3.279 \(\int \frac {a+b x^2+c x^4}{\sqrt {d+e x^2}} \, dx\)

Optimal. Leaf size=97 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac {x \sqrt {d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac {c x^3 \sqrt {d+e x^2}}{4 e} \]

[Out]

1/8*(8*a*e^2-4*b*d*e+3*c*d^2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(5/2)-1/8*(-4*b*e+3*c*d)*x*(e*x^2+d)^(1/2)/
e^2+1/4*c*x^3*(e*x^2+d)^(1/2)/e

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Rubi [A]  time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1159, 388, 217, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )}{8 e^{5/2}}-\frac {x \sqrt {d+e x^2} (3 c d-4 b e)}{8 e^2}+\frac {c x^3 \sqrt {d+e x^2}}{4 e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/Sqrt[d + e*x^2],x]

[Out]

-((3*c*d - 4*b*e)*x*Sqrt[d + e*x^2])/(8*e^2) + (c*x^3*Sqrt[d + e*x^2])/(4*e) + ((3*c*d^2 - 4*b*d*e + 8*a*e^2)*
ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*e^(5/2))

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 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 1159

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

Rubi steps

\begin {align*} \int \frac {a+b x^2+c x^4}{\sqrt {d+e x^2}} \, dx &=\frac {c x^3 \sqrt {d+e x^2}}{4 e}+\frac {\int \frac {4 a e-(3 c d-4 b e) x^2}{\sqrt {d+e x^2}} \, dx}{4 e}\\ &=-\frac {(3 c d-4 b e) x \sqrt {d+e x^2}}{8 e^2}+\frac {c x^3 \sqrt {d+e x^2}}{4 e}-\frac {1}{8} \left (-8 a-\frac {d (3 c d-4 b e)}{e^2}\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {(3 c d-4 b e) x \sqrt {d+e x^2}}{8 e^2}+\frac {c x^3 \sqrt {d+e x^2}}{4 e}-\frac {1}{8} \left (-8 a-\frac {d (3 c d-4 b e)}{e^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )\\ &=-\frac {(3 c d-4 b e) x \sqrt {d+e x^2}}{8 e^2}+\frac {c x^3 \sqrt {d+e x^2}}{4 e}+\frac {\left (3 c d^2-4 b d e+8 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 e^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 82, normalized size = 0.85 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (8 a e^2-4 b d e+3 c d^2\right )+\sqrt {e} x \sqrt {d+e x^2} \left (4 b e-3 c d+2 c e x^2\right )}{8 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/Sqrt[d + e*x^2],x]

[Out]

(Sqrt[e]*x*Sqrt[d + e*x^2]*(-3*c*d + 4*b*e + 2*c*e*x^2) + (3*c*d^2 - 4*b*d*e + 8*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sq
rt[d + e*x^2]])/(8*e^(5/2))

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fricas [A]  time = 1.07, size = 174, normalized size = 1.79 \[ \left [\frac {{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (2 \, c e^{2} x^{3} - {\left (3 \, c d e - 4 \, b e^{2}\right )} x\right )} \sqrt {e x^{2} + d}}{16 \, e^{3}}, -\frac {{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (2 \, c e^{2} x^{3} - {\left (3 \, c d e - 4 \, b e^{2}\right )} x\right )} \sqrt {e x^{2} + d}}{8 \, e^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/16*((3*c*d^2 - 4*b*d*e + 8*a*e^2)*sqrt(e)*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(2*c*e^2*x^3
- (3*c*d*e - 4*b*e^2)*x)*sqrt(e*x^2 + d))/e^3, -1/8*((3*c*d^2 - 4*b*d*e + 8*a*e^2)*sqrt(-e)*arctan(sqrt(-e)*x/
sqrt(e*x^2 + d)) - (2*c*e^2*x^3 - (3*c*d*e - 4*b*e^2)*x)*sqrt(e*x^2 + d))/e^3]

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giac [A]  time = 0.19, size = 79, normalized size = 0.81 \[ -\frac {1}{8} \, {\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -x e^{\frac {1}{2}} + \sqrt {x^{2} e + d} \right |}\right ) + \frac {1}{8} \, {\left (2 \, c x^{2} e^{\left (-1\right )} - {\left (3 \, c d e - 4 \, b e^{2}\right )} e^{\left (-3\right )}\right )} \sqrt {x^{2} e + d} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

-1/8*(3*c*d^2 - 4*b*d*e + 8*a*e^2)*e^(-5/2)*log(abs(-x*e^(1/2) + sqrt(x^2*e + d))) + 1/8*(2*c*x^2*e^(-1) - (3*
c*d*e - 4*b*e^2)*e^(-3))*sqrt(x^2*e + d)*x

