3.1.29 \(\int \frac {1}{(a+b x^2)^2 (c+d x^2) \sqrt {e+f x^2}} \, dx\)

Optimal. Leaf size=203 \[ \frac {b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac {x \sqrt {b e-a f}}{\sqrt {a} \sqrt {e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac {b^2 x \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac {d^2 \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d)^2 \sqrt {d e-c f}} \]

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Rubi [A]  time = 0.27, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {546, 377, 205, 527, 12} \begin {gather*} \frac {b \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right ) \tan ^{-1}\left (\frac {x \sqrt {b e-a f}}{\sqrt {a} \sqrt {e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac {b^2 x \sqrt {e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}+\frac {d^2 \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d)^2 \sqrt {d e-c f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 205

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

Rule 377

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 527

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]

Rule 546

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[b^2/(b*c
- a*d)^2, Int[((c + d*x^2)^(q + 2)*(e + f*x^2)^r)/(a + b*x^2), x], x] - Dist[d/(b*c - a*d)^2, Int[(c + d*x^2)^
q*(e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LtQ[q, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx &=-\frac {b \int \frac {-b c+2 a d+b d x^2}{\left (a+b x^2\right )^2 \sqrt {e+f x^2}} \, dx}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx}{(b c-a d)^2}\\ &=\frac {b^2 x \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{c-(-d e+c f) x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{(b c-a d)^2}+\frac {b \int \frac {b^2 c e-3 a b d e-2 a b c f+4 a^2 d f}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx}{2 a (b c-a d)^2 (b e-a f)}\\ &=\frac {b^2 x \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {d^2 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d)^2 \sqrt {d e-c f}}+\frac {\left (b \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right )\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {e+f x^2}} \, dx}{2 a (b c-a d)^2 (b e-a f)}\\ &=\frac {b^2 x \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {d^2 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d)^2 \sqrt {d e-c f}}+\frac {\left (b \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-(-b e+a f) x^2} \, dx,x,\frac {x}{\sqrt {e+f x^2}}\right )}{2 a (b c-a d)^2 (b e-a f)}\\ &=\frac {b^2 x \sqrt {e+f x^2}}{2 a (b c-a d) (b e-a f) \left (a+b x^2\right )}+\frac {b \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right ) \tan ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac {d^2 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d)^2 \sqrt {d e-c f}}\\ \end {align*}

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Mathematica [C]  time = 2.64, size = 531, normalized size = 2.62 \begin {gather*} \frac {\frac {b x \sqrt {e+f x^2} (b c-a d) \left (-30 f x^2 \sqrt {\frac {a x^2 \left (e+f x^2\right ) (b e-a f)}{e^2 \left (a+b x^2\right )^2}}-45 e \sqrt {\frac {a x^2 \left (e+f x^2\right ) (b e-a f)}{e^2 \left (a+b x^2\right )^2}}+16 f x^2 \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \left (\frac {x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b e-a f) x^2}{e \left (b x^2+a\right )}\right )+16 e \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \left (\frac {x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{5/2} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b e-a f) x^2}{e \left (b x^2+a\right )}\right )+30 f x^2 \sin ^{-1}\left (\sqrt {\frac {x^2 (b e-a f)}{e \left (a+b x^2\right )}}\right )+45 e \sin ^{-1}\left (\sqrt {\frac {x^2 (b e-a f)}{e \left (a+b x^2\right )}}\right )\right )}{e^2 \left (a+b x^2\right )^2 \left (\frac {x^2 (b e-a f)}{e \left (a+b x^2\right )}\right )^{3/2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}-\frac {30 b d \tan ^{-1}\left (\frac {x \sqrt {b e-a f}}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} \sqrt {b e-a f}}+\frac {30 d^2 \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} \sqrt {d e-c f}}}{30 (b c-a d)^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-30*b*d*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*Sqrt[b*e - a*f]) + (30*d^2*ArcTan[(S
qrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])])/(Sqrt[c]*Sqrt[d*e - c*f]) + (b*(b*c - a*d)*x*Sqrt[e + f*x^2]*(-4
5*e*Sqrt[(a*(b*e - a*f)*x^2*(e + f*x^2))/(e^2*(a + b*x^2)^2)] - 30*f*x^2*Sqrt[(a*(b*e - a*f)*x^2*(e + f*x^2))/
(e^2*(a + b*x^2)^2)] + 45*e*ArcSin[Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]] + 30*f*x^2*ArcSin[Sqrt[((b*e - a*f
)*x^2)/(e*(a + b*x^2))]] + 16*e*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(5/2)*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))
]*Hypergeometric2F1[2, 3, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))] + 16*f*x^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)
))^(5/2)*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*Hypergeometric2F1[2, 3, 7/2, ((b*e - a*f)*x^2)/(e*(a + b*x^2))]
))/(e^2*(((b*e - a*f)*x^2)/(e*(a + b*x^2)))^(3/2)*(a + b*x^2)^2*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]))/(30*(b
*c - a*d)^2)

