3.3.54 \(\int \frac {x^2}{(a+b x^2) (c+d x^2)^3} \, dx\) [254]

Optimal. Leaf size=155 \[ \frac {x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {a} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d} (b c-a d)^3} \]

[Out]

1/4*x/(-a*d+b*c)/(d*x^2+c)^2+1/8*(a*d+3*b*c)*x/c/(-a*d+b*c)^2/(d*x^2+c)-b^(3/2)*arctan(x*b^(1/2)/a^(1/2))*a^(1
/2)/(-a*d+b*c)^3+1/8*(-a^2*d^2+6*a*b*c*d+3*b^2*c^2)*arctan(x*d^(1/2)/c^(1/2))/c^(3/2)/(-a*d+b*c)^3/d^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {482, 541, 536, 211} \begin {gather*} \frac {\left (-a^2 d^2+6 a b c d+3 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d} (b c-a d)^3}-\frac {\sqrt {a} b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {x (a d+3 b c)}{8 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {x}{4 \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(4*(b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(8*c*(b*c - a*d)^2*(c + d*x^2)) - (Sqrt[a]*b^(3/2)*ArcTan[
(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(3/
2)*Sqrt[d]*(b*c - a*d)^3)

Rule 211

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

Rule 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

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 541

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 {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=\frac {x}{4 (b c-a d) \left (c+d x^2\right )^2}-\frac {\int \frac {a-3 b x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{4 (b c-a d)}\\ &=\frac {x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\int \frac {a (5 b c-a d)-b (3 b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 c (b c-a d)^2}\\ &=\frac {x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (a b^2\right ) \int \frac {1}{a+b x^2} \, dx}{(b c-a d)^3}+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 c (b c-a d)^3}\\ &=\frac {x}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(3 b c+a d) x}{8 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\sqrt {a} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{3/2} \sqrt {d} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 151, normalized size = 0.97 \begin {gather*} \frac {1}{8} \left (\frac {2 x}{(b c-a d) \left (c+d x^2\right )^2}+\frac {(3 b c+a d) x}{c (b c-a d)^2 \left (c+d x^2\right )}+\frac {8 \sqrt {a} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(-b c+a d)^3}+\frac {\left (3 b^2 c^2+6 a b c d-a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} \sqrt {d} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*x)/((b*c - a*d)*(c + d*x^2)^2) + ((3*b*c + a*d)*x)/(c*(b*c - a*d)^2*(c + d*x^2)) + (8*Sqrt[a]*b^(3/2)*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/(-(b*c) + a*d)^3 + ((3*b^2*c^2 + 6*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c
^(3/2)*Sqrt[d]*(b*c - a*d)^3))/8

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Maple [A]
time = 0.20, size = 151, normalized size = 0.97

