3.2.6 \(\int \frac {1}{(a+b x+c x^2)^{3/2} (d-f x^2)} \, dx\) [106]

Optimal. Leaf size=310 \[ -\frac {2 \left (b \left (b^2 f-c (c d+3 a f)\right )-c \left (2 c^2 d-b^2 f+2 a c f\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {f \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {989, 1047, 738, 212} \begin {gather*} -\frac {2 \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {f \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {d} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {f \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {d} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 989

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b^3*f + b*c*(c*d
- 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b
^2*d*f + (c*d - a*f)^2)*(p + 1))), x] - Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x
+ c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[2*c*(b^2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*
f*(p + 1) - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(
p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x]
&& NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &
&  !IGtQ[q, 0]

Rule 1047

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.72, size = 379, normalized size = 1.22 \begin {gather*} \frac {-4 b^3 f+4 b c (c d+3 a f)-4 b^2 c f x+8 c^2 (c d+a f) x+\left (b^2-4 a c\right ) f \sqrt {a+x (b+c x)} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 a b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 \left (b^2-4 a c\right ) \left (-c^2 d^2-2 a c d f+f \left (b^2 d-a^2 f\right )\right ) \sqrt {a+x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b^3*f + 4*b*c*(c*d + 3*a*f) - 4*b^2*c*f*x + 8*c^2*(c*d + a*f)*x + (b^2 - 4*a*c)*f*Sqrt[a + x*(b + c*x)]*Ro
otSum[b^2*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b*c*d*Log[-(Sqrt[c]*x) + Sqrt[a
+ b*x + c*x^2] - #1] + 2*a*b*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*c^(3/2)*d*Log[-(Sqrt[c]*x)
+ Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a*Sqrt[c]*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - b*f*Log[-
(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#1^3) & ])/(2*(b^2 - 4*a*
c)*(-(c^2*d^2) - 2*a*c*d*f + f*(b^2*d - a^2*f))*Sqrt[a + x*(b + c*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(902\) vs. \(2(252)=504\).
time = 0.13, size = 903, normalized size = 2.91

method	result	size
default	\(\frac {\frac {f}{\left (-b \sqrt {d f}+f a +c d \right ) \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (2 c \left (x +\frac {\sqrt {d f}}{f}\right )+\frac {-2 c \sqrt {d f}+b f}{f}\right )}{\left (-b \sqrt {d f}+f a +c d \right ) \left (\frac {4 c \left (-b \sqrt {d f}+f a +c d \right )}{f}-\frac {\left (-2 c \sqrt {d f}+b f \right )^{2}}{f^{2}}\right ) \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {f \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\left (-b \sqrt {d f}+f a +c d \right ) \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 \sqrt {d f}}-\frac {\frac {f}{\left (b \sqrt {d f}+f a +c d \right ) \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (2 c \sqrt {d f}+b f \right ) \left (2 c \left (x -\frac {\sqrt {d f}}{f}\right )+\frac {2 c \sqrt {d f}+b f}{f}\right )}{\left (b \sqrt {d f}+f a +c d \right ) \left (\frac {4 c \left (b \sqrt {d f}+f a +c d \right )}{f}-\frac {\left (2 c \sqrt {d f}+b f \right )^{2}}{f^{2}}\right ) \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}-\frac {f \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\left (b \sqrt {d f}+f a +c d \right ) \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 \sqrt {d f}}\)	\(903\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+b*x+a)^(3/2)/(-f*x^2+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(((c*sqrt(4*d*f))/(2*f^2)>0)',
see `assume?

