\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^3} \, dx\) [88]

Optimal result

Integrand size = 21, antiderivative size = 225 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) (2 b c+3 a d) x \sqrt {a+b x^2}}{8 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{7/2}} \]

[Out]

-3/8*d*(a^2*d^2-4*a*b*c*d+8*b^2*c^2)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(5/2)/(-a*d+b*c)^(7
/2)-1/4*d*x/c/(-a*d+b*c)/(d*x^2+c)^2/(b*x^2+a)^(1/2)+1/4*b*(a*d+4*b*c)*x/a/c/(-a*d+b*c)^2/(d*x^2+c)/(b*x^2+a)^
(1/2)+1/8*d*(-a*d+4*b*c)*(3*a*d+2*b*c)*x*(b*x^2+a)^(1/2)/a/c^2/(-a*d+b*c)^3/(d*x^2+c)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12, 385, 214} \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=-\frac {3 d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{7/2}}+\frac {d x \sqrt {a+b x^2} (4 b c-a d) (3 a d+2 b c)}{8 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac {b x (a d+4 b c)}{4 a c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x}{4 c \sqrt {a+b x^2} \left (c+d x^2\right )^2 (b c-a d)} \]

[In]

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

[Out]

-1/4*(d*x)/(c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^2) + (b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*Sqrt[a + b
*x^2]*(c + d*x^2)) + (d*(4*b*c - a*d)*(2*b*c + 3*a*d)*x*Sqrt[a + b*x^2])/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) -
(3*d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*
c - a*d)^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 214

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, 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*} \text {integral}& = -\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {\int \frac {4 b c-3 a d-4 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \, dx}{4 c (b c-a d)} \\ & = -\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {\int \frac {a d (8 b c-3 a d)-2 b d (4 b c+a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{4 a c (b c-a d)^2} \\ & = -\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) (2 b c+3 a d) x \sqrt {a+b x^2}}{8 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\int \frac {3 a d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 a c^2 (b c-a d)^3} \\ & = -\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) (2 b c+3 a d) x \sqrt {a+b x^2}}{8 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (3 d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{8 c^2 (b c-a d)^3} \\ & = -\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) (2 b c+3 a d) x \sqrt {a+b x^2}}{8 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (3 d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 c^2 (b c-a d)^3} \\ & = -\frac {d x}{4 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^2}+\frac {b (4 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) (2 b c+3 a d) x \sqrt {a+b x^2}}{8 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 d \left (8 b^2 c^2-4 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{7/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 13.40 (sec) , antiderivative size = 1392, normalized size of antiderivative = 6.19 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {x \left (-108045 \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}-\frac {324135 d x^2 \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}{c}-\frac {324135 d^2 x^4 \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}{c^2}-\frac {103320 d^3 x^6 \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}{c^3}+42735 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{3/2}+\frac {128205 d x^2 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{3/2}}{c}+\frac {139545 d^2 x^4 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{3/2}}{c^2}+\frac {46200 d^3 x^6 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{3/2}}{c^3}-3864 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{5/2}-\frac {4032 d x^2 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{5/2}}{c}-\frac {4032 d^2 x^4 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{5/2}}{c^2}-\frac {1344 d^3 x^6 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{5/2}}{c^3}+108045 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )+\frac {324135 d x^2 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c}+\frac {324135 d^2 x^4 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^2}+\frac {103320 d^3 x^6 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^3}+\frac {8505 (b c-a d)^2 x^4 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^2 \left (a+b x^2\right )^2}+\frac {17955 d (b c-a d)^2 x^6 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^3 \left (a+b x^2\right )^2}+\frac {21735 d^2 (b c-a d)^2 x^8 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^4 \left (a+b x^2\right )^2}+\frac {7560 d^3 (b c-a d)^2 x^{10} \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^5 \left (a+b x^2\right )^2}-\frac {78750 (b c-a d) x^2 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c \left (a+b x^2\right )}+\frac {236250 d (-b c+a d) x^4 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^2 \left (a+b x^2\right )}+\frac {247590 d^2 (-b c+a d) x^6 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^3 \left (a+b x^2\right )}+\frac {80640 d^3 (-b c+a d) x^8 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c^4 \left (a+b x^2\right )}+64 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{9/2} \, _4F_3\left (2,2,2,\frac {5}{2};1,1,\frac {11}{2};\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )+\frac {192 d x^2 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{9/2} \, _4F_3\left (2,2,2,\frac {5}{2};1,1,\frac {11}{2};\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{c}+\frac {192 d^2 x^4 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{9/2} \, _4F_3\left (2,2,2,\frac {5}{2};1,1,\frac {11}{2};\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{c^2}+\frac {64 d^3 x^6 \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{9/2} \, _4F_3\left (2,2,2,\frac {5}{2};1,1,\frac {11}{2};\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{c^3}\right )}{2520 c \left (\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )^{7/2} \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2} \]

