3.1.40 \(\int \frac {1}{(a+b x^2)^3 (c+d x^2)} \, dx\)

Optimal. Leaf size=161 \[ \frac {b x (3 b c-7 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}+\frac {b x}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]

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Rubi [A]  time = 0.20, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {414, 527, 522, 205} \begin {gather*} \frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^3}+\frac {b x (3 b c-7 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}+\frac {b x}{4 a \left (a+b x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

(b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)) + (Sqrt[b]*(3*
b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*ArcTan[(S
qrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^3)

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}-\frac {\int \frac {-3 b c+4 a d-3 b d x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx}{4 a (b c-a d)}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\int \frac {3 b^2 c^2-7 a b c d+8 a^2 d^2+b d (3 b c-7 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3 \int \frac {1}{c+d x^2} \, dx}{(b c-a d)^3}+\frac {\left (b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 (b c-a d)^3}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 158, normalized size = 0.98 \begin {gather*} \frac {1}{8} \left (\frac {b x (3 b c-7 a d)}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (a d-b c)^3}-\frac {8 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}-\frac {2 b x}{a \left (a+b x^2\right )^2 (a d-b c)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

((-2*b*x)/(a*(-(b*c) + a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(a^2*(b*c - a*d)^2*(a + b*x^2)) - (Sqrt[b]*
(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*(-(b*c) + a*d)^3) - (8*d^(5/2)*Arc
Tan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*(b*c - a*d)^3))/8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

IntegrateAlgebraic[1/((a + b*x^2)^3*(c + d*x^2)), x]

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fricas [B]  time = 2.48, size = 1587, normalized size = 9.86

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c),x, algorithm="fricas")

[Out]

[1/16*(2*(3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 - (3*a^2*b^2*c^2 - 10*a^3*b*c*d + 15*a^4*d^2 + (3*b^4*
c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(-b/a)*log
((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 8*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2 + a^4*d^2)*sqrt(-d/c)*log(
(d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^3*c^2 - 14*a^2*b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 -
3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3 + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x
^4 + 2*(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x^2), 1/16*(2*(3*b^4*c^2 - 10*a*b^3*c*d +
7*a^2*b^2*d^2)*x^3 - 16*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2 + a^4*d^2)*sqrt(d/c)*arctan(x*sqrt(d/c)) - (3*a^2*
b^2*c^2 - 10*a^3*b*c*d + 15*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 - 10*a^
2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(5*a*b^3*c^2 - 1
4*a^2*b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3 + (a^2*b^5*c^3 - 3*a^
3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^4 + 2*(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*
d^3)*x^2), 1/8*((3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 + (3*a^2*b^2*c^2 - 10*a^3*b*c*d + 15*a^4*d^2 +
(3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(b/
a)*arctan(x*sqrt(b/a)) - 4*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2 + a^4*d^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d
/c) - c)/(d*x^2 + c)) + (5*a*b^3*c^2 - 14*a^2*b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6
*b*c*d^2 - a^7*d^3 + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^4 + 2*(a^3*b^4*c^3 - 3*
a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x^2), 1/8*((3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^3 + (3*a^
2*b^2*c^2 - 10*a^3*b*c*d + 15*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 15*a^2*b^2*d^2)*x^4 + 2*(3*a*b^3*c^2 - 10*
a^2*b^2*c*d + 15*a^3*b*d^2)*x^2)*sqrt(b/a)*arctan(x*sqrt(b/a)) - 8*(a^2*b^2*d^2*x^4 + 2*a^3*b*d^2*x^2 + a^4*d^
2)*sqrt(d/c)*arctan(x*sqrt(d/c)) + (5*a*b^3*c^2 - 14*a^2*b^2*c*d + 9*a^3*b*d^2)*x)/(a^4*b^3*c^3 - 3*a^5*b^2*c^
2*d + 3*a^6*b*c*d^2 - a^7*d^3 + (a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x^4 + 2*(a^3*b
^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x^2)]

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giac [A]  time = 0.58, size = 218, normalized size = 1.35 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{3} c^{2} - 10 \, a b^{2} c d + 15 \, a^{2} b d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} + \frac {3 \, b^{3} c x^{3} - 7 \, a b^{2} d x^{3} + 5 \, a b^{2} c x - 9 \, a^{2} b d x}{8 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} {\left (b x^{2} + a\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^3/(d*x^2+c),x, algorithm="giac")

