3.193 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{(a-b x^4)^2} \, dx\)

Optimal. Leaf size=203 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {(b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

[Out]

1/4*x*(b*c+a*g+(a*h+b*d)*x+(a*i+b*e)*x^2+b*f*x^3)/a/b/(-b*x^4+a)+1/4*(-a*h+b*d)*arctanh(x^2*b^(1/2)/a^(1/2))/a
^(3/2)/b^(3/2)-1/8*arctan(b^(1/4)*x/a^(1/4))*(b*e-3*a*i-(-a*g+3*b*c)*b^(1/2)/a^(1/2))/a^(5/4)/b^(7/4)+1/8*arct
anh(b^(1/4)*x/a^(1/4))*(b*e-3*a*i+(-a*g+3*b*c)*b^(1/2)/a^(1/2))/a^(5/4)/b^(7/4)

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Rubi [A]  time = 0.27, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1858, 1876, 275, 208, 1167, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac {(b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^2,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + b*f*x^3))/(4*a*b*(a - b*x^4)) - ((b*e - (Sqrt[b]*(3*b*c - a*
g))/Sqrt[a] - 3*a*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(5/4)*b^(7/4)) + ((b*e + (Sqrt[b]*(3*b*c - a*g))/Sqrt[a
] - 3*a*i)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(5/4)*b^(7/4)) + ((b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(4
*a^(3/2)*b^(3/2))

Rule 205

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

Rule 208

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rubi steps

\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+193 x^6}{\left (a-b x^4\right )^2} \, dx &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \frac {-b (3 b c-a g)-2 b (b d-a h) x+b (579 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \left (-\frac {2 b (b d-a h) x}{a-b x^4}+\frac {-b (3 b c-a g)+b (579 a-b e) x^2}{a-b x^4}\right ) \, dx}{4 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\int \frac {-b (3 b c-a g)+b (579 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}+\frac {(b d-a h) \int \frac {x}{a-b x^4} \, dx}{2 a b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac {\left (579 a-b e-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a b}-\frac {\left (579 a-b e+\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a b}+\frac {(b d-a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{4 a b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {\left (579 a-b e+\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{5/4} b^{7/4}}-\frac {\left (579 a-b e-\frac {\sqrt {b} (3 b c-a g)}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{5/4} b^{7/4}}+\frac {(b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 302, normalized size = 1.49 \[ \frac {\frac {4 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt {a} b e+a \sqrt {b} g-3 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt {a} b e-a \sqrt {b} g+3 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-\sqrt {a} b e-a \sqrt {b} g+3 b^{3/2} c\right )-2 \sqrt [4]{a} \sqrt [4]{b} (a h-b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{16 a^{7/4} b^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^2,x]

[Out]

((4*a^(3/4)*b^(3/4)*(b*x*(c + x*(d + e*x)) + a*(f + x*(g + x*(h + i*x)))))/(a - b*x^4) + 2*(3*b^(3/2)*c - Sqrt
[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a^(1/4)] + (-3*b^(3/2)*c - 2*a^(1/4)*b^(5/4)*d - Sqrt[
a]*b*e + a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h + 3*a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x] + (3*b^(3/2)*c - 2*a^(1/4)*
b^(5/4)*d + Sqrt[a]*b*e - a*Sqrt[b]*g + 2*a^(5/4)*b^(1/4)*h - 3*a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x] - 2*a^(1/4
)*b^(1/4)*(-(b*d) + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(7/4))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.20, size = 583, normalized size = 2.87 \[ -\frac {3}{32} \, i {\left (\frac {2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4}} - \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{a b^{4}}\right )} - \frac {3}{32} \, i {\left (\frac {2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4}} + \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{a b^{4}}\right )} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g + 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} + \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {a i x^{3} + b x^{3} e + b d x^{2} + a h x^{2} + b c x + a g x + a f}{4 \, {\left (b x^{4} - a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="giac")

[Out]

-3/32*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a*b^4) - sqrt
(2)*(-a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^4)) - 3/32*i*(2*sqrt(2)*(-a*b^3)^(3/4)*
arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a*b^4) + sqrt(2)*(-a*b^3)^(3/4)*log(x^2 - sqrt(
2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^4)) - 1/16*sqrt(2)*(3*b^2*c - a*b*g - 2*sqrt(2)*(-a*b^3)^(1/4)*b*d + 2*sq
rt(2)*(-a*b^3)^(1/4)*a*h + sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*
b^3)^(3/4)*a) - 1/16*sqrt(2)*(3*b^2*c - a*b*g + 2*sqrt(2)*(-a*b^3)^(1/4)*b*d - 2*sqrt(2)*(-a*b^3)^(1/4)*a*h -
sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/4)*a) - 1/32*sqrt(2
)*(3*b^2*c - a*b*g - sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a) + 1/32*
sqrt(2)*(3*b^2*c - a*b*g - sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a) -
1/4*(a*i*x^3 + b*x^3*e + b*d*x^2 + a*h*x^2 + b*c*x + a*g*x + a*f)/((b*x^4 - a)*a*b)