[In]

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

[Out]

(x*(-108045*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))] - (324135*d*x^2*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c
- (324135*d^2*x^4*Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 - (103320*d^3*x^6*Sqrt[((b*c - a*d)*x^2)/(c*(a
+ b*x^2))])/c^3 + 42735*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2) + (128205*d*x^2*(((b*c - a*d)*x^2)/(c*(a +
b*x^2)))^(3/2))/c + (139545*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c^2 + (46200*d^3*x^6*(((b*c - a
*d)*x^2)/(c*(a + b*x^2)))^(3/2))/c^3 - 3864*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2) - (4032*d*x^2*(((b*c - a
*d)*x^2)/(c*(a + b*x^2)))^(5/2))/c - (4032*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2))/c^2 - (1344*d^3*
x^6*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(5/2))/c^3 + 108045*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]] +
(324135*d*x^2*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c + (324135*d^2*x^4*ArcTanh[Sqrt[((b*c - a*d)
*x^2)/(c*(a + b*x^2))]])/c^2 + (103320*d^3*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/c^3 + (8505*(
b*c - a*d)^2*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^2*(a + b*x^2)^2) + (17955*d*(b*c - a*d)^
2*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^3*(a + b*x^2)^2) + (21735*d^2*(b*c - a*d)^2*x^8*Arc
Tanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^4*(a + b*x^2)^2) + (7560*d^3*(b*c - a*d)^2*x^10*ArcTanh[Sqrt
[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^5*(a + b*x^2)^2) - (78750*(b*c - a*d)*x^2*ArcTanh[Sqrt[((b*c - a*d)*x
^2)/(c*(a + b*x^2))]])/(c*(a + b*x^2)) + (236250*d*(-(b*c) + a*d)*x^4*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b
*x^2))]])/(c^2*(a + b*x^2)) + (247590*d^2*(-(b*c) + a*d)*x^6*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])
/(c^3*(a + b*x^2)) + (80640*d^3*(-(b*c) + a*d)*x^8*ArcTanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c^4*(a +
b*x^2)) + 64*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c
- a*d)*x^2)/(c*(a + b*x^2))] + (192*d*x^2*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2
, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c + (192*d^2*x^4*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))
^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/c^2 + (64*d^3*x^6*(
((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(9/2)*HypergeometricPFQ[{2, 2, 2, 5/2}, {1, 1, 11/2}, ((b*c - a*d)*x^2)/(c*
(a + b*x^2))])/c^3))/(2520*c*(((b*c - a*d)*x^2)/(c*(a + b*x^2)))^(7/2)*(a + b*x^2)^(3/2)*(c + d*x^2)^2)

Maple [A] (verified)

