3.159 \(\int \frac {(a+b x^4)^2}{(c+d x^4)^3} \, dx\)

Optimal. Leaf size=349 \[ -\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]

[Out]

-1/8*(-a*d+b*c)*x*(b*x^4+a)/c/d/(d*x^4+c)^2-1/32*(-a*d+b*c)*(7*a*d+5*b*c)*x/c^2/d^2/(d*x^4+c)+1/128*(21*a^2*d^
2+6*a*b*c*d+5*b^2*c^2)*arctan(-1+d^(1/4)*x*2^(1/2)/c^(1/4))/c^(11/4)/d^(9/4)*2^(1/2)+1/128*(21*a^2*d^2+6*a*b*c
*d+5*b^2*c^2)*arctan(1+d^(1/4)*x*2^(1/2)/c^(1/4))/c^(11/4)/d^(9/4)*2^(1/2)-1/256*(21*a^2*d^2+6*a*b*c*d+5*b^2*c
^2)*ln(-c^(1/4)*d^(1/4)*x*2^(1/2)+c^(1/2)+x^2*d^(1/2))/c^(11/4)/d^(9/4)*2^(1/2)+1/256*(21*a^2*d^2+6*a*b*c*d+5*
b^2*c^2)*ln(c^(1/4)*d^(1/4)*x*2^(1/2)+c^(1/2)+x^2*d^(1/2))/c^(11/4)/d^(9/4)*2^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {413, 385, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {x (b c-a d) (7 a d+5 b c)}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {x \left (a+b x^4\right ) (b c-a d)}{8 c d \left (c+d x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/(c + d*x^4)^3,x]

[Out]

-((b*c - a*d)*x*(a + b*x^4))/(8*c*d*(c + d*x^4)^2) - ((b*c - a*d)*(5*b*c + 7*a*d)*x)/(32*c^2*d^2*(c + d*x^4))
- ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/4)*d^(9/4))
+ ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)])/(64*Sqrt[2]*c^(11/4)*d^(9/4)
) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(128*Sqrt[2]
*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x
^2])/(128*Sqrt[2]*c^(11/4)*d^(9/4))

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])

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 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 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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 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 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^4\right )^2}{\left (c+d x^4\right )^3} \, dx &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}+\frac {\int \frac {a (b c+7 a d)+b (5 b c+3 a d) x^4}{\left (c+d x^4\right )^2} \, dx}{8 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{c+d x^4} \, dx}{32 c^2 d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d^2}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{64 c^{5/2} d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{5/2}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{128 c^{5/2} d^{5/2}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{128 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{128 \sqrt {2} c^{11/4} d^{9/4}}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}\\ &=-\frac {(b c-a d) x \left (a+b x^4\right )}{8 c d \left (c+d x^4\right )^2}-\frac {(b c-a d) (5 b c+7 a d) x}{32 c^2 d^2 \left (c+d x^4\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{128 \sqrt {2} c^{11/4} d^{9/4}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 319, normalized size = 0.91 \[ \frac {-\sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )+\sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )-2 \sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt {2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )-\frac {8 c^{3/4} \sqrt [4]{d} x \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right )}{c+d x^4}+\frac {32 c^{7/4} \sqrt [4]{d} x (b c-a d)^2}{\left (c+d x^4\right )^2}}{256 c^{11/4} d^{9/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*x)/(c + d*x^4)^2 - (8*c^(3/4)*d^(1/4)*(9*b^2*c^2 - 2*a*b*c*d - 7*a^2*d^2)*x
)/(c + d*x^4) - 2*Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*x)/c^(1/4)] + 2*Sqr
t[2]*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*x)/c^(1/4)] - Sqrt[2]*(5*b^2*c^2 + 6*a*b
*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2] + Sqrt[2]*(5*b^2*c^2 + 6*a*b*c*d + 2
1*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*x + Sqrt[d]*x^2])/(256*c^(11/4)*d^(9/4))