[Out]

-d^3*arctan(d*x/sqrt(c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)) + 1/8*(3*b^3*c^2 -
10*a*b^2*c*d + 15*a^2*b*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*
sqrt(a*b)) + 1/8*(3*b^3*c*x^3 - 7*a*b^2*d*x^3 + 5*a*b^2*c*x - 9*a^2*b*d*x)/((a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d
^2)*(b*x^2 + a)^2)

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maple [B]  time = 0.01, size = 309, normalized size = 1.92 \begin {gather*} \frac {5 b^{3} c d \,x^{3}}{4 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )^{2} a}-\frac {3 b^{4} c^{2} x^{3}}{8 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {7 b^{2} d^{2} x^{3}}{8 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )^{2}}-\frac {9 a b \,d^{2} x}{8 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )^{2}}-\frac {5 b^{3} c^{2} x}{8 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )^{2} a}+\frac {7 b^{2} c d x}{4 \left (a d -b c \right )^{3} \left (b \,x^{2}+a \right )^{2}}+\frac {5 b^{2} c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{4 \left (a d -b c \right )^{3} \sqrt {a b}\, a}-\frac {3 b^{3} c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \left (a d -b c \right )^{3} \sqrt {a b}\, a^{2}}-\frac {15 b \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \left (a d -b c \right )^{3} \sqrt {a b}}+\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{3} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^3/(d*x^2+c),x)

[Out]

d^3/(a*d-b*c)^3/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)-7/8*b^2/(a*d-b*c)^3/(b*x^2+a)^2*x^3*d^2+5/4*b^3/(a*d-b*c
)^3/(b*x^2+a)^2/a*x^3*c*d-3/8*b^4/(a*d-b*c)^3/(b*x^2+a)^2/a^2*x^3*c^2-9/8*b/(a*d-b*c)^3/(b*x^2+a)^2*x*a*d^2+7/
4*b^2/(a*d-b*c)^3/(b*x^2+a)^2*x*c*d-5/8*b^3/(a*d-b*c)^3/(b*x^2+a)^2*x/a*c^2-15/8*b/(a*d-b*c)^3/(a*b)^(1/2)*arc
tan(1/(a*b)^(1/2)*b*x)*d^2+5/4*b^2/(a*d-b*c)^3/a/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c*d-3/8*b^3/(a*d-b*c)^3
/a^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c^2

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maxima [A]  time = 3.16, size = 278, normalized size = 1.73 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{3} c^{2} - 10 \, a b^{2} c d + 15 \, a^{2} b d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} + \frac {{\left (3 \, b^{3} c - 7 \, a b^{2} d\right )} x^{3} + {\left (5 \, a b^{2} c - 9 \, a^{2} b d\right )} x}{8 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} + {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^3*arctan(d*x/sqrt(c*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c*d)) + 1/8*(3*b^3*c^2 -
10*a*b^2*c*d + 15*a^2*b*d^2)*arctan(b*x/sqrt(a*b))/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*
sqrt(a*b)) + 1/8*((3*b^3*c - 7*a*b^2*d)*x^3 + (5*a*b^2*c - 9*a^2*b*d)*x)/(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2
+ (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x^4 + 2*(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*x^2)

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mupad [B]  time = 6.89, size = 6033, normalized size = 37.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)^3*(c + d*x^2)),x)

[Out]

