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

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

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Rubi [A]  time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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 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]

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*}

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Mathematica [A]  time = 0.04, size = 66, normalized size = 0.84 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 325, normalized size = 4.11 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.17, size = 59, normalized size = 0.75 \begin {gather*} -\frac {a \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

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]

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

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maple [A]  time = 0.06, size = 60, normalized size = 0.76 \begin {gather*} \frac {a \arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \left (a d -b c \right ) \sqrt {a b}}-\frac {c \arctan \left (\frac {d \,x^{2}}{\sqrt {c d}}\right )}{2 \left (a d -b c \right ) \sqrt {c d}} \end {gather*}

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

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maxima [A]  time = 1.16, size = 59, normalized size = 0.75 \begin {gather*} -\frac {a \arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x^{2}}{\sqrt {c d}}\right )}{2 \, {\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

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]

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

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mupad [B]  time = 5.34, size = 379, normalized size = 4.80 \begin {gather*} \frac {\ln \left (d^2\,{\left (-a\,b\right )}^{5/2}+b^4\,c^2\,\sqrt {-a\,b}-b^5\,c^2\,x^2+2\,b^2\,c\,d\,{\left (-a\,b\right )}^{3/2}-a^2\,b^3\,d^2\,x^2+2\,a\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,b^2\,c-4\,a\,b\,d}-\frac {\ln \left (d^2\,{\left (-a\,b\right )}^{5/2}+b^4\,c^2\,\sqrt {-a\,b}+b^5\,c^2\,x^2+2\,b^2\,c\,d\,{\left (-a\,b\right )}^{3/2}+a^2\,b^3\,d^2\,x^2-2\,a\,b^4\,c\,d\,x^2\right )\,\sqrt {-a\,b}}{4\,\left (b^2\,c-a\,b\,d\right )}-\frac {\ln \left (b^2\,{\left (-c\,d\right )}^{5/2}+a^2\,d^4\,\sqrt {-c\,d}+a^2\,d^5\,x^2+2\,a\,b\,d^2\,{\left (-c\,d\right )}^{3/2}+b^2\,c^2\,d^3\,x^2-2\,a\,b\,c\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,\left (a\,d^2-b\,c\,d\right )}+\frac {\ln \left (b^2\,{\left (-c\,d\right )}^{5/2}+a^2\,d^4\,\sqrt {-c\,d}-a^2\,d^5\,x^2+2\,a\,b\,d^2\,{\left (-c\,d\right )}^{3/2}-b^2\,c^2\,d^3\,x^2+2\,a\,b\,c\,d^4\,x^2\right )\,\sqrt {-c\,d}}{4\,a\,d^2-4\,b\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((a + b*x^4)*(c + d*x^4)),x)

[Out]

(log(d^2*(-a*b)^(5/2) + b^4*c^2*(-a*b)^(1/2) - b^5*c^2*x^2 + 2*b^2*c*d*(-a*b)^(3/2) - a^2*b^3*d^2*x^2 + 2*a*b^
4*c*d*x^2)*(-a*b)^(1/2))/(4*b^2*c - 4*a*b*d) - (log(d^2*(-a*b)^(5/2) + b^4*c^2*(-a*b)^(1/2) + b^5*c^2*x^2 + 2*
b^2*c*d*(-a*b)^(3/2) + a^2*b^3*d^2*x^2 - 2*a*b^4*c*d*x^2)*(-a*b)^(1/2))/(4*(b^2*c - a*b*d)) - (log(b^2*(-c*d)^
(5/2) + a^2*d^4*(-c*d)^(1/2) + a^2*d^5*x^2 + 2*a*b*d^2*(-c*d)^(3/2) + b^2*c^2*d^3*x^2 - 2*a*b*c*d^4*x^2)*(-c*d
)^(1/2))/(4*(a*d^2 - b*c*d)) + (log(b^2*(-c*d)^(5/2) + a^2*d^4*(-c*d)^(1/2) - a^2*d^5*x^2 + 2*a*b*d^2*(-c*d)^(
3/2) - b^2*c^2*d^3*x^2 + 2*a*b*c*d^4*x^2)*(-c*d)^(1/2))/(4*a*d^2 - 4*b*c*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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