a^3*x/(b*x^4+a)*c*d^4-5/2/b^3*a^2
*x/(b*x^4+a)*c^2*d^3+5/2/b^2*a*x/(b*x^4+a)*c^3*d^2-4*d^5/b^5*a^3*x+10*d^2/b^2*c^3*x-25/8/b^2*(1/b*a)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x-1)*c^3*d^2+17/32/b^5*a^3*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(
1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*d^5-25/16/b^2*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(
1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2)))*c^3*d^2-2*d^4/b^3*x^5*a*c-1
/4/b^5*a^4*x/(b*x^4+a)*d^5-5/4/b*x/(b*x^4+a)*c^4*d+3/16/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)
*x-1)*c^5+3/32/a^2*(1/b*a)^(1/4)*2^(1/2)*ln((x^2+(1/b*a)^(1/4)*x*2^(1/2)+(1/b*a)^(1/2))/(x^2-(1/b*a)^(1/4)*x*2
^(1/2)+(1/b*a)^(1/2)))*c^5+15*d^4/b^4*a^2*c*x+5/9*d^4/b^2*x^9*c+3/5*d^5/b^4*x^5*a^2+2*d^3/b^2*x^5*c^2+1/4/a*x/
(b*x^4+a)*c^5-2/9*d^5/b^3*x^9*a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^5/(b*x^4+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.34698, size = 8400, normalized size = 20.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^5/(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/9360*(720*a*b^4*d^5*x^17 + 80*(65*a*b^4*c*d^4 - 17*a^2*b^3*d^5)*x^13 + 208*(90*a*b^4*c^2*d^3 - 65*a^2*b^3*c*
d^4 + 17*a^3*b^2*d^5)*x^9 + 1872*(50*a*b^4*c^3*d^2 - 90*a^2*b^3*c^2*d^3 + 65*a^3*b^2*c*d^4 - 17*a^4*b*d^5)*x^5
+ 2340*(a*b^6*x^4 + a^2*b^5)*(-(81*b^20*c^20 + 540*a*b^19*c^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^
17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^1
3*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11
*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 2660
40144*a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18
- 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21))^(1/4)*arctan(-(a^5*b^16*x*(-(81*b^20*c^20 + 540*a*b^19
*c^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^
5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*
d^9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^
13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39
944900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21))^(3
/4) - a^5*b^16*sqrt((a^4*b^10*sqrt(-(81*b^20*c^20 + 540*a*b^19*c^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^1
7*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13
*c^13*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*
a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 -
266040144*a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*
d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21)) + (9*b^10*c^10 + 30*a*b^9*c^9*d - 275*a^2*b^8*c^8*
d^2 + 40*a^3*b^7*c^7*d^3 + 3010*a^4*b^6*c^6*d^4 - 9548*a^5*b^5*c^5*d^5 + 14770*a^6*b^4*c^4*d^6 - 13400*a^7*b^3
*c^3*d^7 + 7285*a^8*b^2*c^2*d^8 - 2210*a^9*b*c*d^9 + 289*a^10*d^10)*x^2)/(9*b^10*c^10 + 30*a*b^9*c^9*d - 275*a
^2*b^8*c^8*d^2 + 40*a^3*b^7*c^7*d^3 + 3010*a^4*b^6*c^6*d^4 - 9548*a^5*b^5*c^5*d^5 + 14770*a^6*b^4*c^4*d^6 - 13
400*a^7*b^3*c^3*d^7 + 7285*a^8*b^2*c^2*d^8 - 2210*a^9*b*c*d^9 + 289*a^10*d^10))*(-(81*b^20*c^20 + 540*a*b^19*c
^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^5
- 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^
9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^13
*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 3994
4900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21))^(3/4
))/(27*b^15*c^15 + 135*a*b^14*c^14*d - 1125*a^2*b^13*c^13*d^2 - 1945*a^3*b^12*c^12*d^3 + 25095*a^4*b^11*c^11*d
^4 - 42141*a^5*b^10*c^10*d^5 - 131945*a^6*b^9*c^9*d^6 + 774675*a^7*b^8*c^8*d^7 - 1837935*a^8*b^7*c^7*d^8 + 270
0885*a^9*b^6*c^6*d^9 - 2702799*a^10*b^5*c^5*d^10 + 1889685*a^11*b^4*c^4*d^11 - 914675*a^12*b^3*c^3*d^12 + 2935
05*a^13*b^2*c^2*d^13 - 56355*a^14*b*c*d^14 + 4913*a^15*d^15)) + 585*(a*b^6*x^4 + a^2*b^5)*(-(81*b^20*c^20 + 54
0*a*b^19*c^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15
