3.1.74 \(\int \frac {1}{(a+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [74]

Optimal. Leaf size=416 \[ \frac {c (a e+(c d-a f) x)}{a \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}-\frac {f \left (2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {f \left (2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]

[Out]

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

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Rubi [A]
time = 0.38, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {990, 1048, 739, 212} \begin {gather*} -\frac {f \left (2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {f \left (2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {c (x (c d-a f)+a e)}{a \sqrt {a+c x^2} \left ((c d-a f)^2+a c e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(a*e + (c*d - a*f)*x))/(a*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[a + c*x^2]) - (f*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqr
t[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt
[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2
- 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + (f*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e
+ Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqr
t[2]*Sqrt[e^2 - 4*d*f]*(a*c*e^2 + (c*d - a*f)^2)*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 990

Int[((a_.) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e + c*(2*
c^2*d - c*(2*a*f))*x)*(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p +
1))), x] - Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si
mp[2*c*((c*d - a*f)^2 - ((-a)*e)*(c*e))*(p + 1) - (2*c^2*d - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(-2*a*
c^2*e)*(p + q + 2) + (2*f*(2*a*c^2*e)*(p + q + 2) - (2*c^2*d - c*(2*a*f))*((-c)*e*(2*p + q + 4)))*x + c*f*(2*c
^2*d - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, q}, x] && NeQ[e^2 - 4*d*f, 0] && Lt
Q[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1048

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c
*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.53, size = 346, normalized size = 0.83 \begin {gather*} \frac {c (c d x+a (e-f x))-a \sqrt {a+c x^2} \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a c e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{3/2} e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 c^{3/2} d f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a \sqrt {c} f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c e f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{a \left (c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )\right ) \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(c*d*x + a*(e - f*x)) - a*Sqrt[a + c*x^2]*RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sq
rt[c]*e*#1^3 + f*#1^4 & , (a*c*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] + 2*c^(3/2)*e^2*Log[-(Sqrt[c]*x) +
Sqrt[a + c*x^2] - #1]*#1 - 2*c^(3/2)*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 2*a*Sqrt[c]*f^2*Log[-(
Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - c*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*
c*d*#1 - 2*a*f*#1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ])/(a*(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))*Sqrt[a + c*x^
2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1456\) vs. \(2(377)=754\).
time = 0.11, size = 1457, normalized size = 3.50

method	result	size
default	\(\text {Expression too large to display}\)	\(1457\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-%e^2>0)', see `assume?`
for more det