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fricas [B]  time = 1.40, size = 1411, normalized size = 4.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/(d*x^4+c)^3,x, algorithm="fricas")

[Out]

-1/128*(4*(9*b^2*c^2*d - 2*a*b*c*d^2 - 7*a^2*d^3)*x^5 - 4*(c^2*d^4*x^8 + 2*c^3*d^3*x^4 + c^4*d^2)*(-(625*b^8*c
^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^
3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)*arctan(-(c^8*d^7*x
*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 +
176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(3/4) - c^8
*d^7*sqrt((c^6*d^4*sqrt(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112
806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(
c^11*d^9)) + (25*b^4*c^4 + 60*a*b^3*c^3*d + 246*a^2*b^2*c^2*d^2 + 252*a^3*b*c*d^3 + 441*a^4*d^4)*x^2)/(25*b^4*
c^4 + 60*a*b^3*c^3*d + 246*a^2*b^2*c^2*d^2 + 252*a^3*b*c*d^3 + 441*a^4*d^4))*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d
+ 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^
6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(3/4))/(125*b^6*c^6 + 450*a*b^5*c^5*d + 2115*
a^2*b^4*c^4*d^2 + 3996*a^3*b^3*c^3*d^3 + 8883*a^4*b^2*c^2*d^4 + 7938*a^5*b*c*d^5 + 9261*a^6*d^6)) - (c^2*d^4*x
^8 + 2*c^3*d^3*x^4 + c^4*d^2)*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^
3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8
*d^8)/(c^11*d^9))^(1/4)*log(c^3*d^2*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*
c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 1944
81*a^8*d^8)/(c^11*d^9))^(1/4) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*x) + (c^2*d^4*x^8 + 2*c^3*d^3*x^4 + c^4*d
^2)*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4
+ 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(1/4)*lo
g(-c^3*d^2*(-(625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*
c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(c^11*d^9))^(
1/4) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*x) + 4*(5*b^2*c^3 + 6*a*b*c^2*d - 11*a^2*c*d^2)*x)/(c^2*d^4*x^8 +
2*c^3*d^3*x^4 + c^4*d^2)

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giac [A]  time = 0.21, size = 407, normalized size = 1.17 \[ \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{128 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{256 \, c^{3} d^{3}} - \frac {9 \, b^{2} c^{2} d x^{5} - 2 \, a b c d^{2} x^{5} - 7 \, a^{2} d^{3} x^{5} + 5 \, b^{2} c^{3} x + 6 \, a b c^{2} d x - 11 \, a^{2} c d^{2} x}{32 \, {\left (d x^{4} + c\right )}^{2} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/(d*x^4+c)^3,x, algorithm="giac")

[Out]

1/128*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2
)*(2*x + sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c^3*d^3) + 1/128*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4
)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(c/d)^(1/4))/(c/d)^(1/4))/(c^3*d^3) +
1/256*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(x^2 + sqrt(2)
*x*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^3) - 1/256*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 + 6*(c*d^3)^(1/4)*a*b*c*d + 21*
(c*d^3)^(1/4)*a^2*d^2)*log(x^2 - sqrt(2)*x*(c/d)^(1/4) + sqrt(c/d))/(c^3*d^3) - 1/32*(9*b^2*c^2*d*x^5 - 2*a*b*
c*d^2*x^5 - 7*a^2*d^3*x^5 + 5*b^2*c^3*x + 6*a*b*c^2*d*x - 11*a^2*c*d^2*x)/((d*x^4 + c)^2*c^2*d^2)