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IntegrateAlgebraic [B]  time = 6.80, size = 2522, normalized size = 12.42 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((a + b*x^2)^2*(c + d*x^2)*Sqrt[e + f*x^2]),x]

[Out]

(-((e*Sqrt[f]*x^2)/a) - (f^(3/2)*x^4)/a + (e*x*Sqrt[e + f*x^2])/(2*a) + (f*x^3*Sqrt[e + f*x^2])/a)/((a + b*x^2
)*(c + d*x^2)*(e + f*x^2)*(e + 2*f*x^2 - 2*Sqrt[f]*x*Sqrt[e + f*x^2])) + ((c*d*e*Sqrt[f])/(2*a*(b*c - a*d)*(d*
e - c*f)) - (d^2*e*Sqrt[f]*x^2)/(2*a*(b*c - a*d)*(d*e - c*f)) + (c*d*f^(3/2)*x^2)/(a*(b*c - a*d)*(d*e - c*f))
+ (d^2*e*x*Sqrt[e + f*x^2])/(2*a*(b*c - a*d)*(d*e - c*f)) - (c*d*f*x*Sqrt[e + f*x^2])/(a*(b*c - a*d)*(d*e - c*
f)) + (d^2*e^2*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(2*Sqrt[a]
*(b*c - a*d)*Sqrt[b*e - a*f]*(d*e - c*f)) + (d^3*e^2*x^2*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^
2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(2*Sqrt[a]*c*(b*c - a*d)*Sqrt[b*e - a*f]*(d*e - c*f)) + (d^2*e*f*x^2*ArcTan[(a
*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(Sqrt[a]*(b*c - a*d)*Sqrt[b*e - a*
f]*(d*e - c*f)) + (d^3*e*f*x^4*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*
f])])/(Sqrt[a]*c*(b*c - a*d)*Sqrt[b*e - a*f]*(d*e - c*f)) - (d^2*e*Sqrt[f]*x*Sqrt[e + f*x^2]*ArcTan[(a*Sqrt[f]
+ b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(Sqrt[a]*(b*c - a*d)*Sqrt[b*e - a*f]*(d*e
- c*f)) - (d^3*e*Sqrt[f]*x^3*Sqrt[e + f*x^2]*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]
*Sqrt[b*e - a*f])])/(Sqrt[a]*c*(b*c - a*d)*Sqrt[b*e - a*f]*(d*e - c*f)))/((c + d*x^2)*(e + 2*f*x^2 - 2*Sqrt[f]
*x*Sqrt[e + f*x^2])) + (-1/2*(e^2*f^(3/2))/(a*(-(b*e) + a*f)*(-(d*e) + c*f)) - (e*f^(5/2)*x^2)/(2*a*(-(b*e) +
a*f)*(-(d*e) + c*f)) + (e*f^2*x*Sqrt[e + f*x^2])/(2*a*(-(b*e) + a*f)*(-(d*e) + c*f)) - (b*e^3*f^2*ArcTan[(a*Sq
rt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(2*Sqrt[a]*(b*e - a*f)^(3/2)*(-(b*e)
+ a*f)*(-(d*e) + c*f)) + (Sqrt[a]*e^2*f^3*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sq
rt[b*e - a*f])])/(2*(b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f)) - (3*b*e^2*f^3*x^2*ArcTan[(a*Sqrt[f] + b*
Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(2*Sqrt[a]*(b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(
d*e) + c*f)) + (3*Sqrt[a]*e*f^4*x^2*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e
- a*f])])/(2*(b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f)) - (b*e*f^4*x^4*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^
2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(Sqrt[a]*(b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f))
+ (Sqrt[a]*f^5*x^4*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/((b*e
- a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f)) + (b*e^2*f^(5/2)*x*Sqrt[e + f*x^2]*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*
x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(Sqrt[a]*(b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f
)) - (Sqrt[a]*e*f^(7/2)*x*Sqrt[e + f*x^2]*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sq
rt[b*e - a*f])])/((b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f)) + (b*e*f^(7/2)*x^3*Sqrt[e + f*x^2]*ArcTan[(
a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/(Sqrt[a]*(b*e - a*f)^(3/2)*(-(b*e
) + a*f)*(-(d*e) + c*f)) - (Sqrt[a]*f^(9/2)*x^3*Sqrt[e + f*x^2]*ArcTan[(a*Sqrt[f] + b*Sqrt[f]*x^2 - b*x*Sqrt[e
+ f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])])/((b*e - a*f)^(3/2)*(-(b*e) + a*f)*(-(d*e) + c*f)))/((e + f*x^2)*(e + 2*
f*x^2 - 2*Sqrt[f]*x*Sqrt[e + f*x^2])) - ArcTan[(Sqrt[a]*Sqrt[f])/Sqrt[b*e - a*f] + (b*Sqrt[f]*x^2)/(Sqrt[a]*Sq
rt[b*e - a*f]) - (b*x*Sqrt[e + f*x^2])/(Sqrt[a]*Sqrt[b*e - a*f])]/(2*a^(3/2)*c*Sqrt[b*e - a*f]) + (Sqrt[a]*d^2
*ArcTan[(Sqrt[a]*Sqrt[f])/Sqrt[b*e - a*f] + (b*Sqrt[f]*x^2)/(Sqrt[a]*Sqrt[b*e - a*f]) - (b*x*Sqrt[e + f*x^2])/
(Sqrt[a]*Sqrt[b*e - a*f])])/(c*(b*c - a*d)^2*Sqrt[b*e - a*f]) - (d^2*ArcTan[(Sqrt[c]*Sqrt[f])/Sqrt[d*e - c*f]
+ (d*Sqrt[f]*x^2)/(Sqrt[c]*Sqrt[d*e - c*f]) - (d*x*Sqrt[e + f*x^2])/(Sqrt[c]*Sqrt[d*e - c*f])])/(Sqrt[c]*(b*c
- a*d)^2*Sqrt[d*e - c*f])

