3.1.87 \(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^2} \, dx\) [87]

Optimal. Leaf size=143 \[ \frac {b (2 b c+a d) x}{2 a c (b c-a d)^2 \sqrt {a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {425, 541, 12, 385, 214} \begin {gather*} -\frac {d (4 b c-a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2}}+\frac {b x (a d+2 b c)}{2 a c \sqrt {a+b x^2} (b c-a d)^2}-\frac {d x}{2 c \sqrt {a+b x^2} \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.66, size = 151, normalized size = 1.06 \begin {gather*} \frac {\frac {\sqrt {c} x \left (a^2 d^2+a b d^2 x^2+2 b^2 c \left (c+d x^2\right )\right )}{a (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {d (4 b c-a d) \tan ^{-1}\left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}}{2 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1905\) vs. \(2(123)=246\).
time = 0.07, size = 1906, normalized size = 13.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1906\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/c/(-c*d)^(1/2)*(1/(a*d-b*c)*d/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(
1/2)-2*b*(-c*d)^(1/2)/(a*d-b*c)*(2*b*(x-(-c*d)^(1/2)/d)+2*b*(-c*d)^(1/2)/d)/(4*b*(a*d-b*c)/d+4*b^2*c/d)/((x-(-
c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-1/(a*d-b*c)*d/((a*d-b*c)/d)^(1/2)*l
n((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d
)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d)))-1/4/d/c*(-1/(a*d-b*c)*d/(x-(-c*d)^(1/2)/
d)/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-3*b*(-c*d)^(1/2)/(a*d-b*c)
*(1/(a*d-b*c)*d/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-2*b*(-c*d)^(1
/2)/(a*d-b*c)*(2*b*(x-(-c*d)^(1/2)/d)+2*b*(-c*d)^(1/2)/d)/(4*b*(a*d-b*c)/d+4*b^2*c/d)/((x-(-c*d)^(1/2)/d)^2*b+
2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-1/(a*d-b*c)*d/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2
*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d
)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d)))-4*b/(a*d-b*c)*d*(2*b*(x-(-c*d)^(1/2)/d)+2*b*(-c*d)^(1/2)/d
)/(4*b*(a*d-b*c)/d+4*b^2*c/d)/((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)
)-1/4/c/(-c*d)^(1/2)*(1/(a*d-b*c)*d/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)
^(1/2)+2*b*(-c*d)^(1/2)/(a*d-b*c)*(2*b*(x+(-c*d)^(1/2)/d)-2*b*(-c*d)^(1/2)/d)/(4*b*(a*d-b*c)/d+4*b^2*c/d)/((x+
(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-1/(a*d-b*c)*d/((a*d-b*c)/d)^(1/2)
*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c
*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d)))-1/4/d/c*(-1/(a*d-b*c)*d/(x+(-c*d)^(1/2
)/d)/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+3*b*(-c*d)^(1/2)/(a*d-b*
c)*(1/(a*d-b*c)*d/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+2*b*(-c*d)^
(1/2)/(a*d-b*c)*(2*b*(x+(-c*d)^(1/2)/d)-2*b*(-c*d)^(1/2)/d)/(4*b*(a*d-b*c)/d+4*b^2*c/d)/((x+(-c*d)^(1/2)/d)^2*
b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)-1/(a*d-b*c)*d/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d
-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c
*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d)))-4*b/(a*d-b*c)*d*(2*b*(x+(-c*d)^(1/2)/d)-2*b*(-c*d)^(1/2)
/d)/(4*b*(a*d-b*c)/d+4*b^2*c/d)/((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/
2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (123) = 246\).
time = 0.77, size = 864, normalized size = 6.04 \begin {gather*} \left [-\frac {{\left (4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 4 \, {\left ({\left (2 \, b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} + {\left (a b^{4} c^{5} d - 3 \, a^{2} b^{3} c^{4} d^{2} + 3 \, a^{3} b^{2} c^{3} d^{3} - a^{4} b c^{2} d^{4}\right )} x^{4} + {\left (a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 2 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4}\right )} x^{2}\right )}}, \frac {{\left (4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (2 \, b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3}\right )} x^{3} + {\left (2 \, b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} + {\left (a b^{4} c^{5} d - 3 \, a^{2} b^{3} c^{4} d^{2} + 3 \, a^{3} b^{2} c^{3} d^{3} - a^{4} b c^{2} d^{4}\right )} x^{4} + {\left (a b^{4} c^{6} - 2 \, a^{2} b^{3} c^{5} d + 2 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (123) = 246\).
time = 1.17, size = 318, normalized size = 2.22 \begin {gather*} \frac {b^{2} x}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b x^{2} + a}} + \frac {{\left (4 \, b^{\frac {3}{2}} c d - a \sqrt {b} d^{2}\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 )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c d - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d^{2} + a^{2} \sqrt {b} d^{2}}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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