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17397 vs. \(2 (252) = 504\).
time = 13.36, size = 17397, normalized size = 56.12 \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+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="fricas")

[Out]

1/4*(((a*b^2*c^2 - 4*a^2*c^3)*d^2 - (a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d*f + (a^3*b^2 - 4*a^4*c)*f^2 + ((b^2*c^
3 - 4*a*c^4)*d^2 - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d*f + (a^2*b^2*c - 4*a^3*c^2)*f^2)*x^2 + ((b^3*c^2 - 4*a*
b*c^3)*d^2 - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d*f + (a^2*b^3 - 4*a^3*b*c)*f^2)*x)*sqrt((c^3*d^3*f^2 + a^3*f^5 +
3*(b^2*c + a*c^2)*d^2*f^3 + 3*(a*b^2 + a^2*c)*d*f^4 + (c^6*d^7 + a^6*d*f^6 - 3*(b^2*c^4 - 2*a*c^5)*d^6*f + 3*
(b^4*c^2 - 4*a*b^2*c^3 + 5*a^2*c^4)*d^5*f^2 - (b^6 - 6*a*b^4*c + 18*a^2*b^2*c^2 - 20*a^3*c^3)*d^4*f^3 + 3*(a^2
*b^4 - 4*a^3*b^2*c + 5*a^4*c^2)*d^3*f^4 - 3*(a^4*b^2 - 2*a^5*c)*d^2*f^5)*sqrt((9*b^2*c^4*d^4*f^5 + 9*a^4*b^2*f
^9 + 6*(b^4*c^2 + 6*a*b^2*c^3)*d^3*f^6 + (b^6 + 12*a*b^4*c + 54*a^2*b^2*c^2)*d^2*f^7 + 6*(a^2*b^4 + 6*a^3*b^2*
c)*d*f^8)/(c^12*d^13 + a^12*d*f^12 - 6*(b^2*c^10 - 2*a*c^11)*d^12*f + 3*(5*b^4*c^8 - 20*a*b^2*c^9 + 22*a^2*c^1
0)*d^11*f^2 - 10*(2*b^6*c^6 - 12*a*b^4*c^7 + 27*a^2*b^2*c^8 - 22*a^3*c^9)*d^10*f^3 + 15*(b^8*c^4 - 8*a*b^6*c^5
+ 28*a^2*b^4*c^6 - 48*a^3*b^2*c^7 + 33*a^4*c^8)*d^9*f^4 - 6*(b^10*c^2 - 10*a*b^8*c^3 + 50*a^2*b^6*c^4 - 140*a
^3*b^4*c^5 + 210*a^4*b^2*c^6 - 132*a^5*c^7)*d^8*f^5 + (b^12 - 12*a*b^10*c + 90*a^2*b^8*c^2 - 400*a^3*b^6*c^3 +
1050*a^4*b^4*c^4 - 1512*a^5*b^2*c^5 + 924*a^6*c^6)*d^7*f^6 - 6*(a^2*b^10 - 10*a^3*b^8*c + 50*a^4*b^6*c^2 - 14
0*a^5*b^4*c^3 + 210*a^6*b^2*c^4 - 132*a^7*c^5)*d^6*f^7 + 15*(a^4*b^8 - 8*a^5*b^6*c + 28*a^6*b^4*c^2 - 48*a^7*b
^2*c^3 + 33*a^8*c^4)*d^5*f^8 - 10*(2*a^6*b^6 - 12*a^7*b^4*c + 27*a^8*b^2*c^2 - 22*a^9*c^3)*d^4*f^9 + 3*(5*a^8*
