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

Optimal. Leaf size=79 \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)} \]

[Out]

-(Sqrt[a]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[b]*(b*c - a*d)) + (Sqrt[c]*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*
Sqrt[d]*(b*c - a*d))

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Rubi [A]  time = 0.0621678, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {465, 481, 205} \[ \frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x^4)*(c + d*x^4)),x]

[Out]

-(Sqrt[a]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[b]*(b*c - a*d)) + (Sqrt[c]*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/(2*
Sqrt[d]*(b*c - a*d))

Rule 465

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

Rule 481

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> -Dist[(a*e^n)/(b*c -
a*d), Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Dist[(c*e^n)/(b*c - a*d), Int[(e*x)^(m - n)/(c + d*x^n), x], x]
/; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]

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]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 (b c-a d)}+\frac{c \operatorname{Subst}\left (\int \frac{1}{c+d x^2} \, dx,x,x^2\right )}{2 (b c-a d)}\\ &=-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{b} (b c-a d)}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{2 \sqrt{d} (b c-a d)}\\ \end{align*}

Mathematica [A]  time = 0.038372, size = 66, normalized size = 0.84 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c}}\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{b}}}{2 b c-2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[a]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[b]) + (Sqrt[c]*ArcTan[(Sqrt[d]*x^2)/Sqrt[c]])/Sqrt[d])/(2*b*c
- 2*a*d)

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Maple [A]  time = 0.005, size = 60, normalized size = 0.8 \begin{align*} -{\frac{c}{2\,ad-2\,bc}\arctan \left ({{x}^{2}d{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{a}{2\,ad-2\,bc}\arctan \left ({b{x}^{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c/(a*d-b*c)/(c*d)^(1/2)*arctan(x^2*d/(c*d)^(1/2))+1/2*a/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x^2/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34681, size = 657, normalized size = 8.32 \begin{align*} \left [-\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{2} \sqrt{\frac{a}{b}}}{a}\right ) + \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{4} - 2 \, d x^{2} \sqrt{-\frac{c}{d}} - c}{d x^{4} + c}\right )}{4 \,{\left (b c - a d\right )}}, \frac{2 \, \sqrt{\frac{c}{d}} \arctan \left (\frac{d x^{2} \sqrt{\frac{c}{d}}}{c}\right ) - \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{4} + 2 \, b x^{2} \sqrt{-\frac{a}{b}} - a}{b x^{4} + a}\right )}{4 \,{\left (b c - a d\right )}}, -\frac{\sqrt{\frac{a}{b}} \arctan \left (\frac{b x^{2} \sqrt{\frac{a}{b}}}{a}\right ) - \sqrt{\frac{c}{d}} \arctan \left (\frac{d x^{2} \sqrt{\frac{c}{d}}}{c}\right )}{2 \,{\left (b c - a d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(sqrt(-a/b)*log((b*x^4 + 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a)) + sqrt(-c/d)*log((d*x^4 - 2*d*x^2*sqrt(-c/
d) - c)/(d*x^4 + c)))/(b*c - a*d), -1/4*(2*sqrt(a/b)*arctan(b*x^2*sqrt(a/b)/a) + sqrt(-c/d)*log((d*x^4 - 2*d*x
^2*sqrt(-c/d) - c)/(d*x^4 + c)))/(b*c - a*d), 1/4*(2*sqrt(c/d)*arctan(d*x^2*sqrt(c/d)/c) - sqrt(-a/b)*log((b*x
^4 + 2*b*x^2*sqrt(-a/b) - a)/(b*x^4 + a)))/(b*c - a*d), -1/2*(sqrt(a/b)*arctan(b*x^2*sqrt(a/b)/a) - sqrt(c/d)*
arctan(d*x^2*sqrt(c/d)/c))/(b*c - a*d)]

