3.1.59 \(\int \frac {(a+b x^2)^2}{(c+d x^2) \sqrt {e+f x^2}} \, dx\) [59]

Optimal. Leaf size=166 \[ \frac {b^2 x \sqrt {e+f x^2}}{2 d f}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d^2 \sqrt {d e-c f}}-\frac {b (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d^2 \sqrt {f}}-\frac {b (b e-2 a f) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{2 d f^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {559, 396, 223, 212, 537, 385, 211} \begin {gather*} \frac {(b c-a d)^2 \text {ArcTan}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d^2 \sqrt {d e-c f}}-\frac {b (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d^2 \sqrt {f}}-\frac {b (b e-2 a f) \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{2 d f^{3/2}}+\frac {b^2 x \sqrt {e+f x^2}}{2 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 559

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

Rubi steps

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

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Mathematica [A]
time = 0.32, size = 151, normalized size = 0.91 \begin {gather*} \frac {\frac {b^2 d x \sqrt {e+f x^2}}{f}-\frac {2 (b c-a d)^2 \tan ^{-1}\left (\frac {c \sqrt {f}+d x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {c} \sqrt {d e-c f}}\right )}{\sqrt {c} \sqrt {d e-c f}}+\frac {b (b d e+2 b c f-4 a d f) \log \left (-\sqrt {f} x+\sqrt {e+f x^2}\right )}{f^{3/2}}}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(138)=276\).
time = 0.14, size = 449, normalized size = 2.70