b^4 - 20*a^9*b^2*c + 22*a^10*c^2)*d^3*f^10 - 6*(a^10*b^2 - 2*a^11*c)*d^2*f^11)))/(c^6*d^7 + a^6*d*f^6 - 3*(b^2
*c^4 - 2*a*c^5)*d^6*f + 3*(b^4*c^2 - 4*a*b^2*c^3 + 5*a^2*c^4)*d^5*f^2 - (b^6 - 6*a*b^4*c + 18*a^2*b^2*c^2 - 20
*a^3*c^3)*d^4*f^3 + 3*(a^2*b^4 - 4*a^3*b^2*c + 5*a^4*c^2)*d^3*f^4 - 3*(a^4*b^2 - 2*a^5*c)*d^2*f^5))*log((3*b^2
*c^2*d^2*f^3 + 3*a^2*b^2*f^5 + (b^4 + 6*a*b^2*c)*d*f^4 + 2*(3*b*c^3*d^2*f^3 + 3*a^2*b*c*f^5 + (b^3*c + 6*a*b*c
^2)*d*f^4)*x + 2*(3*b*c^4*d^4*f^2 + 3*a^4*b*f^6 + 4*(b^3*c^2 + 3*a*b*c^3)*d^3*f^3 + (b^5 + 8*a*b^3*c + 18*a^2*
b*c^2)*d^2*f^4 + 4*(a^2*b^3 + 3*a^3*b*c)*d*f^5 - 2*(b*c^7*d^8 + a^7*b*d*f^7 - (3*b^3*c^5 - 7*a*b*c^6)*d^7*f +
3*(b^5*c^3 - 5*a*b^3*c^4 + 7*a^2*b*c^5)*d^6*f^2 - (b^7*c - 9*a*b^5*c^2 + 30*a^2*b^3*c^3 - 35*a^3*b*c^4)*d^5*f^
3 - (a*b^7 - 9*a^2*b^5*c + 30*a^3*b^3*c^2 - 35*a^4*b*c^3)*d^4*f^4 + 3*(a^3*b^5 - 5*a^4*b^3*c + 7*a^5*b*c^2)*d^
3*f^5 - (3*a^5*b^3 - 7*a^6*b*c)*d^2*f^6)*sqrt((9*b^2*c^4*d^4*f^5 + 9*a^4*b^2*f^9 + 6*(b^4*c^2 + 6*a*b^2*c^3)*d
^3*f^6 + (b^6 + 12*a*b^4*c + 54*a^2*b^2*c^2)*d^2*f^7 + 6*(a^2*b^4 + 6*a^3*b^2*c)*d*f^8)/(c^12*d^13 + a^12*d*f^
12 - 6*(b^2*c^10 - 2*a*c^11)*d^12*f + 3*(5*b^4*c^8 - 20*a*b^2*c^9 + 22*a^2*c^10)*d^11*f^2 - 10*(2*b^6*c^6 - 12
*a*b^4*c^7 + 27*a^2*b^2*c^8 - 22*a^3*c^9)*d^10*f^3 + 15*(b^8*c^4 - 8*a*b^6*c^5 + 28*a^2*b^4*c^6 - 48*a^3*b^2*c
^7 + 33*a^4*c^8)*d^9*f^4 - 6*(b^10*c^2 - 10*a*b^8*c^3 + 50*a^2*b^6*c^4 - 140*a^3*b^4*c^5 + 210*a^4*b^2*c^6 - 1
32*a^5*c^7)*d^8*f^5 + (b^12 - 12*a*b^10*c + 90*a^2*b^8*c^2 - 400*a^3*b^6*c^3 + 1050*a^4*b^4*c^4 - 1512*a^5*b^2
*c^5 + 924*a^6*c^6)*d^7*f^6 - 6*(a^2*b^10 - 10*a^3*b^8*c + 50*a^4*b^6*c^2 - 140*a^5*b^4*c^3 + 210*a^6*b^2*c^4
- 132*a^7*c^5)*d^6*f^7 + 15*(a^4*b^8 - 8*a^5*b^6*c + 28*a^6*b^4*c^2 - 48*a^7*b^2*c^3 + 33*a^8*c^4)*d^5*f^8 - 1
0*(2*a^6*b^6 - 12*a^7*b^4*c + 27*a^8*b^2*c^2 - 22*a^9*c^3)*d^4*f^9 + 3*(5*a^8*b^4 - 20*a^9*b^2*c + 22*a^10*c^2
)*d^3*f^10 - 6*(a^10*b^2 - 2*a^11*c)*d^2*f^11)))*sqrt(c*x^2 + b*x + a)*sqrt((c^3*d^3*f^2 + a^3*f^5 + 3*(b^2*c