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maple [B]  time = 0.05, size = 409, normalized size = 2.01 \[ -\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{8 \sqrt {a b}\, a}+\frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{8 \sqrt {a b}\, b}-\frac {e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a^{2}}+\frac {3 i \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}-\frac {3 i \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {-\frac {\left (a i +b e \right ) x^{3}}{4 a b}-\frac {\left (a h +b d \right ) x^{2}}{4 a b}-\frac {f}{4 b}-\frac {\left (a g +b c \right ) x}{4 a b}}{b \,x^{4}-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x)

[Out]

(-1/4*(a*i+b*e)/a/b*x^3-1/4*(a*h+b*d)/a/b*x^2-1/4*(a*g+b*c)/a/b*x-1/4/b*f)/(b*x^4-a)-1/8*(a/b)^(1/4)/a/b*g*arc
tan(1/(a/b)^(1/4)*x)+3/8*(a/b)^(1/4)/a^2*c*arctan(1/(a/b)^(1/4)*x)-1/16*(a/b)^(1/4)/a/b*g*ln((x+(a/b)^(1/4))/(
x-(a/b)^(1/4)))+3/16*(a/b)^(1/4)/a^2*c*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/8/(a*b)^(1/2)/b*h*ln(((a*b)^(1/2)
*x^2-a)/(-(a*b)^(1/2)*x^2-a))-1/8/(a*b)^(1/2)/a*d*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))+3/8/b^2/(a/b)^(
1/4)*arctan(1/(a/b)^(1/4)*x)*i-1/8/(a/b)^(1/4)/a/b*e*arctan(1/(a/b)^(1/4)*x)-3/16/b^2/(a/b)^(1/4)*ln((x+(a/b)^
(1/4))/(x-(a/b)^(1/4)))*i+1/16/(a/b)^(1/4)/a/b*e*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))

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maxima [A]  time = 3.06, size = 260, normalized size = 1.28 \[ -\frac {{\left (b e + a i\right )} x^{3} + {\left (b d + a h\right )} x^{2} + a f + {\left (b c + a g\right )} x}{4 \, {\left (a b^{2} x^{4} - a^{2} b\right )}} + \frac {\frac {2 \, {\left (b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {2 \, {\left (b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e - a \sqrt {b} g + 3 \, a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (3 \, b^{\frac {3}{2}} c + \sqrt {a} b e - a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{16 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="maxima")

[Out]

-1/4*((b*e + a*i)*x^3 + (b*d + a*h)*x^2 + a*f + (b*c + a*g)*x)/(a*b^2*x^4 - a^2*b) + 1/16*(2*(b*d - a*h)*log(s
qrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 2*(b*d - a*h)*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) + 2*(3*b^
(3/2)*c - sqrt(a)*b*e - a*sqrt(b)*g + 3*a^(3/2)*i)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(
a)*sqrt(b))*sqrt(b)) - (3*b^(3/2)*c + sqrt(a)*b*e - a*sqrt(b)*g - 3*a^(3/2)*i)*log((sqrt(b)*x - sqrt(sqrt(a)*s
qrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/(a*b)

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mupad [B]  time = 5.67, size = 2611, normalized size = 12.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4)^2,x)

[Out]

