3.194 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{(a-b x^4)^2} \, dx\)

Optimal. Leaf size=225 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{j \log \left (a-b x^4\right )}{4 b^2}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{4 a b \left (a-b x^4\right )} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + (b*e + a*i)*x^2 + (b*f + a*j)*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)) + (j*Log[a - b*x^4])/(4*b^2)

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Rubi [A]  time = 0.310455, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1858, 1876, 1167, 205, 208, 1248, 635, 260} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{j \log \left (a-b x^4\right )}{4 b^2}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+x^3 (a j+b f)+a g+b c\right )}{4 a b \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(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 + a*j)*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)) + (j*Log[a - b*x^4])/(4*b^2)

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

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 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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+194 x^6+j x^7}{\left (a-b x^4\right )^2} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+(194 a+b e) x^2+(b f+a j) 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 (582 a-b e) x^2+4 a b j x^3}{a-b x^4} \, dx}{4 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(194 a+b e) x^2+(b f+a j) x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\int \left (\frac{-b (3 b c-a g)+b (582 a-b e) x^2}{a-b x^4}+\frac{x \left (-2 b (b d-a h)+4 a b j x^2\right )}{a-b x^4}\right ) \, dx}{4 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(194 a+b e) x^2+(b f+a j) x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\int \frac{-b (3 b c-a g)+b (582 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}-\frac{\int \frac{x \left (-2 b (b d-a h)+4 a b j x^2\right )}{a-b x^4} \, dx}{4 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(194 a+b e) x^2+(b f+a j) x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-2 b (b d-a h)+4 a b j x}{a-b x^2} \, dx,x,x^2\right )}{8 a b^2}-\frac{\left (582 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 (582 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{x \left (b c+a g+(b d+a h) x+(194 a+b e) x^2+(b f+a j) x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\left (582 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 (582 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) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{4 a b}-\frac{j \operatorname{Subst}\left (\int \frac{x}{a-b x^2} \, dx,x,x^2\right )}{2 b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(194 a+b e) x^2+(b f+a j) x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\left (582 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 (582 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}}+\frac{j \log \left (a-b x^4\right )}{4 b^2}\\ \end{align*}

Mathematica [A]  time = 0.239555, size = 338, normalized size = 1.5 \[ \frac{\frac{4 \left (a^2 j+a b (f+x (g+x (h+i x)))+b^2 x (c+x (d+e x))\right )}{a \left (a-b x^4\right )}+\frac{\sqrt [4]{b} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt{a} b e+a \sqrt{b} g-3 b^{3/2} c\right )}{a^{7/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right )}{a^{7/4}}+\frac{2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right )}{a^{7/4}}+\frac{2 \sqrt{b} (b d-a h) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{a^{3/2}}+4 j \log \left (a-b x^4\right )}{16 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6 + j*x^7)/(a - b*x^4)^2,x]

[Out]

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

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Maple [B]  time = 0.011, size = 431, normalized size = 1.9 \begin{align*}{\frac{1}{b{x}^{4}-a} \left ( -{\frac{ \left ( ai+be \right ){x}^{3}}{4\,ab}}-{\frac{ \left ( ah+bd \right ){x}^{2}}{4\,ab}}-{\frac{ \left ( ag+bc \right ) x}{4\,ab}}-{\frac{aj+bf}{4\,{b}^{2}}} \right ) }-{\frac{g}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{g}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,c}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{h}{8\,b}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{d}{8\,a}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,i}{8\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{3\,i}{16\,{b}^{2}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{j\ln \left ( b{x}^{4}-a \right ) }{4\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((j*x^7+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*(a*j+b*f)/b^2)/(b*x^4-a)-1/8/b/a*(1/b*a)
^(1/4)*arctan(x/(1/b*a)^(1/4))*g+3/8*c/a^2*(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))-1/16/b/a*(1/b*a)^(1/4)*ln((x+
(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))*g+3/16*c/a^2*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))+1/8/b/(a*
b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))*h-1/8*d/a/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(
a*b)^(1/2)))+3/8/b^2/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))*i-1/8*e/a/b/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))-3
/16/b^2/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))*i+1/16*e/a/b/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/
(x-(1/b*a)^(1/4)))+1/4/b^2*j*ln(b*x^4-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((j*x^7+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]

Exception raised: ValueError

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Fricas [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((j*x^7+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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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((j*x**7+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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Giac [B]  time = 1.11957, size = 884, normalized size = 3.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((j*x^7+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)) + 1/4*j*log(abs(b*x^4 - a))/b^2 - 1/4*((a*i + b*e)*x^3 + (b*d + a*h)*
x^2 + (b*c + a*g)*x + (a*b*f + a^2*j)/b)/((b*x^4 - a)*a*b) - 1/16*sqrt(2)*(2*sqrt(2)*sqrt(-a*b)*b^2*d - 2*sqrt
(2)*sqrt(-a*b)*a*b*h - 3*(-a*b^3)^(1/4)*b^2*c + (-a*b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2
*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^3) - 1/16*sqrt(2)*(2*sqrt(2)*sqrt(-a*b)*b^2*d - 2*sqrt(2)*sqrt
(-a*b)*a*b*h - 3*(-a*b^3)^(1/4)*b^2*c + (-a*b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqr
t(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^3) + 1/32*sqrt(2)*(3*(-a*b^3)^(1/4)*b^2*c - (-a*b^3)^(1/4)*a*b*g - (-a
*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^3) - 1/32*sqrt(2)*(3*(-a*b^3)^(1/4)*b^2*c
- (-a*b^3)^(1/4)*a*b*g - (-a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^3)