+ a*c^2)*d^2*f^3 + 3*(a*b^2 + a^2*c)*d*f^4 + (c^6*d^7 + a^6*d*f^6 - 3*(b^2*c^4 - 2*a*c^5)*d^6*f + 3*(b^4*c^2 -
4*a*b^2*c^3 + 5*a^2*c^4)*d^5*f^2 - (b^6 - 6*a*b^4*c + 18*a^2*b^2*c^2 - 20*a^3*c^3)*d^4*f^3 + 3*(a^2*b^4 - 4*a
^3*b^2*c + 5*a^4*c^2)*d^3*f^4 - 3*(a^4*b^2 - 2*a^5*c)*d^2*f^5)*sqrt((9*b^2*c^4*d^4*f^5 + 9*a^4*b^2*f^9 + 6*(b^
4*c^2 + 6*a*b^2*c^3)*d^3*f^6 + (b^6 + 12*a*b^4*c + 54*a^2*b^2*c^2)*d^2*f^7 + 6*(a^2*b^4 + 6*a^3*b^2*c)*d*f^8)/
(c^12*d^13 + a^12*d*f^12 - 6*(b^2*c^10 - 2*a*c^11)*d^12*f + 3*(5*b^4*c^8 - 20*a*b^2*c^9 + 22*a^2*c^10)*d^11*f^
2 - 10*(2*b^6*c^6 - 12*a*b^4*c^7 + 27*a^2*b^2*c^8 - 22*a^3*c^9)*d^10*f^3 + 15*(b^8*c^4 - 8*a*b^6*c^5 + 28*a^2*
b^4*c^6 - 48*a^3*b^2*c^7 + 33*a^4*c^8)*d^9*f^4 - 6*(b^10*c^2 - 10*a*b^8*c^3 + 50*a^2*b^6*c^4 - 140*a^3*b^4*c^5
+ 210*a^4*b^2*c^6 - 132*a^5*c^7)*d^8*f^5 + (b^12 - 12*a*b^10*c + 90*a^2*b^8*c^2 - 400*a^3*b^6*c^3 + 1050*a^4*
b^4*c^4 - 1512*a^5*b^2*c^5 + 924*a^6*c^6)*d^7*f^6 - 6*(a^2*b^10 - 10*a^3*b^8*c + 50*a^4*b^6*c^2 - 140*a^5*b^4*
c^3 + 210*a^6*b^2*c^4 - 132*a^7*c^5)*d^6*f^7 + 15*(a^4*b^8 - 8*a^5*b^6*c + 28*a^6*b^4*c^2 - 48*a^7*b^2*c^3 + 3
3*a^8*c^4)*d^5*f^8 - 10*(2*a^6*b^6 - 12*a^7*b^4*c + 27*a^8*b^2*c^2 - 22*a^9*c^3)*d^4*f^9 + 3*(5*a^8*b^4 - 20*a
^9*b^2*c + 22*a^10*c^2)*d^3*f^10 - 6*(a^10*b^2 - 2*a^11*c)*d^2*f^11)))/(c^6*d^7 + a^6*d*f^6 - 3*(b^2*c^4 - 2*a
*c^5)*d^6*f + 3*(b^4*c^2 - 4*a*b^2*c^3 + 5*a^2*c^4)*d^5*f^2 - (b^6 - 6*a*b^4*c + 18*a^2*b^2*c^2 - 20*a^3*c^3)*
d^4*f^3 + 3*(a^2*b^4 - 4*a^3*b^2*c + 5*a^4*c^2)...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- a d \sqrt {a + b x + c x^{2}} + a f x^{2} \sqrt {a + b x + c x^{2}} - b d x \sqrt {a + b x + c x^{2}} + b f x^{3} \sqrt {a + b x + c x^{2}} - c d x^{2} \sqrt {a + b x + c x^{2}} + c f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueDone

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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