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

Optimal. Leaf size=91 \[ \frac {b \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d \sqrt {f}}-\frac {(b c-a d) \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d \sqrt {d e-c f}} \]

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Rubi [A]  time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {523, 217, 206, 377, 205} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{d \sqrt {f}}-\frac {(b c-a d) \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} d \sqrt {d e-c f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 88, normalized size = 0.97 \begin {gather*} \frac {\frac {(a d-b c) \tan ^{-1}\left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} \sqrt {d e-c f}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e+f x^2}}\right )}{\sqrt {f}}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.25, size = 114, normalized size = 1.25 \begin {gather*} \frac {(b c-a d) \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 \sqrt {d e-c f}}-\frac {b \log \left (\sqrt {e+f x^2}-\sqrt {f} x\right )}{d \sqrt {f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)*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*S
qrt[d*e - c*f]) - (b*Log[-(Sqrt[f]*x) + Sqrt[e + f*x^2]])/(d*Sqrt[f])

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fricas [B]  time = 4.14, size = 737, normalized size = 8.10 \begin {gather*} \left [\frac {\sqrt {-c d e + c^{2} f} {\left (b c - a d\right )} f \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 c d e - b c^{2} f\right )} \sqrt {f} \log \left (-2 \, f x^{2} - 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right )}{4 \, {\left (c d^{2} e f - c^{2} d f^{2}\right )}}, -\frac {\sqrt {c d e - c^{2} f} {\left (b c - a d\right )} f \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 c d e - b c^{2} f\right )} \sqrt {f} \log \left (-2 \, f x^{2} - 2 \, \sqrt {f x^{2} + e} \sqrt {f} x - e\right )}{2 \, {\left (c d^{2} e f - c^{2} d f^{2}\right )}}, \frac {\sqrt {-c d e + c^{2} f} {\left (b c - a d\right )} f \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 ) - 4 \, {\left (b c d e - b c^{2} f\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right )}{4 \, {\left (c d^{2} e f - c^{2} d f^{2}\right )}}, -\frac {\sqrt {c d e - c^{2} f} {\left (b c - a d\right )} f \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 c d e - b c^{2} f\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {-f} x}{\sqrt {f x^{2} + e}}\right )}{2 \, {\left (c d^{2} e f - c^{2} d f^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/(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 = 0.01, size = 646, normalized size = 7.10 \begin {gather*} -\frac {a \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 \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 {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 )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}\, d}-\frac {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 )}{2 \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}\, d}+\frac {b \ln \left (\sqrt {f}\, x +\sqrt {f \,x^{2}+e}\right )}{d \sqrt {f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b/d*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+1/2/(-c*d)^(1/2)/d/(-(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*c+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-1/2/(-c*d)^(1/2)/d/(-(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*c

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

[Out]

integrate((b*x^2 + a)/((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 {b\,x^2+a}{\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)/((c + d*x^2)*(e + f*x^2)^(1/2)),x)

[Out]

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

[Out]

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

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