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maple [A]  time = 0.06, size = 499, normalized size = 1.43 \[ \frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{128 c^{3}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{128 c^{3}}+\frac {21 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} \ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{256 c^{3}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 c^{2} d}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 c^{2} d}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, a b \ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{128 c^{2} d}+\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{128 c \,d^{2}}+\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{128 c \,d^{2}}+\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, b^{2} \ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {c}{d}}}\right )}{256 c \,d^{2}}+\frac {\frac {\left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{5}}{32 c^{2} d}+\frac {\left (11 a^{2} d^{2}-6 a b c d -5 b^{2} c^{2}\right ) x}{32 c \,d^{2}}}{\left (d \,x^{4}+c \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/32*(7*a^2*d^2+2*a*b*c*d-9*b^2*c^2)/c^2/d*x^5+1/32*(11*a^2*d^2-6*a*b*c*d-5*b^2*c^2)/d^2/c*x)/(d*x^4+c)^2+21/
128/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*a^2+3/64/c^2/d*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(c/d)^(1/4)*x-1)*a*b+5/128/c/d^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x-1)*b^2+21/256/c^3*(c/d)^(1/4
)*2^(1/2)*ln((x^2+(c/d)^(1/4)*2^(1/2)*x+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*2^(1/2)*x+(c/d)^(1/2)))*a^2+3/128/c^2/d*
(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*2^(1/2)*x+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*2^(1/2)*x+(c/d)^(1/2)))*a*b+5/
256/c/d^2*(c/d)^(1/4)*2^(1/2)*ln((x^2+(c/d)^(1/4)*2^(1/2)*x+(c/d)^(1/2))/(x^2-(c/d)^(1/4)*2^(1/2)*x+(c/d)^(1/2
)))*b^2+21/128/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*a^2+3/64/c^2/d*(c/d)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(c/d)^(1/4)*x+1)*a*b+5/128/c/d^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x+1)*b^2

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maxima [A]  time = 1.22, size = 361, normalized size = 1.03 \[ -\frac {{\left (9 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 7 \, a^{2} d^{3}\right )} x^{5} + {\left (5 \, b^{2} c^{3} + 6 \, a b c^{2} d - 11 \, a^{2} c d^{2}\right )} x}{32 \, {\left (c^{2} d^{4} x^{8} + 2 \, c^{3} d^{3} x^{4} + c^{4} d^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{256 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*((9*b^2*c^2*d - 2*a*b*c*d^2 - 7*a^2*d^3)*x^5 + (5*b^2*c^3 + 6*a*b*c^2*d - 11*a^2*c*d^2)*x)/(c^2*d^4*x^8
+ 2*c^3*d^3*x^4 + c^4*d^2) + 1/256*(2*sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*arctan(1/2*sqrt(2)*(2*sqrt(
d)*x + sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(5*b^2*c^2
+ 6*a*b*c*d + 21*a^2*d^2)*arctan(1/2*sqrt(2)*(2*sqrt(d)*x - sqrt(2)*c^(1/4)*d^(1/4))/sqrt(sqrt(c)*sqrt(d)))/(s
qrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(d)*x^2 + sqrt(2)*c^(1/4)
*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(d)*x^2 - sqrt(
2)*c^(1/4)*d^(1/4)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(c^2*d^2)

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mupad [B]  time = 1.66, size = 1401, normalized size = 4.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2/(c + d*x^4)^3,x)

[Out]