Time = 2.66 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {3 \left (\sqrt {b \,x^{2}+a}\, a d \left (d \,x^{2}+c \right )^{2} \left (a^{2} d^{2}-4 a b c d +8 b^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )-\frac {5 x \left (d^{3} \left (\frac {3 d \,x^{2}}{5}+c \right ) a^{3}-\frac {12 \left (-\frac {d \,x^{2}}{3}+c \right ) b \,d^{2} \left (\frac {3 d \,x^{2}}{4}+c \right ) a^{2}}{5}-\frac {12 x^{2} \left (\frac {5 d \,x^{2}}{6}+c \right ) b^{2} d^{2} c a}{5}-\frac {8 b^{3} c^{2} \left (d \,x^{2}+c \right )^{2}}{5}\right ) \sqrt {\left (a d -b c \right ) c}}{3}\right )}{8 \sqrt {b \,x^{2}+a}\, \sqrt {\left (a d -b c \right ) c}\, \left (d \,x^{2}+c \right )^{2} c^{2} \left (a d -b c \right )^{3} a}\)	\(214\)
default	\(\text {Expression too large to display}\)	\(4035\)

[In]

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

[Out]

-3/8*((b*x^2+a)^(1/2)*a*d*(d*x^2+c)^2*(a^2*d^2-4*a*b*c*d+8*b^2*c^2)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(
1/2))-5/3*x*(d^3*(3/5*d*x^2+c)*a^3-12/5*(-1/3*d*x^2+c)*b*d^2*(3/4*d*x^2+c)*a^2-12/5*x^2*(5/6*d*x^2+c)*b^2*d^2*
c*a-8/5*b^3*c^2*(d*x^2+c)^2)*((a*d-b*c)*c)^(1/2))/(b*x^2+a)^(1/2)/((a*d-b*c)*c)^(1/2)/(d*x^2+c)^2/c^2/(a*d-b*c
)^3/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (201) = 402\).

Time = 0.88 (sec) , antiderivative size = 1482, normalized size of antiderivative = 6.59 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/32*(3*(8*a^2*b^2*c^4*d - 4*a^3*b*c^3*d^2 + a^4*c^2*d^3 + (8*a*b^3*c^2*d^3 - 4*a^2*b^2*c*d^4 + a^3*b*d^5)*x
^6 + (16*a*b^3*c^3*d^2 - 2*a^3*b*c*d^4 + a^4*d^5)*x^4 + (8*a*b^3*c^4*d + 12*a^2*b^2*c^3*d^2 - 7*a^3*b*c^2*d^3
+ 2*a^4*c*d^4)*x^2)*sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 -
3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2
)) - 4*((8*b^4*c^4*d^2 + 2*a*b^3*c^3*d^3 - 13*a^2*b^2*c^2*d^4 + 3*a^3*b*c*d^5)*x^5 + (16*b^4*c^5*d - 4*a*b^3*c
^4*d^2 - 7*a^2*b^2*c^3*d^3 - 8*a^3*b*c^2*d^4 + 3*a^4*c*d^5)*x^3 + (8*b^4*c^6 - 8*a*b^3*c^5*d + 12*a^2*b^2*c^4*
d^2 - 17*a^3*b*c^3*d^3 + 5*a^4*c^2*d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^4*c^9 - 4*a^3*b^3*c^8*d + 6*a^4*b^2*c^7*d^2
- 4*a^5*b*c^6*d^3 + a^6*c^5*d^4 + (a*b^5*c^7*d^2 - 4*a^2*b^4*c^6*d^3 + 6*a^3*b^3*c^5*d^4 - 4*a^4*b^2*c^4*d^5
+ a^5*b*c^3*d^6)*x^6 + (2*a*b^5*c^8*d - 7*a^2*b^4*c^7*d^2 + 8*a^3*b^3*c^6*d^3 - 2*a^4*b^2*c^5*d^4 - 2*a^5*b*c^
4*d^5 + a^6*c^3*d^6)*x^4 + (a*b^5*c^9 - 2*a^2*b^4*c^8*d - 2*a^3*b^3*c^7*d^2 + 8*a^4*b^2*c^6*d^3 - 7*a^5*b*c^5*
d^4 + 2*a^6*c^4*d^5)*x^2), 1/16*(3*(8*a^2*b^2*c^4*d - 4*a^3*b*c^3*d^2 + a^4*c^2*d^3 + (8*a*b^3*c^2*d^3 - 4*a^2
*b^2*c*d^4 + a^3*b*d^5)*x^6 + (16*a*b^3*c^3*d^2 - 2*a^3*b*c*d^4 + a^4*d^5)*x^4 + (8*a*b^3*c^4*d + 12*a^2*b^2*c
^3*d^2 - 7*a^3*b*c^2*d^3 + 2*a^4*c*d^4)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*
d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*((8*b^4*c^4*d^2 + 2*a*b^3
*c^3*d^3 - 13*a^2*b^2*c^2*d^4 + 3*a^3*b*c*d^5)*x^5 + (16*b^4*c^5*d - 4*a*b^3*c^4*d^2 - 7*a^2*b^2*c^3*d^3 - 8*a
^3*b*c^2*d^4 + 3*a^4*c*d^5)*x^3 + (8*b^4*c^6 - 8*a*b^3*c^5*d + 12*a^2*b^2*c^4*d^2 - 17*a^3*b*c^3*d^3 + 5*a^4*c
^2*d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^4*c^9 - 4*a^3*b^3*c^8*d + 6*a^4*b^2*c^7*d^2 - 4*a^5*b*c^6*d^3 + a^6*c^5*d^4
+ (a*b^5*c^7*d^2 - 4*a^2*b^4*c^6*d^3 + 6*a^3*b^3*c^5*d^4 - 4*a^4*b^2*c^4*d^5 + a^5*b*c^3*d^6)*x^6 + (2*a*b^5*
c^8*d - 7*a^2*b^4*c^7*d^2 + 8*a^3*b^3*c^6*d^3 - 2*a^4*b^2*c^5*d^4 - 2*a^5*b*c^4*d^5 + a^6*c^3*d^6)*x^4 + (a*b^
5*c^9 - 2*a^2*b^4*c^8*d - 2*a^3*b^3*c^7*d^2 + 8*a^4*b^2*c^6*d^3 - 7*a^5*b*c^5*d^4 + 2*a^6*c^4*d^5)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (201) = 402\).