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maple [A]  time = 0.01, size = 122, normalized size = 1.26 \[ \frac {\sqrt {e \,x^{2}+d}\, c \,x^{3}}{4 e}+\frac {a \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{\sqrt {e}}-\frac {b d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}+\frac {3 c \,d^{2} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{8 e^{\frac {5}{2}}}+\frac {\sqrt {e \,x^{2}+d}\, b x}{2 e}-\frac {3 \sqrt {e \,x^{2}+d}\, c d x}{8 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x)

[Out]

1/4*c*x^3*(e*x^2+d)^(1/2)/e-3/8*c*d/e^2*x*(e*x^2+d)^(1/2)+3/8*c*d^2/e^(5/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))+1/2*
b*x/e*(e*x^2+d)^(1/2)-1/2*b*d/e^(3/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))+a*ln(e^(1/2)*x+(e*x^2+d)^(1/2))/e^(1/2)

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maxima [A]  time = 1.07, size = 100, normalized size = 1.03 \[ \frac {\sqrt {e x^{2} + d} c x^{3}}{4 \, e} - \frac {3 \, \sqrt {e x^{2} + d} c d x}{8 \, e^{2}} + \frac {\sqrt {e x^{2} + d} b x}{2 \, e} + \frac {3 \, c d^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{8 \, e^{\frac {5}{2}}} - \frac {b d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{2 \, e^{\frac {3}{2}}} + \frac {a \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(e*x^2 + d)*c*x^3/e - 3/8*sqrt(e*x^2 + d)*c*d*x/e^2 + 1/2*sqrt(e*x^2 + d)*b*x/e + 3/8*c*d^2*arcsinh(e*
x/sqrt(d*e))/e^(5/2) - 1/2*b*d*arcsinh(e*x/sqrt(d*e))/e^(3/2) + a*arcsinh(e*x/sqrt(d*e))/sqrt(e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c\,x^4+b\,x^2+a}{\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/(d + e*x^2)^(1/2),x)

[Out]

int((a + b*x^2 + c*x^4)/(d + e*x^2)^(1/2), x)

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sympy [A]  time = 7.05, size = 230, normalized size = 2.37 \[ a \left (\begin {cases} \frac {\sqrt {- \frac {d}{e}} \operatorname {asin}{\left (x \sqrt {- \frac {e}{d}} \right )}}{\sqrt {d}} & \text {for}\: d > 0 \wedge e < 0 \\\frac {\sqrt {\frac {d}{e}} \operatorname {asinh}{\left (x \sqrt {\frac {e}{d}} \right )}}{\sqrt {d}} & \text {for}\: d > 0 \wedge e > 0 \\\frac {\sqrt {- \frac {d}{e}} \operatorname {acosh}{\left (x \sqrt {- \frac {e}{d}} \right )}}{\sqrt {- d}} & \text {for}\: e > 0 \wedge d < 0 \end {cases}\right ) + \frac {b \sqrt {d} x \sqrt {1 + \frac {e x^{2}}{d}}}{2 e} - \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{2 e^{\frac {3}{2}}} - \frac {3 c d^{\frac {3}{2}} x}{8 e^{2} \sqrt {1 + \frac {e x^{2}}{d}}} - \frac {c \sqrt {d} x^{3}}{8 e \sqrt {1 + \frac {e x^{2}}{d}}} + \frac {3 c d^{2} \operatorname {asinh}{\left (\frac {\sqrt {e} x}{\sqrt {d}} \right )}}{8 e^{\frac {5}{2}}} + \frac {c x^{5}}{4 \sqrt {d} \sqrt {1 + \frac {e x^{2}}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

a*Piecewise((sqrt(-d/e)*asin(x*sqrt(-e/d))/sqrt(d), (d > 0) & (e < 0)), (sqrt(d/e)*asinh(x*sqrt(e/d))/sqrt(d),
(d > 0) & (e > 0)), (sqrt(-d/e)*acosh(x*sqrt(-e/d))/sqrt(-d), (e > 0) & (d < 0))) + b*sqrt(d)*x*sqrt(1 + e*x*
*2/d)/(2*e) - b*d*asinh(sqrt(e)*x/sqrt(d))/(2*e**(3/2)) - 3*c*d**(3/2)*x/(8*e**2*sqrt(1 + e*x**2/d)) - c*sqrt(
d)*x**3/(8*e*sqrt(1 + e*x**2/d)) + 3*c*d**2*asinh(sqrt(e)*x/sqrt(d))/(8*e**(5/2)) + c*x**5/(4*sqrt(d)*sqrt(1 +
e*x**2/d))

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