3.3.52 \(\int \frac {x^4}{(a+b x^2) (c+d x^2)^3} \, dx\) [252]

Optimal. Leaf size=157 \[ -\frac {c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^{3/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{3/2} (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 157, 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 = {481, 541, 536, 211} \begin {gather*} \frac {a^{3/2} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{3/2} (b c-a d)^3}+\frac {x (b c-5 a d)}{8 d \left (c+d x^2\right ) (b c-a d)^2}-\frac {c x}{4 d \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-1/4*(c*x)/(d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(8*d*(b*c - a*d)^2*(c + d*x^2)) + (a^(3/2)*Sqrt[b
]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])
/(8*Sqrt[c]*d^(3/2)*(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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=-\frac {c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {\int \frac {a c+(b c-4 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{4 d (b c-a d)}\\ &=-\frac {c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\int \frac {a c (b c+3 a d)+b c (b c-5 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 c d (b c-a d)^2}\\ &=-\frac {c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a^2 b\right ) \int \frac {1}{a+b x^2} \, dx}{(b c-a d)^3}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac {1}{c+d x^2} \, dx}{8 d (b c-a d)^3}\\ &=-\frac {c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac {(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^{3/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 \sqrt {c} d^{3/2} (b c-a d)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 154, normalized size = 0.98 \begin {gather*} \frac {1}{8} \left (\frac {2 c x}{d (-b c+a d) \left (c+d x^2\right )^2}+\frac {(b c-5 a d) x}{d (b c-a d)^2 \left (c+d x^2\right )}+\frac {8 a^{3/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{(b c-a d)^3}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{3/2} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((2*c*x)/(d*(-(b*c) + a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(d*(b*c - a*d)^2*(c + d*x^2)) + (8*a^(3/2)*Sqrt[
b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]
)/(Sqrt[c]*d^(3/2)*(b*c - a*d)^3))/8

________________________________________________________________________________________

Maple [A]
time = 0.19, size = 154, normalized size = 0.98

method	result	size
default	\(-\frac {a^{2} b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right )^{3} \sqrt {a b}}+\frac {\frac {\left (-\frac {5}{8} a^{2} d^{2}+\frac {3}{4} a b c d -\frac {1}{8} b^{2} c^{2}\right ) x^{3}-\frac {c \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x}{8 d}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+6 a b c d -b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 d \sqrt {c d}}}{\left (a d -b c \right )^{3}}\)	\(154\)
risch	\(\frac {-\frac {\left (5 a d -b c \right ) x^{3}}{8 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {c \left (3 a d +b c \right ) x}{8 d \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}\, a \ln \left (\left (-64 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}-64 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}-73 a^{4} \sqrt {-a b}\, d^{4} b -36 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}-30 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}+12 \sqrt {-a b}\, a \,b^{4} c^{3} d -b^{5} c^{4} \sqrt {-a b}\right ) x -9 d^{4} a^{5} b +28 a^{4} b^{2} c \,d^{3}-30 a^{3} b^{3} c^{2} d^{2}+12 a^{2} b^{4} c^{3} d -a \,b^{5} c^{4}\right )}{2 \left (a d -b c \right )^{3}}-\frac {\sqrt {-a b}\, a \ln \left (\left (64 \left (-a b \right )^{\frac {3}{2}} a^{3} d^{4}+64 \left (-a b \right )^{\frac {3}{2}} a^{2} b c \,d^{3}+73 a^{4} \sqrt {-a b}\, d^{4} b +36 \sqrt {-a b}\, a^{3} b^{2} c \,d^{3}+30 \sqrt {-a b}\, a^{2} b^{3} c^{2} d^{2}-12 \sqrt {-a b}\, a \,b^{4} c^{3} d +b^{5} c^{4} \sqrt {-a b}\right ) x -9 d^{4} a^{5} b +28 a^{4} b^{2} c \,d^{3}-30 a^{3} b^{3} c^{2} d^{2}+12 a^{2} b^{4} c^{3} d -a \,b^{5} c^{4}\right )}{2 \left (a d -b c \right )^{3}}-\frac {3 d \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3}}-\frac {3 \ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b c}{8 \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 ) b^{2} c^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3} d}+\frac {3 d \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3}}+\frac {3 \ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a b c}{8 \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 ) b^{2} c^{2}}{16 \sqrt {-c d}\, \left (a d -b c \right )^{3} d}\)	\(712\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.55, size = 264, normalized size = 1.68 \begin {gather*} \frac {a^{2} b \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 (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt {c d}} + \frac {{\left (b c d - 5 \, a d^{2}\right )} x^{3} - {\left (b c^{2} + 3 \, a c d\right )} x}{8 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + {\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (135) = 270\).
time = 1.70, size = 1573, normalized size = 10.02 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c^{3} d^{2} - 6 \, a b c^{2} d^{3} + 5 \, a^{2} c d^{4}\right )} x^{3} - 8 \, {\left (a c d^{4} x^{4} + 2 \, a c^{2} d^{3} x^{2} + a c^{3} d^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (b^{2} c^{4} - 6 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - 3 \, 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 (b^{2} c^{4} d + 2 \, a b c^{3} d^{2} - 3 \, a^{2} c^{2} d^{3}\right )} x}{16 \, {\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} + {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 2 \, {\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^{2}\right )}}, \frac {{\left (b^{2} c^{3} d^{2} - 6 \, a b c^{2} d^{3} + 5 \, a^{2} c d^{4}\right )} x^{3} + {\left (b^{2} c^{4} - 6 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - 4 \, {\left (a c d^{4} x^{4} + 2 \, a c^{2} d^{3} x^{2} + a c^{3} d^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - {\left (b^{2} c^{4} d + 2 \, a b c^{3} d^{2} - 3 \, a^{2} c^{2} d^{3}\right )} x}{8 \, {\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} + {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 2 \, {\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^{2}\right )}}, \frac {2 \, {\left (b^{2} c^{3} d^{2} - 6 \, a b c^{2} d^{3} + 5 \, a^{2} c d^{4}\right )} x^{3} + 16 \, {\left (a c d^{4} x^{4} + 2 \, a c^{2} d^{3} x^{2} + a c^{3} d^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (b^{2} c^{4} - 6 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - 3 \, 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 (b^{2} c^{4} d + 2 \, a b c^{3} d^{2} - 3 \, a^{2} c^{2} d^{3}\right )} x}{16 \, {\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} + {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 2 \, {\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^{2}\right )}}, \frac {{\left (b^{2} c^{3} d^{2} - 6 \, a b c^{2} d^{3} + 5 \, a^{2} c d^{4}\right )} x^{3} + 8 \, {\left (a c d^{4} x^{4} + 2 \, a c^{2} d^{3} x^{2} + a c^{3} d^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (b^{2} c^{4} - 6 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 6 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b^{2} c^{4} d + 2 \, a b c^{3} d^{2} - 3 \, a^{2} c^{2} d^{3}\right )} x}{8 \, {\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} + {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} x^{4} + 2 \, {\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^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.90, size = 204, normalized size = 1.30 \begin {gather*} \frac {a^{2} b \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 (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt {c d}} + \frac {b c d x^{3} - 5 \, a d^{2} x^{3} - b c^{2} x - 3 \, a c d x}{8 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.07, size = 2500, normalized size = 15.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x^2)*(c + d*x^2)^3),x)

