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]

-((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))

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Rubi [A]  time = 0.265617, antiderivative size = 349, normalized size of antiderivative = 1., 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 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 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 (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*}

Mathematica [A]  time = 0.186252, size = 319, normalized size = 0.91 \[ \frac{-\frac{8 c^{3/4} \sqrt [4]{d} x \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right )}{c+d x^4}-\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )+\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt{c}+\sqrt{d} x^2\right )-2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )+\frac{32 c^{7/4} \sqrt [4]{d} x (b c-a d)^2}{\left (c+d x^4\right )^2}}{256 c^{11/4} d^{9/4}} \]

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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Maple [A]  time = 0.008, size = 499, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( d{x}^{4}+c \right ) ^{2}} \left ({\frac{ \left ( 7\,{a}^{2}{d}^{2}+2\,cabd-9\,{b}^{2}{c}^{2} \right ){x}^{5}}{32\,{c}^{2}d}}+{\frac{ \left ( 11\,{a}^{2}{d}^{2}-6\,cabd-5\,{b}^{2}{c}^{2} \right ) x}{32\,{d}^{2}c}} \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}ab}{64\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{256\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}ab}{128\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{256\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{c}{d}}}x\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}ab}{64\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.5754, size = 3302, normalized size = 9.46 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 8.0575, size = 264, normalized size = 0.76 \begin{align*} \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 )} \end{align*}

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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Giac [A]  time = 1.14108, size = 549, normalized size = 1.57 \begin{align*} \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}} \end{align*}

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)