3.1.80 \(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^3} \, dx\) [80]

Optimal. Leaf size=163 \[ -\frac {d x \sqrt {a+b x^2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {3 d (2 b c-a d) x \sqrt {a+b x^2}}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 163, 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 {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{5/2}}-\frac {3 d x \sqrt {a+b x^2} (2 b c-a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d x \sqrt {a+b x^2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.75, size = 160, normalized size = 0.98 \begin {gather*} \frac {d x \sqrt {a+b x^2} \left (-2 b c \left (4 c+3 d x^2\right )+a d \left (5 c+3 d x^2\right )\right )}{8 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \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 )}{8 c^{5/2} (-b c+a d)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1842\) vs. \(2(143)=286\).
time = 0.07, size = 1843, normalized size = 11.31

method	result	size
default	\(\text {Expression too large to display}\)	\(1843\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d/c/(-c*d)^(1/2)*(-1/2/(a*d-b*c)*d/(x-(-c*d)^(1/2)/d)^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)-3/2*b*(-c*d)^(1/2)/(a*d-b*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)+b*(-c*d)^(1/2)/(a*d-b*c)/((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)))+1/2*b/(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)))-3/16/(-c*d)^(1/2)/c^2/((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))+3/16/(-c*d)^(1/2)/c^2/((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))-3/16/d/c^2*(-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)+b*(-c*d)^(1/2)/(a
*d-b*c)/((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)))-3/16/d/c^2*
(-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)-b*(-c*d)^(1/2)/(a*d-b*c)/((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)))+1/8/d/c/(-c*d)^(1/2)*(-1/2/(a*d-b*c)*d/(x+(-c*d)^(1/2)/d)^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)+3/2*b*(-c*d)^(1/2)/(a*d-b*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)-b*(-c*d)^(1/2)/(a*d-b*c)/((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)))+1/2*b/(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)))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (143) = 286\).
time = 0.76, size = 864, normalized size = 5.30 \begin {gather*} \left [\frac {{\left (8 \, b^{2} c^{4} - 8 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 8 \, a b c^{2} d^{2} + 3 \, a^{2} c 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 (3 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d - 13 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\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^{4} + 2 \, {\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}\right )} x^{2}\right )}}, -\frac {{\left (8 \, b^{2} c^{4} - 8 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + {\left (8 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (8 \, b^{2} c^{3} d - 8 \, a b c^{2} d^{2} + 3 \, a^{2} c 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 (3 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d - 13 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{3} c^{8} - 3 \, a b^{2} c^{7} d + 3 \, a^{2} b c^{6} d^{2} - a^{3} c^{5} d^{3} + {\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^{4} + 2 \, {\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}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (143) = 286\).
time = 2.31, size = 538, normalized size = 3.30 \begin {gather*} -\frac {1}{8} \, b^{\frac {5}{2}} {\left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} 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 )}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} + \frac {2 \, {\left (8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{2} c^{2} d - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b c d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} d^{3} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{3} c^{3} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{2} c^{2} d + 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} d^{3} + 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{2} c^{2} d - 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} d^{3} + 6 \, a^{4} b c d^{2} - 3 \, a^{5} d^{3}\right )}}{{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} 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 )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*b^(5/2)*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*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^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*d^2)*sqrt(-b^2*c^2 + a*b*c*d)) + 2*(8*(sqrt
(b)*x - sqrt(b*x^2 + a))^6*b^2*c^2*d - 8*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b*c*d^2 + 3*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*a^2*d^3 + 48*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^3*c^3 - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^2*c^2*d
+ 42*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b*c*d^2 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*d^3 + 40*(sqrt(b)*x
- sqrt(b*x^2 + a))^2*a^2*b^2*c^2*d - 40*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b*c*d^2 + 9*(sqrt(b)*x - sqrt(b*x^
2 + a))^2*a^4*d^3 + 6*a^4*b*c*d^2 - 3*a^5*d^3)/((b^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*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)^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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