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fricas [F(-1)]  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(1/(b*x^2+a)^2/(d*x^2+c)/(f*x^2+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 26.05, size = 479, normalized size = 2.36 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, d^{2} \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt {-c^{2} f^{2} + c d f e}}\right )}{{\left (b^{2} c^{2} f^{2} - 2 \, a b c d f^{2} + a^{2} d^{2} f^{2}\right )} \sqrt {-c^{2} f^{2} + c d f e}} + \frac {{\left (2 \, a b^{2} c f - 4 \, a^{2} b d f - b^{3} c e + 3 \, a b^{2} d e\right )} \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} b + 2 \, a f - b e}{2 \, \sqrt {-a^{2} f^{2} + a b f e}}\right )}{{\left (a^{2} b^{2} c^{2} f^{3} - 2 \, a^{3} b c d f^{3} + a^{4} d^{2} f^{3} - a b^{3} c^{2} f^{2} e + 2 \, a^{2} b^{2} c d f^{2} e - a^{3} b d^{2} f^{2} e\right )} \sqrt {-a^{2} f^{2} + a b f e}} + \frac {2 \, {\left (2 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} a b f - {\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} b^{2} e + b^{2} e^{2}\right )}}{{\left (a^{2} b c f^{3} - a^{3} d f^{3} - a b^{2} c f^{2} e + a^{2} b d f^{2} e\right )} {\left ({\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{4} b + 4 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} a f - 2 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} b e + b e^{2}\right )}}\right )} f^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 1865, normalized size = 9.19

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )^{2} \left (c + d x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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