symsum(log((27*a^4*i^3 - a*b^3*e^3 - 12*b^4*c*d^2 + 9*b^4*c^2*e - 12*a^2*b^2*c*h^2 + a^2*b^2*e*g^2 + 9*a^2*b^2
*e^2*i + 4*a*b^3*d^2*g - 27*a*b^3*c^2*i - 27*a^3*b*e*i^2 + 4*a^3*b*g*h^2 - 3*a^3*b*g^2*i + 18*a^2*b^2*c*g*i -
8*a^2*b^2*d*g*h + 24*a*b^3*c*d*h - 6*a*b^3*c*e*g)/(64*a^3*b^2) - root(65536*a^7*b^7*z^4 - 3072*a^6*b^4*g*i*z^2
+ 9216*a^5*b^5*c*i*z^2 + 4096*a^5*b^5*d*h*z^2 + 1024*a^5*b^5*e*g*z^2 - 3072*a^4*b^6*c*e*z^2 - 2048*a^6*b^4*h^
2*z^2 - 2048*a^4*b^6*d^2*z^2 + 768*a^5*b^3*e*h*i*z - 768*a^4*b^4*d*e*i*z + 768*a^4*b^4*c*g*h*z - 768*a^3*b^5*c
*d*g*z - 1152*a^6*b^2*h*i^2*z - 128*a^5*b^3*g^2*h*z + 1152*a^5*b^3*d*i^2*z - 128*a^4*b^4*e^2*h*z - 1152*a^3*b^
5*c^2*h*z + 128*a^4*b^4*d*g^2*z + 128*a^3*b^5*d*e^2*z + 1152*a^2*b^6*c^2*d*z + 96*a^4*b^2*d*g*h*i - 288*a^3*b^
3*c*d*h*i + 72*a^3*b^3*c*e*g*i - 32*a^3*b^3*d*e*g*h + 96*a^2*b^4*c*d*e*h - 12*a^4*b^2*e*g^2*i + 144*a^4*b^2*c*
h^2*i - 48*a^3*b^3*d^2*g*i + 16*a^4*b^2*e*g*h^2 - 108*a^4*b^2*c*g*i^2 - 108*a^2*b^4*c^2*e*i + 144*a^2*b^4*c*d^
2*i - 48*a^3*b^3*c*e*h^2 + 16*a^2*b^4*d^2*e*g - 12*a^2*b^4*c*e^2*g - 48*a^5*b*g*h^2*i - 48*a*b^5*c*d^2*e + 108
*a^5*b*e*i^3 + 108*a*b^5*c^3*g - 54*a^4*b^2*e^2*i^2 + 162*a^3*b^3*c^2*i^2 + 96*a^3*b^3*d^2*h^2 + 2*a^3*b^3*e^2
*g^2 - 54*a^2*b^4*c^2*g^2 + 18*a^5*b*g^2*i^2 + 12*a^3*b^3*e^3*i - 64*a^4*b^2*d*h^3 - 64*a^2*b^4*d^3*h + 12*a^3
*b^3*c*g^3 + 18*a*b^5*c^2*e^2 + 16*a^5*b*h^4 + 16*a*b^5*d^4 - 81*a^6*i^4 - 81*b^6*c^4 - a^4*b^2*g^4 - a^2*b^4*
e^4, z, l)*(root(65536*a^7*b^7*z^4 - 3072*a^6*b^4*g*i*z^2 + 9216*a^5*b^5*c*i*z^2 + 4096*a^5*b^5*d*h*z^2 + 1024
*a^5*b^5*e*g*z^2 - 3072*a^4*b^6*c*e*z^2 - 2048*a^6*b^4*h^2*z^2 - 2048*a^4*b^6*d^2*z^2 + 768*a^5*b^3*e*h*i*z -
768*a^4*b^4*d*e*i*z + 768*a^4*b^4*c*g*h*z - 768*a^3*b^5*c*d*g*z - 1152*a^6*b^2*h*i^2*z - 128*a^5*b^3*g^2*h*z +
1152*a^5*b^3*d*i^2*z - 128*a^4*b^4*e^2*h*z - 1152*a^3*b^5*c^2*h*z + 128*a^4*b^4*d*g^2*z + 128*a^3*b^5*d*e^2*z
+ 1152*a^2*b^6*c^2*d*z + 96*a^4*b^2*d*g*h*i - 288*a^3*b^3*c*d*h*i + 72*a^3*b^3*c*e*g*i - 32*a^3*b^3*d*e*g*h +
96*a^2*b^4*c*d*e*h - 12*a^4*b^2*e*g^2*i + 144*a^4*b^2*c*h^2*i - 48*a^3*b^3*d^2*g*i + 16*a^4*b^2*e*g*h^2 - 108
*a^4*b^2*c*g*i^2 - 108*a^2*b^4*c^2*e*i + 144*a^2*b^4*c*d^2*i - 48*a^3*b^3*c*e*h^2 + 16*a^2*b^4*d^2*e*g - 12*a^