((x^3*(3*b^3*c - 7*a*b^2*d))/(8*a^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(5*b^2*c - 9*a*b*d))/(8*a*(a^2*d^2 +
b^2*c^2 - 2*a*b*c*d)))/(a^2 + b^2*x^4 + 2*a*b*x^2) + (atan(((((x*(289*a^4*b^3*d^7 + 9*b^7*c^4*d^3 - 60*a*b^6*
c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c
^2*d^2 - 4*a^7*b*c*d^3)) - ((-c*d^5)^(1/2)*((256*a^10*b^2*d^10 - 1760*a^9*b^3*c*d^9 + 96*a^2*b^10*c^8*d^2 - 80
0*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d^6 - 9056*a^7*b^5*c^3*d^7
+ 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^
3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) - (x*(-c*d^5)^(1/2)*(256*a^11*b^2*d^9 - 1280*a^10*b^3*c*d^8 + 256*a^4
*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280*a^7*b^6*c^4*d^5 - 1280*a^8*b^5*c^3*d^6 + 230
4*a^9*b^4*c^2*d^7))/(64*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5
*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3))))/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d))
)*(-c*d^5)^(1/2)*1i)/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)) + (((x*(289*a^4*b^3*d^7 + 9*b
^7*c^4*d^3 - 60*a*b^6*c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b
^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)) + ((-c*d^5)^(1/2)*((256*a^10*b^2*d^10 - 1760*a^9*b^3*c*d^9 + 96
*a^2*b^10*c^8*d^2 - 800*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d^6 -
9056*a^7*b^5*c^3*d^7 + 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d
^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) + (x*(-c*d^5)^(1/2)*(256*a^11*b^2*d^9 - 1280*a^
10*b^3*c*d^8 + 256*a^4*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280*a^7*b^6*c^4*d^5 - 1280
*a^8*b^5*c^3*d^6 + 2304*a^9*b^4*c^2*d^7))/(64*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(a^8*d^4
+ a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3))))/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2
*d^2 - 3*a*b^2*c^3*d)))*(-c*d^5)^(1/2)*1i)/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)))/((105*
a^3*b^3*d^8 - 9*b^6*c^3*d^5 + 51*a*b^5*c^2*d^6 - 115*a^2*b^4*c*d^7)/(32*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^
5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) - (((x*(289*a^4*b^3*d^7 +
9*b^7*c^4*d^3 - 60*a*b^6*c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b^4*c^4 - 4*a
^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)) - ((-c*d^5)^(1/2)*((256*a^10*b^2*d^10 - 1760*a^9*b^3*c*d^9
+ 96*a^2*b^10*c^8*d^2 - 800*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d
^6 - 9056*a^7*b^5*c^3*d^7 + 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c
^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) - (x*(-c*d^5)^(1/2)*(256*a^11*b^2*d^9 - 128
0*a^10*b^3*c*d^8 + 256*a^4*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280*a^7*b^6*c^4*d^5 -
1280*a^8*b^5*c^3*d^6 + 2304*a^9*b^4*c^2*d^7))/(64*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)*(a^8
*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3))))/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b
*c^2*d^2 - 3*a*b^2*c^3*d)))*(-c*d^5)^(1/2))/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)) + (((x
*(289*a^4*b^3*d^7 + 9*b^7*c^4*d^3 - 60*a*b^6*c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4
+ a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)) + ((-c*d^5)^(1/2)*((256*a^10*b^2*d^10 -
1760*a^9*b^3*c*d^9 + 96*a^2*b^10*c^8*d^2 - 800*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*b^7*c^5*d^5 +
9760*a^6*b^6*c^4*d^6 - 9056*a^7*b^5*c^3*d^7 + 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c
^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) + (x*(-c*d^5)^(1/2)*(256
*a^11*b^2*d^9 - 1280*a^10*b^3*c*d^8 + 256*a^4*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280
*a^7*b^6*c^4*d^5 - 1280*a^8*b^5*c^3*d^6 + 2304*a^9*b^4*c^2*d^7))/(64*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 -
3*a*b^2*c^3*d)*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3))))/(2*(b^3*c^4 -
a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)))*(-c*d^5)^(1/2))/(2*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a
*b^2*c^3*d))))*(-c*d^5)^(1/2)*1i)/(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d) + (atan(((((x*(289*a
^4*b^3*d^7 + 9*b^7*c^4*d^3 - 60*a*b^6*c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b
^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)) - (((256*a^10*b^2*d^10 - 1760*a^9*b^3*c*d^9 + 9
6*a^2*b^10*c^8*d^2 - 800*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d^6
- 9056*a^7*b^5*c^3*d^7 + 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*