[Out]

(atan((((-c*d^3)^(1/2)*((x*(b^7*c^4 + 73*a^4*b^3*d^4 + 36*a^3*b^4*c*d^3 + 30*a^2*b^5*c^2*d^2 - 12*a*b^6*c^3*d)
)/(32*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)) - (((96*a^8*b^2*d^9 + 32*a*
b^9*c^7*d^2 - 544*a^7*b^3*c*d^8 - 96*a^2*b^8*c^6*d^3 - 96*a^3*b^7*c^5*d^4 + 800*a^4*b^6*c^4*d^5 - 1440*a^5*b^5
*c^3*d^6 + 1248*a^6*b^4*c^2*d^7)/(64*(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*
c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6)) - (x*(-c*d^3)^(1/2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d)*(256*a^7*
b^2*d^10 + 256*b^9*c^7*d^3 - 1280*a*b^8*c^6*d^4 - 1280*a^6*b^3*c*d^9 + 2304*a^2*b^7*c^5*d^5 - 1280*a^3*b^6*c^4
*d^6 - 1280*a^4*b^5*c^3*d^7 + 2304*a^5*b^4*c^2*d^8))/(512*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b
*c^2*d^5)*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a^2*
d^2 - b^2*c^2 + 6*a*b*c*d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)))*(3*a^2*d^2 - b
^2*c^2 + 6*a*b*c*d)*1i)/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)) + ((-c*d^3)^(1/2)*(
(x*(b^7*c^4 + 73*a^4*b^3*d^4 + 36*a^3*b^4*c*d^3 + 30*a^2*b^5*c^2*d^2 - 12*a*b^6*c^3*d))/(32*(a^4*d^5 + b^4*c^4
*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)) + (((96*a^8*b^2*d^9 + 32*a*b^9*c^7*d^2 - 544*a^7*b^
3*c*d^8 - 96*a^2*b^8*c^6*d^3 - 96*a^3*b^7*c^5*d^4 + 800*a^4*b^6*c^4*d^5 - 1440*a^5*b^5*c^3*d^6 + 1248*a^6*b^4*
c^2*d^7)/(64*(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2
*d^5 - 6*a^5*b*c*d^6)) + (x*(-c*d^3)^(1/2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d)*(256*a^7*b^2*d^10 + 256*b^9*c^7*d
^3 - 1280*a*b^8*c^6*d^4 - 1280*a^6*b^3*c*d^9 + 2304*a^2*b^7*c^5*d^5 - 1280*a^3*b^6*c^4*d^6 - 1280*a^4*b^5*c^3*
d^7 + 2304*a^5*b^4*c^2*d^8))/(512*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)*(a^4*d^5 + b^4
*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*
d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)))*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d)*1i)/
(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)))/((a^2*b^6*c^3 + 15*a^5*b^3*d^3 - 11*a^3*b^
5*c^2*d + 27*a^4*b^4*c*d^2)/(32*(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d
^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6)) - ((-c*d^3)^(1/2)*((x*(b^7*c^4 + 73*a^4*b^3*d^4 + 36*a^3*b^4*c*d^3 +
30*a^2*b^5*c^2*d^2 - 12*a*b^6*c^3*d))/(32*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*
b*c*d^4)) - (((96*a^8*b^2*d^9 + 32*a*b^9*c^7*d^2 - 544*a^7*b^3*c*d^8 - 96*a^2*b^8*c^6*d^3 - 96*a^3*b^7*c^5*d^4
+ 800*a^4*b^6*c^4*d^5 - 1440*a^5*b^5*c^3*d^6 + 1248*a^6*b^4*c^2*d^7)/(64*(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d
^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6)) - (x*(-c*d^3)^(1/2)*(3*a^2
*d^2 - b^2*c^2 + 6*a*b*c*d)*(256*a^7*b^2*d^10 + 256*b^9*c^7*d^3 - 1280*a*b^8*c^6*d^4 - 1280*a^6*b^3*c*d^9 + 23
04*a^2*b^7*c^5*d^5 - 1280*a^3*b^6*c^4*d^6 - 1280*a^4*b^5*c^3*d^7 + 2304*a^5*b^4*c^2*d^8))/(512*(a^3*c*d^6 - b^
3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*
a^3*b*c*d^4)))*(-c*d^3)^(1/2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^
4 - 3*a^2*b*c^2*d^5)))*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a
^2*b*c^2*d^5)) + ((-c*d^3)^(1/2)*((x*(b^7*c^4 + 73*a^4*b^3*d^4 + 36*a^3*b^4*c*d^3 + 30*a^2*b^5*c^2*d^2 - 12*a*
b^6*c^3*d))/(32*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)) + (((96*a^8*b^2*d
^9 + 32*a*b^9*c^7*d^2 - 544*a^7*b^3*c*d^8 - 96*a^2*b^8*c^6*d^3 - 96*a^3*b^7*c^5*d^4 + 800*a^4*b^6*c^4*d^5 - 14
40*a^5*b^5*c^3*d^6 + 1248*a^6*b^4*c^2*d^7)/(64*(a^6*d^7 + b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 2
0*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6)) + (x*(-c*d^3)^(1/2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d)
*(256*a^7*b^2*d^10 + 256*b^9*c^7*d^3 - 1280*a*b^8*c^6*d^4 - 1280*a^6*b^3*c*d^9 + 2304*a^2*b^7*c^5*d^5 - 1280*a
^3*b^6*c^4*d^6 - 1280*a^4*b^5*c^3*d^7 + 2304*a^5*b^4*c^2*d^8))/(512*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4
- 3*a^2*b*c^2*d^5)*(a^4*d^5 + b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4)))*(-c*d^3)^(1/
2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)))*(3*a
^2*d^2 - b^2*c^2 + 6*a*b*c*d))/(16*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5))))*(-c*d^3)^(
1/2)*(3*a^2*d^2 - b^2*c^2 + 6*a*b*c*d)*1i)/(8*(a^3*c*d^6 - b^3*c^4*d^3 + 3*a*b^2*c^3*d^4 - 3*a^2*b*c^2*d^5)) -
(atan((((-a^3*b)^(1/2)*(((-a^3*b)^(1/2)*((96*a^8*b^2*d^9 + 32*a*b^9*c^7*d^2 - 544*a^7*b^3*c*d^8 - 96*a^2*b^8*
c^6*d^3 - 96*a^3*b^7*c^5*d^4 + 800*a^4*b^6*c^4*d^5 - 1440*a^5*b^5*c^3*d^6 + 1248*a^6*b^4*c^2*d^7)/(64*(a^6*d^7
+ b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6)
) - (x*(-a^3*b)^(1/2)*(256*a^7*b^2*d^10 + 256*b...

________________________________________________________________________________________