Time = 1.21 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.86 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=\frac {b^{3} x}{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {b x^{2} + a}} + \frac {3 \, {\left (8 \, b^{\frac {5}{2}} c^{2} d - 4 \, a b^{\frac {3}{2}} c d^{2} + a^{2} \sqrt {b} d^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {3}{2}} c d^{3} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{4} + 80 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} d - 104 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d^{2} + 54 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{3} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{4} + 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d^{2} - 52 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{3} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{4} + 10 \, a^{4} b^{\frac {3}{2}} c d^{3} - 3 \, a^{5} \sqrt {b} d^{4}}{4 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2}} \]

[In]

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

[Out]

b^3*x/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*sqrt(b*x^2 + a)) + 3/8*(8*b^(5/2)*c^2*d - 4*a*b
^(3/2)*c*d^2 + a^2*sqrt(b)*d^3)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a
*b*c*d))/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*sqrt(-b^2*c^2 + a*b*c*d)) + 1/4*(16*(sqrt(
b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^2*d^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(3/2)*c*d^3 + 3*(sqrt(b)*x
- sqrt(b*x^2 + a))^6*a^2*sqrt(b)*d^4 + 80*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3*d - 104*(sqrt(b)*x - sqr
t(b*x^2 + a))^4*a*b^(5/2)*c^2*d^2 + 54*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2)*c*d^3 - 9*(sqrt(b)*x - sqrt
(b*x^2 + a))^4*a^3*sqrt(b)*d^4 + 64*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d^2 - 52*(sqrt(b)*x - sqrt
(b*x^2 + a))^2*a^3*b^(3/2)*c*d^3 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*sqrt(b)*d^4 + 10*a^4*b^(3/2)*c*d^3 -
3*a^5*sqrt(b)*d^4)/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4
*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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