3.1.63 \(\int \frac {1}{(a+b x^2)^2 (c+d x^2) \sqrt {e+f x^2}} \, dx\) [63]

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

[Out]

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

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Rubi [A]
time = 0.19, 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 = {560, 385, 211, 541, 12} \begin {gather*} \frac {b \text {ArcTan}\left (\frac {x \sqrt {b e-a f}}{\sqrt {a} \sqrt {e+f x^2}}\right ) \left (4 a^2 d f-2 a b c f-3 a b d e+b^2 c e\right )}{2 a^{3/2} (b c-a d)^2 (b e-a f)^{3/2}}+\frac {d^2 \text {ArcTan}\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}}+\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)} \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 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 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 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]

Rule 560

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 \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 14.00, size = 531, normalized size = 2.62 \begin {gather*} \frac {-\frac {30 b d \tan ^{-1}\left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} \sqrt {b e-a f}}+\frac {30 d^2 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} \sqrt {d e-c f}}+\frac {b (b c-a d) x \sqrt {e+f x^2} \left (-45 e \sqrt {\frac {a (b e-a f) x^2 \left (e+f x^2\right )}{e^2 \left (a+b x^2\right )^2}}-30 f x^2 \sqrt {\frac {a (b e-a f) x^2 \left (e+f x^2\right )}{e^2 \left (a+b x^2\right )^2}}+45 e \sin ^{-1}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )+30 f x^2 \sin ^{-1}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )+16 e \left (\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}\right )^{5/2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}\right )+16 f x^2 \left (\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}\right )^{5/2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}\right )\right )}{e^2 \left (\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}\right )^{3/2} \left (a+b x^2\right )^2 \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}}}{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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1399\) vs. \(2(177)=354\).
time = 0.12, size = 1400, normalized size = 6.90

method	result	size
default	\(-\frac {b^{2} d^{4} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}+\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right )^{2} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {b^{3} d^{2} \left (3 a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}+\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right )^{2} \sqrt {-\frac {a f -b e}{b}}}-\frac {b^{3} d^{2} \left (3 a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}-\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 a \sqrt {-a b}\, \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right )^{2} \sqrt {-\frac {a f -b e}{b}}}+\frac {b^{2} d^{4} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}-\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right )^{2} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {b d \left (\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{\left (a f -b e \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {f \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}+\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a f -b e \right ) \sqrt {-\frac {a f -b e}{b}}}\right )}{4 a \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right )}-\frac {b d \left (\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{\left (a f -b e \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {f \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}-\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a f -b e \right ) \sqrt {-\frac {a f -b e}{b}}}\right )}{4 a \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (\sqrt {-a b}\, d -b \sqrt {-c d}\right )}\)	\(1400\)

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,method=_RETURNVERBOSE)

[Out]

-1/2*b^2*d^4/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)^2/((-a*b)^(1/2)*d-b*(-c*d)^(1/2))^2/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(
1/2)*ln((-2*(c*f-d*e)/d+2*f*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+2*(-(c*f-d*e)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*f+2
*f*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(x-(-c*d)^(1/2)/d))+1/4*b^3*d^2*(3*a*d-b*c)/a/(-a*b)^
(1/2)/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)^2/((-a*b)^(1/2)*d-b*(-c*d)^(1/2))^2/(-(a*f-b*e)/b)^(1/2)*ln((-2*(a*f-b*e
)/b+2*f*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*f-b*e)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*f+2*f*(-a*b)^(1/2)/
b*(x-1/b*(-a*b)^(1/2))-(a*f-b*e)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))-1/4*b^3*d^2*(3*a*d-b*c)/a/(-a*b)^(1/2)/(b*(-c
*d)^(1/2)+(-a*b)^(1/2)*d)^2/((-a*b)^(1/2)*d-b*(-c*d)^(1/2))^2/(-(a*f-b*e)/b)^(1/2)*ln((-2*(a*f-b*e)/b-2*f*(-a*
b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*f-b*e)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*f-2*f*(-a*b)^(1/2)/b*(x+1/b*(-a
*b)^(1/2))-(a*f-b*e)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))+1/2*b^2*d^4/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)^2/((-a*b)^(1/
2)*d-b*(-c*d)^(1/2))^2/(-c*d)^(1/2)/(-(c*f-d*e)/d)^(1/2)*ln((-2*(c*f-d*e)/d-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)
/d)+2*(-(c*f-d*e)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*f-2*f*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)-(c*f-d*e)/d)^(1/2))/(
x+(-c*d)^(1/2)/d))-1/4*b*d/a/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/((-a*b)^(1/2)*d-b*(-c*d)^(1/2))*(1/(a*f-b*e)*b/(x
-1/b*(-a*b)^(1/2))*((x-1/b*(-a*b)^(1/2))^2*f+2*f*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*f-b*e)/b)^(1/2)-f*(-a*
b)^(1/2)/(a*f-b*e)/(-(a*f-b*e)/b)^(1/2)*ln((-2*(a*f-b*e)/b+2*f*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*f-b*
e)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*f+2*f*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*f-b*e)/b)^(1/2))/(x-1/b*(-a*b
)^(1/2))))-1/4*b*d/a/(b*(-c*d)^(1/2)+(-a*b)^(1/2)*d)/((-a*b)^(1/2)*d-b*(-c*d)^(1/2))*(1/(a*f-b*e)*b/(x+1/b*(-a
*b)^(1/2))*((x+1/b*(-a*b)^(1/2))^2*f-2*f*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*f-b*e)/b)^(1/2)+f*(-a*b)^(1/2)
/(a*f-b*e)/(-(a*f-b*e)/b)^(1/2)*ln((-2*(a*f-b*e)/b-2*f*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*f-b*e)/b)^(1
/2)*((x+1/b*(-a*b)^(1/2))^2*f-2*f*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*f-b*e)/b)^(1/2))/(x+1/b*(-a*b)^(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)^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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Fricas [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(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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (188) = 376\).
time = 7.96, 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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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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