method	result	size
default	\(\frac {b \left (b d \left (\frac {x \sqrt {f \,x^{2}+e}}{2 f}-\frac {e \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{2 f^{\frac {3}{2}}}\right )+\frac {2 a d \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{\sqrt {f}}-\frac {b c \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{\sqrt {f}}\right )}{d^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}+\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 d^{2} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {\left (-a^{2} d^{2}+2 a b c d -b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}-\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 d^{2} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}\)	\(449\)
risch	\(\frac {b^{2} x \sqrt {f \,x^{2}+e}}{2 d f}+\frac {2 b \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right ) a}{d \sqrt {f}}-\frac {b^{2} \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right ) c}{d^{2} \sqrt {f}}-\frac {b^{2} \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right ) e}{2 d \,f^{\frac {3}{2}}}-\frac {\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 ) a^{2}}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {\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 ) a b c}{d \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {\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 ) b^{2} c^{2}}{2 d^{2} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {\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 ) a^{2}}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}-\frac {\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 ) a b c}{d \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {\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 ) b^{2} c^{2}}{2 d^{2} \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}\)	\(1052\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 2.52, size = 1166, normalized size = 7.02 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c^{2} f - c d e} f^{2} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - {\left (b^{2} c d^{2} e^{2} - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} f e\right )} \sqrt {f} \log \left (-2 \, f x^{2} + 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right ) + 2 \, {\left (b^{2} c^{2} d f^{2} x - b^{2} c d^{2} f x e\right )} \sqrt {f x^{2} + e}}{4 \, {\left (c^{2} d^{2} f^{3} - c d^{3} f^{2} e\right )}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c^{2} f - c d e} f^{2} \log \left (\frac {8 \, c^{2} f^{2} x^{4} + 4 \, {\left (2 \, c f x^{3} - {\left (d x^{3} - c x\right )} e\right )} \sqrt {c^{2} f - c d e} \sqrt {f x^{2} + e} + {\left (d^{2} x^{4} - 6 \, c d x^{2} + c^{2}\right )} e^{2} - 8 \, {\left (c d f x^{4} - c^{2} f x^{2}\right )} e}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 2 \, {\left (b^{2} c d^{2} e^{2} - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} f e\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right ) + 2 \, {\left (b^{2} c^{2} d f^{2} x - b^{2} c d^{2} f x e\right )} \sqrt {f x^{2} + e}}{4 \, {\left (c^{2} d^{2} f^{3} - c d^{3} f^{2} e\right )}}, -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c^{2} f + c d e} f^{2} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) + {\left (b^{2} c d^{2} e^{2} - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} f e\right )} \sqrt {f} \log \left (-2 \, f x^{2} + 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right ) - 2 \, {\left (b^{2} c^{2} d f^{2} x - b^{2} c d^{2} f x e\right )} \sqrt {f x^{2} + e}}{4 \, {\left (c^{2} d^{2} f^{3} - c d^{3} f^{2} e\right )}}, -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c^{2} f + c d e} f^{2} \arctan \left (\frac {{\left (2 \, c f x^{2} - {\left (d x^{2} - c\right )} e\right )} \sqrt {-c^{2} f + c d e} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} f^{2} x^{3} - c d x e^{2} - {\left (c d f x^{3} - c^{2} f x\right )} e\right )}}\right ) + {\left (b^{2} c d^{2} e^{2} - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} f e\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right ) - {\left (b^{2} c^{2} d f^{2} x - b^{2} c d^{2} f x e\right )} \sqrt {f x^{2} + e}}{2 \, {\left (c^{2} d^{2} f^{3} - c d^{3} f^{2} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c^2*f - c*d*e)*f^2*log((8*c^2*f^2*x^4 + 4*(2*c*f*x^3 - (d*x^3 - c*x
)*e)*sqrt(c^2*f - c*d*e)*sqrt(f*x^2 + e) + (d^2*x^4 - 6*c*d*x^2 + c^2)*e^2 - 8*(c*d*f*x^4 - c^2*f*x^2)*e)/(d^2
*x^4 + 2*c*d*x^2 + c^2)) - (b^2*c*d^2*e^2 - 2*(b^2*c^3 - 2*a*b*c^2*d)*f^2 + (b^2*c^2*d - 4*a*b*c*d^2)*f*e)*sqr
t(f)*log(-2*f*x^2 + 2*sqrt(f*x^2 + e)*sqrt(f)*x - e) + 2*(b^2*c^2*d*f^2*x - b^2*c*d^2*f*x*e)*sqrt(f*x^2 + e))/
(c^2*d^2*f^3 - c*d^3*f^2*e), 1/4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c^2*f - c*d*e)*f^2*log((8*c^2*f^2*x^4 +
4*(2*c*f*x^3 - (d*x^3 - c*x)*e)*sqrt(c^2*f - c*d*e)*sqrt(f*x^2 + e) + (d^2*x^4 - 6*c*d*x^2 + c^2)*e^2 - 8*(c*
d*f*x^4 - c^2*f*x^2)*e)/(d^2*x^4 + 2*c*d*x^2 + c^2)) - 2*(b^2*c*d^2*e^2 - 2*(b^2*c^3 - 2*a*b*c^2*d)*f^2 + (b^2
*c^2*d - 4*a*b*c*d^2)*f*e)*sqrt(-f)*arctan(sqrt(-f)*x/sqrt(f*x^2 + e)) + 2*(b^2*c^2*d*f^2*x - b^2*c*d^2*f*x*e)
*sqrt(f*x^2 + e))/(c^2*d^2*f^3 - c*d^3*f^2*e), -1/4*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-c^2*f + c*d*e)*f^
2*arctan(1/2*(2*c*f*x^2 - (d*x^2 - c)*e)*sqrt(-c^2*f + c*d*e)*sqrt(f*x^2 + e)/(c^2*f^2*x^3 - c*d*x*e^2 - (c*d*
f*x^3 - c^2*f*x)*e)) + (b^2*c*d^2*e^2 - 2*(b^2*c^3 - 2*a*b*c^2*d)*f^2 + (b^2*c^2*d - 4*a*b*c*d^2)*f*e)*sqrt(f)
*log(-2*f*x^2 + 2*sqrt(f*x^2 + e)*sqrt(f)*x - e) - 2*(b^2*c^2*d*f^2*x - b^2*c*d^2*f*x*e)*sqrt(f*x^2 + e))/(c^2
*d^2*f^3 - c*d^3*f^2*e), -1/2*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-c^2*f + c*d*e)*f^2*arctan(1/2*(2*c*f*x^2
- (d*x^2 - c)*e)*sqrt(-c^2*f + c*d*e)*sqrt(f*x^2 + e)/(c^2*f^2*x^3 - c*d*x*e^2 - (c*d*f*x^3 - c^2*f*x)*e)) + (
b^2*c*d^2*e^2 - 2*(b^2*c^3 - 2*a*b*c^2*d)*f^2 + (b^2*c^2*d - 4*a*b*c*d^2)*f*e)*sqrt(-f)*arctan(sqrt(-f)*x/sqrt
(f*x^2 + e)) - (b^2*c^2*d*f^2*x - b^2*c*d^2*f*x*e)*sqrt(f*x^2 + e))/(c^2*d^2*f^3 - c*d^3*f^2*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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((b*x**2+a)**2/(d*x**2+c)/(f*x**2+e)**(1/2),x)

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\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((a + b*x^2)^2/((c + d*x^2)*(e + f*x^2)^(1/2)),x)

[Out]

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

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