2*b^4*c*e^2*g - 48*a^5*b*g*h^2*i - 48*a*b^5*c*d^2*e + 108*a^5*b*e*i^3 + 108*a*b^5*c^3*g - 54*a^4*b^2*e^2*i^2 +
162*a^3*b^3*c^2*i^2 + 96*a^3*b^3*d^2*h^2 + 2*a^3*b^3*e^2*g^2 - 54*a^2*b^4*c^2*g^2 + 18*a^5*b*g^2*i^2 + 12*a^3
*b^3*e^3*i - 64*a^4*b^2*d*h^3 - 64*a^2*b^4*d^3*h + 12*a^3*b^3*c*g^3 + 18*a*b^5*c^2*e^2 + 16*a^5*b*h^4 + 16*a*b
^5*d^4 - 81*a^6*i^4 - 81*b^6*c^4 - a^4*b^2*g^4 - a^2*b^4*e^4, z, l)*((768*a^3*b^5*c - 256*a^4*b^4*g)/(64*a^3*b
^2) - (x*(128*a^3*b^4*d - 128*a^4*b^3*h))/(16*a^3*b)) - (64*a^2*b^4*d*e - 192*a^3*b^3*d*i - 64*a^3*b^3*e*h + 1
92*a^4*b^2*h*i)/(64*a^3*b^2) + (x*(36*a*b^4*c^2 + 36*a^4*b*i^2 + 4*a^2*b^3*e^2 + 4*a^3*b^2*g^2 - 24*a^2*b^3*c*
g - 24*a^3*b^2*e*i))/(16*a^3*b)) - (x*(2*b^3*d^3 - 2*a^3*h^3 - 3*b^3*c*d*e + 3*a^3*g*h*i - 6*a*b^2*d^2*h + 6*a
^2*b*d*h^2 + 9*a*b^2*c*d*i + 3*a*b^2*c*e*h + a*b^2*d*e*g - 9*a^2*b*c*h*i - 3*a^2*b*d*g*i - a^2*b*e*g*h))/(16*a
^3*b))*root(65536*a^7*b^7*z^4 - 3072*a^6*b^4*g*i*z^2 + 9216*a^5*b^5*c*i*z^2 + 4096*a^5*b^5*d*h*z^2 + 1024*a^5*
b^5*e*g*z^2 - 3072*a^4*b^6*c*e*z^2 - 2048*a^6*b^4*h^2*z^2 - 2048*a^4*b^6*d^2*z^2 + 768*a^5*b^3*e*h*i*z - 768*a
^4*b^4*d*e*i*z + 768*a^4*b^4*c*g*h*z - 768*a^3*b^5*c*d*g*z - 1152*a^6*b^2*h*i^2*z - 128*a^5*b^3*g^2*h*z + 1152
*a^5*b^3*d*i^2*z - 128*a^4*b^4*e^2*h*z - 1152*a^3*b^5*c^2*h*z + 128*a^4*b^4*d*g^2*z + 128*a^3*b^5*d*e^2*z + 11
52*a^2*b^6*c^2*d*z + 96*a^4*b^2*d*g*h*i - 288*a^3*b^3*c*d*h*i + 72*a^3*b^3*c*e*g*i - 32*a^3*b^3*d*e*g*h + 96*a
^2*b^4*c*d*e*h - 12*a^4*b^2*e*g^2*i + 144*a^4*b^2*c*h^2*i - 48*a^3*b^3*d^2*g*i + 16*a^4*b^2*e*g*h^2 - 108*a^4*
b^2*c*g*i^2 - 108*a^2*b^4*c^2*e*i + 144*a^2*b^4*c*d^2*i - 48*a^3*b^3*c*e*h^2 + 16*a^2*b^4*d^2*e*g - 12*a^2*b^4
*c*e^2*g - 48*a^5*b*g*h^2*i - 48*a*b^5*c*d^2*e + 108*a^5*b*e*i^3 + 108*a*b^5*c^3*g - 54*a^4*b^2*e^2*i^2 + 162*
a^3*b^3*c^2*i^2 + 96*a^3*b^3*d^2*h^2 + 2*a^3*b^3*e^2*g^2 - 54*a^2*b^4*c^2*g^2 + 18*a^5*b*g^2*i^2 + 12*a^3*b^3*
e^3*i - 64*a^4*b^2*d*h^3 - 64*a^2*b^4*d^3*h + 12*a^3*b^3*c*g^3 + 18*a*b^5*c^2*e^2 + 16*a^5*b*h^4 + 16*a*b^5*d^
4 - 81*a^6*i^4 - 81*b^6*c^4 - a^4*b^2*g^4 - a^2*b^4*e^4, z, l), l, 1, 4) + (f/(4*b) + (x*(b*c + a*g))/(4*a*b)
+ (x^2*(b*d + a*h))/(4*a*b) + (x^3*(b*e + a*i))/(4*a*b))/(a - b*x^4)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**2,x)

[Out]

Timed out

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