method	result	size
default	\(\frac {a \,b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{3} \sqrt {a b}}+\frac {\frac {\frac {d \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) x^{3}}{8 c}+\left (\frac {3}{4} a b c d -\frac {5}{8} b^{2} c^{2}-\frac {1}{8} a^{2} d^{2}\right ) x}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (a^{2} d^{2}-6 a b c d -3 b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 c \sqrt {c d}}}{\left (a d -b c \right )^{3}}\)	\(151\)
risch	\(\frac {\frac {d \left (a d +3 b c \right ) x^{3}}{8 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a d -5 b c \right ) x}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\sqrt {-a b}\, b \ln \left (\left (-64 \left (-a b \right )^{\frac {3}{2}} a b \,c^{2} d^{2}-64 \left (-a b \right )^{\frac {3}{2}} b^{2} c^{3} d -\sqrt {-a b}\, a^{4} d^{4}+12 \sqrt {-a b}\, a^{3} b c \,d^{3}-94 \sqrt {-a b}\, a^{2} b^{2} c^{2} d^{2}-36 \sqrt {-a b}\, a \,b^{3} c^{3} d -9 \sqrt {-a b}\, b^{4} c^{4}\right ) x +d^{4} a^{5}-12 c \,d^{3} b \,a^{4}+30 b^{2} c^{2} d^{2} a^{3}-28 b^{3} c^{3} d \,a^{2}+9 b^{4} c^{4} a \right )}{2 \left (a d -b c \right )^{3}}-\frac {\sqrt {-a b}\, b \ln \left (\left (64 \left (-a b \right )^{\frac {3}{2}} a b \,c^{2} d^{2}+64 \left (-a b \right )^{\frac {3}{2}} b^{2} c^{3} d +\sqrt {-a b}\, a^{4} d^{4}-12 \sqrt {-a b}\, a^{3} b c \,d^{3}+94 \sqrt {-a b}\, a^{2} b^{2} c^{2} d^{2}+36 \sqrt {-a b}\, a \,b^{3} c^{3} d +9 \sqrt {-a b}\, b^{4} c^{4}\right ) x +d^{4} a^{5}-12 c \,d^{3} b \,a^{4}+30 b^{2} c^{2} d^{2} a^{3}-28 b^{3} c^{3} d \,a^{2}+9 b^{4} c^{4} a \right )}{2 \left (a d -b c \right )^{3}}-\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2} d^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3} c}+\frac {3 \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b d}{8 \sqrt {-c d}\, \left (a d -b c \right )^{3}}+\frac {3 c \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3}}+\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2} d^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3} c}-\frac {3 \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b d}{8 \sqrt {-c d}\, \left (a d -b c \right )^{3}}-\frac {3 c \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3}}\)	\(699\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.55, size = 266, normalized size = 1.72 \begin {gather*} -\frac {a b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\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 {a b}} + \frac {{\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt {c d}} + \frac {{\left (3 \, b c d + a d^{2}\right )} x^{3} + {\left (5 \, b c^{2} - a c d\right )} x}{8 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (133) = 266\).
time = 1.48, size = 1587, normalized size = 10.24 \begin {gather*} \left [\frac {2 \, {\left (3 \, b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} - a^{2} c d^{4}\right )} x^{3} - 8 \, {\left (b c^{2} d^{3} x^{4} + 2 \, b c^{3} d^{2} x^{2} + b c^{4} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (3 \, b^{2} c^{4} + 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (3 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (5 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x}{16 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4} + {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2}\right )}}, \frac {{\left (3 \, b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} - a^{2} c d^{4}\right )} x^{3} + {\left (3 \, b^{2} c^{4} + 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (3 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - 4 \, {\left (b c^{2} d^{3} x^{4} + 2 \, b c^{3} d^{2} x^{2} + b c^{4} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (5 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x}{8 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4} + {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2}\right )}}, \frac {2 \, {\left (3 \, b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} - a^{2} c d^{4}\right )} x^{3} - 16 \, {\left (b c^{2} d^{3} x^{4} + 2 \, b c^{3} d^{2} x^{2} + b c^{4} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (3 \, b^{2} c^{4} + 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (3 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (5 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x}{16 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4} + {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2}\right )}}, \frac {{\left (3 \, b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} - a^{2} c d^{4}\right )} x^{3} - 8 \, {\left (b c^{2} d^{3} x^{4} + 2 \, b c^{3} d^{2} x^{2} + b c^{4} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, b^{2} c^{4} + 6 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (3 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (5 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x}{8 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4} + {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
