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

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

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Rubi [A]  time = 0.11, 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 = {545, 388, 217, 206, 523, 377, 205} \begin {gather*} \frac {(b c-a d)^2 \tan ^{-1}\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 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 206

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

Rule 217

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 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 388

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 523

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 545

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)) \operatorname {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 \operatorname {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)) \operatorname {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 = 0.34, size = 150, normalized size = 0.90 \begin {gather*} \frac {\frac {2 f (b c-a d)^2 \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )+b^2 \sqrt {c} d x \sqrt {e+f x^2} \sqrt {d e-c f}}{\sqrt {c} f \sqrt {d e-c f}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right ) (-4 a d f+2 b c f+b d e)}{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*Sqrt[c]*d*Sqrt[d*e - c*f]*x*Sqrt[e + f*x^2] + 2*(b*c - a*d)^2*f*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt
[e + f*x^2])])/(Sqrt[c]*f*Sqrt[d*e - c*f]) - (b*(b*d*e + 2*b*c*f - 4*a*d*f)*ArcTanh[(Sqrt[f]*x)/Sqrt[e + f*x^2
]])/f^(3/2))/(2*d^2)

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IntegrateAlgebraic [A]  time = 0.36, size = 175, normalized size = 1.05 \begin {gather*} \frac {\left (-a^2 d^2+2 a b c d-b^2 c^2\right ) \tan ^{-1}\left (\frac {c \sqrt {f}-d x \sqrt {e+f x^2}+d \sqrt {f} x^2}{\sqrt {c} \sqrt {d e-c f}}\right )}{\sqrt {c} d^2 \sqrt {d e-c f}}+\frac {\log \left (\sqrt {e+f x^2}-\sqrt {f} x\right ) \left (-4 a b d f+2 b^2 c f+b^2 d e\right )}{2 d^2 f^{3/2}}+\frac {b^2 x \sqrt {e+f x^2}}{2 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 6.22, size = 1111, normalized size = 6.69 \begin {gather*} \left [-\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c d e + c^{2} f} f^{2} \log \left (\frac {{\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \, {\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2} - 4 \, {\left ({\left (d e - 2 \, c f\right )} x^{3} - c e x\right )} \sqrt {-c d e + c^{2} f} \sqrt {f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 2 \, {\left (b^{2} c d^{2} e f - b^{2} c^{2} d f^{2}\right )} \sqrt {f x^{2} + e} x + {\left (b^{2} c d^{2} e^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} e f - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2}\right )} \sqrt {f} \log \left (-2 \, f x^{2} - 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right )}{4 \, {\left (c d^{3} e f^{2} - c^{2} d^{2} f^{3}\right )}}, \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d e - c^{2} f} f^{2} \arctan \left (\frac {\sqrt {c d e - c^{2} f} {\left ({\left (d e - 2 \, c f\right )} x^{2} - c e\right )} \sqrt {f x^{2} + e}}{2 \, {\left ({\left (c d e f - c^{2} f^{2}\right )} x^{3} + {\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right ) + 2 \, {\left (b^{2} c d^{2} e f - b^{2} c^{2} d f^{2}\right )} \sqrt {f x^{2} + e} x - {\left (b^{2} c d^{2} e^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} e f - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2}\right )} \sqrt {f} \log \left (-2 \, f x^{2} - 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right )}{4 \, {\left (c d^{3} e f^{2} - c^{2} d^{2} f^{3}\right )}}, -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c d e + c^{2} f} f^{2} \log \left (\frac {{\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \, {\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2} - 4 \, {\left ({\left (d e - 2 \, c f\right )} x^{3} - c e x\right )} \sqrt {-c d e + c^{2} f} \sqrt {f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) - 2 \, {\left (b^{2} c d^{2} e f - b^{2} c^{2} d f^{2}\right )} \sqrt {f x^{2} + e} x - 2 \, {\left (b^{2} c d^{2} e^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} e f - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2}\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right )}{4 \, {\left (c d^{3} e f^{2} - c^{2} d^{2} f^{3}\right )}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d e - c^{2} f} f^{2} \arctan \left (\frac {\sqrt {c d e - c^{2} f} {\left ({\left (d e - 2 \, c f\right )} x^{2} - c e\right )} \sqrt {f x^{2} + e}}{2 \, {\left ({\left (c d e f - c^{2} f^{2}\right )} x^{3} + {\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right ) + {\left (b^{2} c d^{2} e f - b^{2} c^{2} d f^{2}\right )} \sqrt {f x^{2} + e} x + {\left (b^{2} c d^{2} e^{2} + {\left (b^{2} c^{2} d - 4 \, a b c d^{2}\right )} e f - 2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d\right )} f^{2}\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right )}{2 \, {\left (c d^{3} e f^{2} - c^{2} d^{2} f^{3}\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*d*e + c^2*f)*f^2*log(((d^2*e^2 - 8*c*d*e*f + 8*c^2*f^2)*x^4 + c
^2*e^2 - 2*(3*c*d*e^2 - 4*c^2*e*f)*x^2 - 4*((d*e - 2*c*f)*x^3 - c*e*x)*sqrt(-c*d*e + c^2*f)*sqrt(f*x^2 + e))/(
d^2*x^4 + 2*c*d*x^2 + c^2)) - 2*(b^2*c*d^2*e*f - b^2*c^2*d*f^2)*sqrt(f*x^2 + e)*x + (b^2*c*d^2*e^2 + (b^2*c^2*
d - 4*a*b*c*d^2)*e*f - 2*(b^2*c^3 - 2*a*b*c^2*d)*f^2)*sqrt(f)*log(-2*f*x^2 - 2*sqrt(f*x^2 + e)*sqrt(f)*x - e))
/(c*d^3*e*f^2 - c^2*d^2*f^3), 1/4*(2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d*e - c^2*f)*f^2*arctan(1/2*sqrt(c
*d*e - c^2*f)*((d*e - 2*c*f)*x^2 - c*e)*sqrt(f*x^2 + e)/((c*d*e*f - c^2*f^2)*x^3 + (c*d*e^2 - c^2*e*f)*x)) + 2
*(b^2*c*d^2*e*f - b^2*c^2*d*f^2)*sqrt(f*x^2 + e)*x - (b^2*c*d^2*e^2 + (b^2*c^2*d - 4*a*b*c*d^2)*e*f - 2*(b^2*c
^3 - 2*a*b*c^2*d)*f^2)*sqrt(f)*log(-2*f*x^2 - 2*sqrt(f*x^2 + e)*sqrt(f)*x - e))/(c*d^3*e*f^2 - c^2*d^2*f^3), -
1/4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-c*d*e + c^2*f)*f^2*log(((d^2*e^2 - 8*c*d*e*f + 8*c^2*f^2)*x^4 + c^2
*e^2 - 2*(3*c*d*e^2 - 4*c^2*e*f)*x^2 - 4*((d*e - 2*c*f)*x^3 - c*e*x)*sqrt(-c*d*e + c^2*f)*sqrt(f*x^2 + e))/(d^
2*x^4 + 2*c*d*x^2 + c^2)) - 2*(b^2*c*d^2*e*f - b^2*c^2*d*f^2)*sqrt(f*x^2 + e)*x - 2*(b^2*c*d^2*e^2 + (b^2*c^2*
d - 4*a*b*c*d^2)*e*f - 2*(b^2*c^3 - 2*a*b*c^2*d)*f^2)*sqrt(-f)*arctan(sqrt(-f)*x/sqrt(f*x^2 + e)))/(c*d^3*e*f^
2 - c^2*d^2*f^3), 1/2*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c*d*e - c^2*f)*f^2*arctan(1/2*sqrt(c*d*e - c^2*f)*
((d*e - 2*c*f)*x^2 - c*e)*sqrt(f*x^2 + e)/((c*d*e*f - c^2*f^2)*x^3 + (c*d*e^2 - c^2*e*f)*x)) + (b^2*c*d^2*e*f
- b^2*c^2*d*f^2)*sqrt(f*x^2 + e)*x + (b^2*c*d^2*e^2 + (b^2*c^2*d - 4*a*b*c*d^2)*e*f - 2*(b^2*c^3 - 2*a*b*c^2*d
)*f^2)*sqrt(-f)*arctan(sqrt(-f)*x/sqrt(f*x^2 + e)))/(c*d^3*e*f^2 - c^2*d^2*f^3)]

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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,x):;OUTPUT:

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maple [B]  time = 0.02, size = 1052, normalized size = 6.34 \begin {gather*} -\frac {a^{2} \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {a^{2} \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {a b c \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{\sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}\, d}-\frac {a b c \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{\sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}\, d}-\frac {b^{2} c^{2} \ln \left (\frac {\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}\, d^{2}}+\frac {b^{2} c^{2} \ln \left (\frac {-\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {2 \left (c f -d e \right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right ) f}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}\, d^{2}}+\frac {2 a b \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{d \sqrt {f}}-\frac {b^{2} c \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{d^{2} \sqrt {f}}-\frac {b^{2} e \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{2 d \,f^{\frac {3}{2}}}+\frac {\sqrt {f \,x^{2}+e}\, b^{2} x}{2 d f} \end {gather*}

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)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \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]

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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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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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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