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 26013 vs. \(2 (383) = 766\).
time = 74.22, size = 26013, normalized size = 62.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2)*(a^2*c^2*d^2 - 2*a^3*c*d*f + a^4*f^2 + (a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*x^2 + (a^2*c^2*x^
2 + a^3*c)*e^2)*sqrt((2*c^3*d^3*f^3 - 6*a*c^2*d^2*f^4 + 6*a^2*c*d*f^5 - 2*a^3*f^6 - c^3*e^6 + 3*(2*c^3*d*f - a
*c^2*f^2)*e^4 - 3*(3*c^3*d^2*f^2 - 4*a*c^2*d*f^3 + a^2*c*f^4)*e^2 + (4*c^6*d^7*f - 24*a*c^5*d^6*f^2 + 60*a^2*c
^4*d^5*f^3 - 80*a^3*c^3*d^4*f^4 + 60*a^4*c^2*d^3*f^5 - 24*a^5*c*d^2*f^6 + 4*a^6*d*f^7 - a^3*c^3*e^8 - (3*a^2*c
^4*d^2 - 10*a^3*c^3*d*f + 3*a^4*c^2*f^2)*e^6 - 3*(a*c^5*d^4 - 8*a^2*c^4*d^3*f + 14*a^3*c^3*d^2*f^2 - 8*a^4*c^2
*d*f^3 + a^5*c*f^4)*e^4 - (c^6*d^6 - 18*a*c^5*d^5*f + 63*a^2*c^4*d^4*f^2 - 92*a^3*c^3*d^3*f^3 + 63*a^4*c^2*d^2
*f^4 - 18*a^5*c*d*f^5 + a^6*f^6)*e^2)*sqrt(-(c^6*e^10 - 2*(4*c^6*d*f - 3*a*c^5*f^2)*e^8 + (22*c^6*d^2*f^2 - 36
*a*c^5*d*f^3 + 15*a^2*c^4*f^4)*e^6 - 6*(4*c^6*d^3*f^3 - 11*a*c^5*d^2*f^4 + 10*a^2*c^4*d*f^5 - 3*a^3*c^3*f^6)*e
^4 + 9*(c^6*d^4*f^4 - 4*a*c^5*d^3*f^5 + 6*a^2*c^4*d^2*f^6 - 4*a^3*c^3*d*f^7 + a^4*c^2*f^8)*e^2)/(4*c^12*d^13*f
- 48*a*c^11*d^12*f^2 + 264*a^2*c^10*d^11*f^3 - 880*a^3*c^9*d^10*f^4 + 1980*a^4*c^8*d^9*f^5 - 3168*a^5*c^7*d^8
*f^6 + 3696*a^6*c^6*d^7*f^7 - 3168*a^7*c^5*d^6*f^8 + 1980*a^8*c^4*d^5*f^9 - 880*a^9*c^3*d^4*f^10 + 264*a^10*c^
2*d^3*f^11 - 48*a^11*c*d^2*f^12 + 4*a^12*d*f^13 - a^6*c^6*e^14 - 2*(3*a^5*c^7*d^2 - 8*a^6*c^6*d*f + 3*a^7*c^5*
f^2)*e^12 - 3*(5*a^4*c^8*d^4 - 28*a^5*c^7*d^3*f + 46*a^6*c^6*d^2*f^2 - 28*a^7*c^5*d*f^3 + 5*a^8*c^4*f^4)*e^10
- 20*(a^3*c^9*d^6 - 9*a^4*c^8*d^5*f + 27*a^5*c^7*d^4*f^2 - 38*a^6*c^6*d^3*f^3 + 27*a^7*c^5*d^2*f^4 - 9*a^8*c^4
*d*f^5 + a^9*c^3*f^6)*e^8 - 5*(3*a^2*c^10*d^8 - 40*a^3*c^9*d^7*f + 180*a^4*c^8*d^6*f^2 - 408*a^5*c^7*d^5*f^3 +
530*a^6*c^6*d^4*f^4 - 408*a^7*c^5*d^3*f^5 + 180*a^8*c^4*d^2*f^6 - 40*a^9*c^3*d*f^7 + 3*a^10*c^2*f^8)*e^6 - 6*
(a*c^11*d^10 - 20*a^2*c^10*d^9*f + 125*a^3*c^9*d^8*f^2 - 400*a^4*c^8*d^7*f^3 + 770*a^5*c^7*d^6*f^4 - 952*a^6*c
^6*d^5*f^5 + 770*a^7*c^5*d^4*f^6 - 400*a^8*c^4*d^3*f^7 + 125*a^9*c^3*d^2*f^8 - 20*a^10*c^2*d*f^9 + a^11*c*f^10
)*e^4 - (c^12*d^12 - 36*a*c^11*d^11*f + 306*a^2*c^10*d^10*f^2 - 1300*a^3*c^9*d^9*f^3 + 3375*a^4*c^8*d^8*f^4 -
5832*a^5*c^7*d^7*f^5 + 6972*a^6*c^6*d^6*f^6 - 5832*a^7*c^5*d^5*f^7 + 3375*a^8*c^4*d^4*f^8 - 1300*a^9*c^3*d^3*f
^9 + 306*a^10*c^2*d^2*f^10 - 36*a^11*c*d*f^11 + a^12*f^12)*e^2)))/(4*c^6*d^7*f - 24*a*c^5*d^6*f^2 + 60*a^2*c^4
*d^5*f^3 - 80*a^3*c^3*d^4*f^4 + 60*a^4*c^2*d^3*f^5 - 24*a^5*c*d^2*f^6 + 4*a^6*d*f^7 - a^3*c^3*e^8 - (3*a^2*c^4
*d^2 - 10*a^3*c^3*d*f + 3*a^4*c^2*f^2)*e^6 - 3*(a*c^5*d^4 - 8*a^2*c^4*d^3*f + 14*a^3*c^3*d^2*f^2 - 8*a^4*c^2*d