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**2/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.77, size = 206, normalized size = 1.33 \begin {gather*} -\frac {a b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\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 {a b}} + \frac {{\left (3 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt {c d}} + \frac {3 \, b c d x^{3} + a d^{2} x^{3} + 5 \, b c^{2} x - a c d x}{8 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.06, size = 2500, normalized size = 16.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-a*b^3)^(1/2)*((((160*a*b^9*c^8*d^2 - 32*a^8*b^2*c*d^9 - 992*a^2*b^8*c^7*d^3 + 2592*a^3*b^7*c^6*d^4 -
3680*a^4*b^6*c^5*d^5 + 3040*a^5*b^5*c^4*d^6 - 1440*a^6*b^4*c^3*d^7 + 352*a^7*b^3*c^2*d^8)/(64*(b^6*c^8 + a^6*
c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (
x*(-a*b^3)^(1/2)*(256*b^9*c^9*d^2 - 1280*a*b^8*c^8*d^3 + 2304*a^2*b^7*c^7*d^4 - 1280*a^3*b^6*c^6*d^5 - 1280*a^
4*b^5*c^5*d^6 + 2304*a^5*b^4*c^4*d^7 - 1280*a^6*b^3*c^3*d^8 + 256*a^7*b^2*c^2*d^9))/(64*(a^3*d^3 - b^3*c^3 + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d)))*
(-a*b^3)^(1/2))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (x*(9*b^7*c^4*d + a^4*b^3*d^5 + 36*a
*b^6*c^3*d^2 - 12*a^3*b^4*c*d^4 + 94*a^2*b^5*c^2*d^3))/(32*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^
2*c^4*d^2 - 4*a*b^3*c^5*d)))*1i)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - ((-a*b^3)^(1/2)*(((
(160*a*b^9*c^8*d^2 - 32*a^8*b^2*c*d^9 - 992*a^2*b^8*c^7*d^3 + 2592*a^3*b^7*c^6*d^4 - 3680*a^4*b^6*c^5*d^5 + 30
40*a^5*b^5*c^4*d^6 - 1440*a^6*b^4*c^3*d^7 + 352*a^7*b^3*c^2*d^8)/(64*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5
+ 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + (x*(-a*b^3)^(1/2)*(256*b^9*
c^9*d^2 - 1280*a*b^8*c^8*d^3 + 2304*a^2*b^7*c^7*d^4 - 1280*a^3*b^6*c^6*d^5 - 1280*a^4*b^5*c^5*d^6 + 2304*a^5*b
^4*c^4*d^7 - 1280*a^6*b^3*c^3*d^8 + 256*a^7*b^2*c^2*d^9))/(64*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d
^2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d)))*(-a*b^3)^(1/2))/(2*(a^3*d^
3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(9*b^7*c^4*d + a^4*b^3*d^5 + 36*a*b^6*c^3*d^2 - 12*a^3*b^4*
c*d^4 + 94*a^2*b^5*c^2*d^3))/(32*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d)
))*1i)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/((3*a^3*b^5*c*d^3 - a^4*b^4*d^4 + 21*a^2*b^6*c
^2*d^2 + 9*a*b^7*c^3*d)/(32*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3
+ 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) + ((-a*b^3)^(1/2)*((((160*a*b^9*c^8*d^2 - 32*a^8*b^2*c*d^9 - 992*a^2*b
^8*c^7*d^3 + 2592*a^3*b^7*c^6*d^4 - 3680*a^4*b^6*c^5*d^5 + 3040*a^5*b^5*c^4*d^6 - 1440*a^6*b^4*c^3*d^7 + 352*a
^7*b^3*c^2*d^8)/(64*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^
4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (x*(-a*b^3)^(1/2)*(256*b^9*c^9*d^2 - 1280*a*b^8*c^8*d^3 + 2304*a^2*b^7*c^7*d
^4 - 1280*a^3*b^6*c^6*d^5 - 1280*a^4*b^5*c^5*d^6 + 2304*a^5*b^4*c^4*d^7 - 1280*a^6*b^3*c^3*d^8 + 256*a^7*b^2*c
^2*d^9))/(64*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*
a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d)))*(-a*b^3)^(1/2))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (
x*(9*b^7*c^4*d + a^4*b^3*d^5 + 36*a*b^6*c^3*d^2 - 12*a^3*b^4*c*d^4 + 94*a^2*b^5*c^2*d^3))/(32*(b^4*c^6 + a^4*c
^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2)) + ((-a*b^3)^(1/2)*((((160*a*b^9*c^8*d^2 - 32*a^8*b^2*c*d^9 - 992*a^2*b^8*c^7*d^3 + 2592*a^3*b^7*c^6
*d^4 - 3680*a^4*b^6*c^5*d^5 + 3040*a^5*b^5*c^4*d^6 - 1440*a^6*b^4*c^3*d^7 + 352*a^7*b^3*c^2*d^8)/(64*(b^6*c^8
+ a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d
)) + (x*(-a*b^3)^(1/2)*(256*b^9*c^9*d^2 - 1280*a*b^8*c^8*d^3 + 2304*a^2*b^7*c^7*d^4 - 1280*a^3*b^6*c^6*d^5 - 1
280*a^4*b^5*c^5*d^6 + 2304*a^5*b^4*c^4*d^7 - 1280*a^6*b^3*c^3*d^8 + 256*a^7*b^2*c^2*d^9))/(64*(a^3*d^3 - b^3*c
^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5
*d)))*(-a*b^3)^(1/2))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(9*b^7*c^4*d + a^4*b^3*d^5
+ 36*a*b^6*c^3*d^2 - 12*a^3*b^4*c*d^4 + 94*a^2*b^5*c^2*d^3))/(32*(b^4*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*
a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d))))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))*(-a*b^3)^(1/2)*1
i)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) - ((x*(a*d - 5*b*c))/(8*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)
) - (x^3*(a*d^2 + 3*b*c*d))/(8*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(c^2 + d^2*x^4 + 2*c*d*x^2) - (atan((((-c^3
*d)^(1/2)*((x*(9*b^7*c^4*d + a^4*b^3*d^5 + 36*a*b^6*c^3*d^2 - 12*a^3*b^4*c*d^4 + 94*a^2*b^5*c^2*d^3))/(32*(b^4
*c^6 + a^4*c^2*d^4 - 4*a^3*b*c^3*d^3 + 6*a^2*b^2*c^4*d^2 - 4*a*b^3*c^5*d)) - (((160*a*b^9*c^8*d^2 - 32*a^8*b^2
*c*d^9 - 992*a^2*b^8*c^7*d^3 + 2592*a^3*b^7*c^6*d^4 - 3680*a^4*b^6*c^5*d^5 + 3040*a^5*b^5*c^4*d^6 - 1440*a^6*b
^4*c^3*d^7 + 352*a^7*b^3*c^2*d^8)/(64*(b^6*c^8 + a^6*c^2*d^6 - 6*a^5*b*c^3*d^5 + 15*a^2*b^4*c^6*d^2 - 20*a^3*b
^3*c^5*d^3 + 15*a^4*b^2*c^4*d^4 - 6*a*b^5*c^7*d)) - (x*(-c^3*d)^(1/2)*(3*b^2*c^2 - a^2*d^2 + 6*a*b*c*d)*(256*b
^9*c^9*d^2 - 1280*a*b^8*c^8*d^3 + 2304*a^2*b^7*c^7*d^4 - 1280*a^3*b^6*c^6*d^5 - 1280*a^4*b^5*c^5*d^6 + 2304*a^
5*b^4*c^4*d^7 - 1280*a^6*b^3*c^3*d^8 + 256*a^7*...

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