- ((x*(5*b^2*c^2 - 11*a^2*d^2 + 6*a*b*c*d))/(32*c*d^2) - (x^5*(7*a^2*d^2 - 9*b^2*c^2 + 2*a*b*c*d))/(32*c^2*d))
/(c^2 + d^2*x^8 + 2*c*d*x^4) - (atan((((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*
b*c*d^2))/(256*(-c)^(15/4)*d^(9/4)) - (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 25
2*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*1i)/(128*(-c)^(11/4)*d^(9/4)) - ((((21*a^2*d
^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2))/(256*(-c)^(15/4)*d^(9/4)) + (x*(441*a^4*
d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c
^2 + 6*a*b*c*d)*1i)/(128*(-c)^(11/4)*d^(9/4)))/(((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^
2*d + 6*a*b*c*d^2))/(256*(-c)^(15/4)*d^(9/4)) - (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*
c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d))/(128*(-c)^(11/4)*d^(9/4)) + ((((2
1*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2))/(256*(-c)^(15/4)*d^(9/4)) + (x*(4
41*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 +
5*b^2*c^2 + 6*a*b*c*d))/(128*(-c)^(11/4)*d^(9/4))))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*1i)/(64*(-c)^(11/4)*d
^(9/4)) - (atan((((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2)*1i)/(256*(-c
)^(15/4)*d^(9/4)) - (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(2
56*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d))/(128*(-c)^(11/4)*d^(9/4)) - ((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*
b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2)*1i)/(256*(-c)^(15/4)*d^(9/4)) + (x*(441*a^4*d^4 + 25*b^4*c^4 +
246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d))/(
128*(-c)^(11/4)*d^(9/4)))/(((((21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2)*1i
)/(256*(-c)^(15/4)*d^(9/4)) - (x*(441*a^4*d^4 + 25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*
c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d)*1i)/(128*(-c)^(11/4)*d^(9/4)) + ((((21*a^2*d^2 + 5*b
^2*c^2 + 6*a*b*c*d)*(21*a^2*d^3 + 5*b^2*c^2*d + 6*a*b*c*d^2)*1i)/(256*(-c)^(15/4)*d^(9/4)) + (x*(441*a^4*d^4 +
25*b^4*c^4 + 246*a^2*b^2*c^2*d^2 + 60*a*b^3*c^3*d + 252*a^3*b*c*d^3))/(256*c^4*d))*(21*a^2*d^2 + 5*b^2*c^2 +
6*a*b*c*d)*1i)/(128*(-c)^(11/4)*d^(9/4))))*(21*a^2*d^2 + 5*b^2*c^2 + 6*a*b*c*d))/(64*(-c)^(11/4)*d^(9/4))

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sympy [A]  time = 5.85, size = 264, normalized size = 0.76 \[ \frac {x^{5} \left (7 a^{2} d^{3} + 2 a b c d^{2} - 9 b^{2} c^{2} d\right ) + x \left (11 a^{2} c d^{2} - 6 a b c^{2} d - 5 b^{2} c^{3}\right )}{32 c^{4} d^{2} + 64 c^{3} d^{3} x^{4} + 32 c^{2} d^{4} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} c^{11} d^{9} + 194481 a^{8} d^{8} + 222264 a^{7} b c d^{7} + 280476 a^{6} b^{2} c^{2} d^{6} + 176904 a^{5} b^{3} c^{3} d^{5} + 112806 a^{4} b^{4} c^{4} d^{4} + 42120 a^{3} b^{5} c^{5} d^{3} + 15900 a^{2} b^{6} c^{6} d^{2} + 3000 a b^{7} c^{7} d + 625 b^{8} c^{8}, \left (t \mapsto t \log {\left (\frac {128 t c^{3} d^{2}}{21 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**2/(d*x**4+c)**3,x)

[Out]

(x**5*(7*a**2*d**3 + 2*a*b*c*d**2 - 9*b**2*c**2*d) + x*(11*a**2*c*d**2 - 6*a*b*c**2*d - 5*b**2*c**3))/(32*c**4
*d**2 + 64*c**3*d**3*x**4 + 32*c**2*d**4*x**8) + RootSum(268435456*_t**4*c**11*d**9 + 194481*a**8*d**8 + 22226
4*a**7*b*c*d**7 + 280476*a**6*b**2*c**2*d**6 + 176904*a**5*b**3*c**3*d**5 + 112806*a**4*b**4*c**4*d**4 + 42120
*a**3*b**5*c**5*d**3 + 15900*a**2*b**6*c**6*d**2 + 3000*a*b**7*c**7*d + 625*b**8*c**8, Lambda(_t, _t*log(128*_
t*c**3*d**2/(21*a**2*d**2 + 6*a*b*c*d + 5*b**2*c**2) + x)))

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