*f^3 + a^5*c*f^4)*e^4 - (c^6*d^6 - 18*a*c^5*d^5*f + 63*a^2*c^4*d^4*f^2 - 92*a^3*c^3*d^3*f^3 + 63*a^4*c^2*d^2*f
^4 - 18*a^5*c*d*f^5 + a^6*f^6)*e^2))*log((4*c^4*d*f^3*x*e^5 - 2*a*c^3*f^3*e^6 - 4*(4*c^4*d^2*f^4 - 3*a*c^3*d*f
^5)*x*e^3 + 12*(c^4*d^3*f^5 - 2*a*c^3*d^2*f^6 + a^2*c^2*d*f^7)*x*e + sqrt(2)*(c^5*d*e^9 - (9*c^5*d^2*f - 5*a*c
^4*d*f^2 + a^2*c^3*f^3)*e^7 + (27*c^5*d^3*f^2 - 37*a*c^4*d^2*f^3 + 17*a^2*c^3*d*f^4 - 3*a^3*c^2*f^5)*e^5 - (31
*c^5*d^4*f^3 - 80*a*c^4*d^3*f^4 + 70*a^2*c^3*d^2*f^5 - 24*a^3*c^2*d*f^6 + 3*a^4*c*f^7)*e^3 + 12*(c^5*d^5*f^4 -
4*a*c^4*d^4*f^5 + 6*a^2*c^3*d^3*f^6 - 4*a^3*c^2*d^2*f^7 + a^4*c*d*f^8)*e - (a^3*c^5*d*e^11 + (3*a^2*c^6*d^3 -
13*a^3*c^5*d^2*f + 5*a^4*c^4*d*f^2 + a^5*c^3*f^3)*e^9 + (3*a*c^7*d^5 - 33*a^2*c^6*d^4*f + 78*a^3*c^5*d^3*f^2
- 50*a^4*c^4*d^2*f^3 - a^5*c^3*d*f^4 + 3*a^6*c^2*f^5)*e^7 + (c^8*d^7 - 27*a*c^7*d^6*f + 141*a^2*c^6*d^5*f^2 -
263*a^3*c^5*d^4*f^3 + 195*a^4*c^4*d^3*f^4 - 33*a^5*c^3*d^2*f^5 - 17*a^6*c^2*d*f^6 + 3*a^7*c*f^7)*e^5 - (7*c^8*
d^8*f - 80*a*c^7*d^7*f^2 + 284*a^2*c^6*d^6*f^3 - 464*a^3*c^5*d^5*f^4 + 370*a^4*c^4*d^4*f^5 - 112*a^5*c^3*d^3*f
^6 - 20*a^6*c^2*d^2*f^7 + 16*a^7*c*d*f^8 - a^8*f^9)*e^3 + 4*(3*c^8*d^9*f^2 - 20*a*c^7*d^8*f^3 + 56*a^2*c^6*d^7
*f^4 - 84*a^3*c^5*d^6*f^5 + 70*a^4*c^4*d^5*f^6 - 28*a^5*c^3*d^4*f^7 + 4*a^7*c*d^2*f^9 - a^8*d*f^10)*e)*sqrt(-(
c^6*e^10 - 2*(4*c^6*d*f - 3*a*c^5*f^2)*e^8 + (22*c^6*d^2*f^2 - 36*a*c^5*d*f^3 + 15*a^2*c^4*f^4)*e^6 - 6*(4*c^6
*d^3*f^3 - 11*a*c^5*d^2*f^4 + 10*a^2*c^4*d*f^5 - 3*a^3*c^3*f^6)*e^4 + 9*(c^6*d^4*f^4 - 4*a*c^5*d^3*f^5 + 6*a^2
*c^4*d^2*f^6 - 4*a^3*c^3*d*f^7 + a^4*c^2*f^8)*e^2)/(4*c^12*d^13*f - 48*a*c^11*d^12*f^2 + 264*a^2*c^10*d^11*f^3
- 880*a^3*c^9*d^10*f^4 + 1980*a^4*c^8*d^9*f^5 - 3168*a^5*c^7*d^8*f^6 + 3696*a^6*c^6*d^7*f^7 - 3168*a^7*c^5*d^
6*f^8 + 1980*a^8*c^4*d^5*f^9 - 880*a^9*c^3*d^4*f^10 + 264*a^10*c^2*d^3*f^11 - 48*a^11*c*d^2*f^12 + 4*a^12*d*f^
13 - a^6*c^6*e^14 - 2*(3*a^5*c^7*d^2 - 8*a^6*c^6*d*f + 3*a^7*c^5*f^2)*e^12 - 3*(5*a^4*c^8*d^4 - 28*a^5*c^7*d^3
*f + 46*a^6*c^6*d^2*f^2 - 28*a^7*c^5*d*f^3 + 5*a^8*c^4*f^4)*e^10 - 20*(a^3*c^9*d^6 - 9*a^4*c^8*d^5*f + 27*a^5*
c^7*d^4*f^2 - 38*a^6*c^6*d^3*f^3 + 27*a^7*c^5*d^2*f^4 - 9*a^8*c^4*d*f^5 + a^9*c^3*f^6)*e^8 - 5*(3*a^2*c^10*d^8
- 40*a^3*c^9*d^7*f + 180*a^4*c^8*d^6*f^2 - 408*a^5*c^7*d^5*f^3 + 530*a^6*c^6*d^4*f^4 - 408*a^7*c^5*d^3*f^5 +
180*a^8*c^4*d^2*f^6 - 40*a^9*c^3*d*f^7 + 3*a^10*c^2*f^8)*e^6 - 6*(a*c^11*d^10 - 20*a^2*c^10*d^9*f + 125*a^3*c^
9*d^8*f^2 - 400*a^4*c^8*d^7*f^3 + 770*a^5*c^7*d...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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/(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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