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Sympy [B]  time = 38.0516, size = 576, normalized size = 7.29 \begin{align*} \frac{\sqrt{- \frac{a}{b}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} - \frac{\sqrt{- \frac{a}{b}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{a}{b}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{a}{b}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{a}{b}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} + \frac{\sqrt{- \frac{c}{d}} \log{\left (- \frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} - \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} - \frac{\sqrt{- \frac{c}{d}} \log{\left (\frac{2 a^{2} b d^{3} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{2} c d^{2} \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{a d \sqrt{- \frac{c}{d}}}{a d - b c} + \frac{2 b^{3} c^{2} d \left (- \frac{c}{d}\right )^{\frac{3}{2}}}{\left (a d - b c\right )^{3}} + \frac{b c \sqrt{- \frac{c}{d}}}{a d - b c} + x^{2} \right )}}{4 \left (a d - b c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a/b)*log(-2*a**2*b*d**3*(-a/b)**(3/2)/(a*d - b*c)**3 + 4*a*b**2*c*d**2*(-a/b)**(3/2)/(a*d - b*c)**3 - a*
d*sqrt(-a/b)/(a*d - b*c) - 2*b**3*c**2*d*(-a/b)**(3/2)/(a*d - b*c)**3 - b*c*sqrt(-a/b)/(a*d - b*c) + x**2)/(4*
(a*d - b*c)) - sqrt(-a/b)*log(2*a**2*b*d**3*(-a/b)**(3/2)/(a*d - b*c)**3 - 4*a*b**2*c*d**2*(-a/b)**(3/2)/(a*d
- b*c)**3 + a*d*sqrt(-a/b)/(a*d - b*c) + 2*b**3*c**2*d*(-a/b)**(3/2)/(a*d - b*c)**3 + b*c*sqrt(-a/b)/(a*d - b*
c) + x**2)/(4*(a*d - b*c)) + sqrt(-c/d)*log(-2*a**2*b*d**3*(-c/d)**(3/2)/(a*d - b*c)**3 + 4*a*b**2*c*d**2*(-c/
d)**(3/2)/(a*d - b*c)**3 - a*d*sqrt(-c/d)/(a*d - b*c) - 2*b**3*c**2*d*(-c/d)**(3/2)/(a*d - b*c)**3 - b*c*sqrt(
-c/d)/(a*d - b*c) + x**2)/(4*(a*d - b*c)) - sqrt(-c/d)*log(2*a**2*b*d**3*(-c/d)**(3/2)/(a*d - b*c)**3 - 4*a*b*
*2*c*d**2*(-c/d)**(3/2)/(a*d - b*c)**3 + a*d*sqrt(-c/d)/(a*d - b*c) + 2*b**3*c**2*d*(-c/d)**(3/2)/(a*d - b*c)*
*3 + b*c*sqrt(-c/d)/(a*d - b*c) + x**2)/(4*(a*d - b*c))

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Giac [B]  time = 1.14814, size = 271, normalized size = 3.43 \begin{align*} -\frac{\sqrt{c d} b x^{4}{\left | d \right |} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, b c + 2 \, a d + \sqrt{-16 \, a b c d + 4 \,{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b c d{\left | b c - a d \right |} + a d^{2}{\left | b c - a d \right |} +{\left (b c - a d\right )}^{2} d} + \frac{\sqrt{a b} d x^{4}{\left | b \right |} \arctan \left (\frac{2 \, x^{2}}{\sqrt{\frac{2 \, b c + 2 \, a d - \sqrt{-16 \, a b c d + 4 \,{\left (b c + a d\right )}^{2}}}{b d}}}\right )}{b^{2} c{\left | b c - a d \right |} + a b d{\left | b c - a d \right |} -{\left (b c - a d\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*d)*b*x^4*abs(d)*arctan(2*x^2/sqrt((2*b*c + 2*a*d + sqrt(-16*a*b*c*d + 4*(b*c + a*d)^2))/(b*d)))/(b*c*d
*abs(b*c - a*d) + a*d^2*abs(b*c - a*d) + (b*c - a*d)^2*d) + sqrt(a*b)*d*x^4*abs(b)*arctan(2*x^2/sqrt((2*b*c +
2*a*d - sqrt(-16*a*b*c*d + 4*(b*c + a*d)^2))/(b*d)))/(b^2*c*abs(b*c - a*d) + a*b*d*abs(b*c - a*d) - (b*c - a*d
)^2*b)