d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) - (x*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 -
10*a*b*c*d)*(256*a^11*b^2*d^9 - 1280*a^10*b^3*c*d^8 + 256*a^4*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^
7*c^5*d^4 - 1280*a^7*b^6*c^4*d^5 - 1280*a^8*b^5*c^3*d^6 + 2304*a^9*b^4*c^2*d^7))/(512*(a^8*d^3 - a^5*b^3*c^3 +
3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3
)))*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d))/(16*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7
*b*c*d^2)))*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d)*1i)/(16*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^
2*d - 3*a^7*b*c*d^2)) + (((x*(289*a^4*b^3*d^7 + 9*b^7*c^4*d^3 - 60*a*b^6*c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2
*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)) + (((256*a^1
0*b^2*d^10 - 1760*a^9*b^3*c*d^9 + 96*a^2*b^10*c^8*d^2 - 800*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*
b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d^6 - 9056*a^7*b^5*c^3*d^7 + 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6
- 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) + (x*(-a^5*
b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d)*(256*a^11*b^2*d^9 - 1280*a^10*b^3*c*d^8 + 256*a^4*b^9*c^7*d^2 -
1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280*a^7*b^6*c^4*d^5 - 1280*a^8*b^5*c^3*d^6 + 2304*a^9*b^4*c^2*
d^7))/(512*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d
+ 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)))*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d))/(16*(a^8*d^3 - a^
5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d)*1i)/(16*(a
^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))/((105*a^3*b^3*d^8 - 9*b^6*c^3*d^5 + 51*a*b^5*c^2*d^6
- 115*a^2*b^4*c*d^7)/(32*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3
+ 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) - (((x*(289*a^4*b^3*d^7 + 9*b^7*c^4*d^3 - 60*a*b^6*c^3*d^4 - 300*a^3*b^
4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d
^3)) - (((256*a^10*b^2*d^10 - 1760*a^9*b^3*c*d^9 + 96*a^2*b^10*c^8*d^2 - 800*a^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^
6*d^4 - 6816*a^5*b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d^6 - 9056*a^7*b^5*c^3*d^7 + 5280*a^8*b^4*c^2*d^8)/(64*(a^10*d
^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 + 15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*
d^5)) - (x*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d)*(256*a^11*b^2*d^9 - 1280*a^10*b^3*c*d^8 + 256*
a^4*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280*a^7*b^6*c^4*d^5 - 1280*a^8*b^5*c^3*d^6 +
2304*a^9*b^4*c^2*d^7))/(512*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)*(a^8*d^4 + a^4*b^4*c^4 -
4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)))*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d))/
(16*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*
b*c*d))/(16*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)) + (((x*(289*a^4*b^3*d^7 + 9*b^7*c^4*d^3
- 60*a*b^6*c^3*d^4 - 300*a^3*b^4*c*d^6 + 190*a^2*b^5*c^2*d^5))/(32*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d +
6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)) + (((256*a^10*b^2*d^10 - 1760*a^9*b^3*c*d^9 + 96*a^2*b^10*c^8*d^2 - 800*a
^3*b^9*c^7*d^3 + 3040*a^4*b^8*c^6*d^4 - 6816*a^5*b^7*c^5*d^5 + 9760*a^6*b^6*c^4*d^6 - 9056*a^7*b^5*c^3*d^7 + 5
280*a^8*b^4*c^2*d^8)/(64*(a^10*d^6 + a^4*b^6*c^6 - 6*a^5*b^5*c^5*d + 15*a^6*b^4*c^4*d^2 - 20*a^7*b^3*c^3*d^3 +
15*a^8*b^2*c^2*d^4 - 6*a^9*b*c*d^5)) + (x*(-a^5*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d)*(256*a^11*b^2*
d^9 - 1280*a^10*b^3*c*d^8 + 256*a^4*b^9*c^7*d^2 - 1280*a^5*b^8*c^6*d^3 + 2304*a^6*b^7*c^5*d^4 - 1280*a^7*b^6*c
^4*d^5 - 1280*a^8*b^5*c^3*d^6 + 2304*a^9*b^4*c^2*d^7))/(512*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b
*c*d^2)*(a^8*d^4 + a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3)))*(-a^5*b)^(1/2)*(15*a^2
*d^2 + 3*b^2*c^2 - 10*a*b*c*d))/(16*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2)))*(-a^5*b)^(1/2)
*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d))/(16*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^2))))*(-a^5
*b)^(1/2)*(15*a^2*d^2 + 3*b^2*c^2 - 10*a*b*c*d)*1i)/(8*(a^8*d^3 - a^5*b^3*c^3 + 3*a^6*b^2*c^2*d - 3*a^7*b*c*d^
2))

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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(1/(b*x**2+a)**3/(d*x**2+c),x)

[Out]

Timed out

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