*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^
11*c^11*d^9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 5482
31440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d
^16 - 39944900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b
^21))^(1/4)*log(a^2*b^5*(-(81*b^20*c^20 + 540*a*b^19*c^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3
+ 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7
- 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11*b^9*c
^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*
a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18 - 127
7380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21))^(1/4) + (3*b^5*c^5 + 5*a*b^4*c^4*d - 50*a^2*b^3*c^3*d^2 + 90
*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 17*a^5*d^5)*x) - 585*(a*b^6*x^4 + a^2*b^5)*(-(81*b^20*c^20 + 540*a*b^19*c^
19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^5 -
1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9
+ 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^13*
b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944
900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21))^(1/4)
*log(-a^2*b^5*(-(81*b^20*c^20 + 540*a*b^19*c^19*d - 4050*a^2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*
a^4*b^16*c^16*d^4 - 13264*a^5*b^15*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7 - 500110*a
^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9 + 174873556*a^10*b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 +
517559250*a^12*b^8*c^8*d^12 - 548231440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*a^15*b^5*c
^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944900*a^17*b^3*c^3*d^17 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*
b*c*d^19 + 83521*a^20*d^20)/(a^7*b^21))^(1/4) + (3*b^5*c^5 + 5*a*b^4*c^4*d - 50*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c
^2*d^3 - 65*a^4*b*c*d^4 + 17*a^5*d^5)*x) + 2340*(b^5*c^5 - 5*a*b^4*c^4*d + 50*a^2*b^3*c^3*d^2 - 90*a^3*b^2*c^2
*d^3 + 65*a^4*b*c*d^4 - 17*a^5*d^5)*x)/(a*b^6*x^4 + a^2*b^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**4+c)**5/(b*x**4+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.11543, size = 1077, normalized size = 2.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^5/(b*x^4+a)^2,x, algorithm="giac")

[Out]

1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4*c^4*d - 50*(a*b^3)^(1/4)*a^2*b^3*c^3*d^2 + 90*(a
*b^3)^(1/4)*a^3*b^2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 + 17*(a*b^3)^(1/4)*a^5*d^5)*arctan(1/2*sqrt(2)*(2*x
+ sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^6) + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4
*c^4*d - 50*(a*b^3)^(1/4)*a^2*b^3*c^3*d^2 + 90*(a*b^3)^(1/4)*a^3*b^2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 +
17*(a*b^3)^(1/4)*a^5*d^5)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^6) + 1/32*sqrt(2)
*(3*(a*b^3)^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4*c^4*d - 50*(a*b^3)^(1/4)*a^2*b^3*c^3*d^2 + 90*(a*b^3)^(1/4)*
a^3*b^2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 + 17*(a*b^3)^(1/4)*a^5*d^5)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + s
qrt(a/b))/(a^2*b^6) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4*c^4*d - 50*(a*b^3)^(1/4)*a
^2*b^3*c^3*d^2 + 90*(a*b^3)^(1/4)*a^3*b^2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 + 17*(a*b^3)^(1/4)*a^5*d^5)*l
og(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^6) + 1/4*(b^5*c^5*x - 5*a*b^4*c^4*d*x + 10*a^2*b^3*c^3*d^2*
x - 10*a^3*b^2*c^2*d^3*x + 5*a^4*b*c*d^4*x - a^5*d^5*x)/((b*x^4 + a)*a*b^5) + 1/585*(45*b^24*d^5*x^13 + 325*b^
24*c*d^4*x^9 - 130*a*b^23*d^5*x^9 + 1170*b^24*c^2*d^3*x^5 - 1170*a*b^23*c*d^4*x^5 + 351*a^2*b^22*d^5*x^5 + 585
0*b^24*c^3*d^2*x - 11700*a*b^23*c^2*d^3*x + 8775*a^2*b^22*c*d^4*x - 2340*a^3*b^21*d^5*x)/b^26