3.2.49 \(\int \frac {c+d x^2+e x^4+f x^6}{\sqrt {a+b x^2}} \, dx\)

Optimal. Leaf size=145 \[ \frac {x \sqrt {a+b x^2} \left (5 a^2 f-6 a b e+8 b^2 d\right )}{16 b^3}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right )}{16 b^{7/2}}+\frac {x^3 \sqrt {a+b x^2} (6 b e-5 a f)}{24 b^2}+\frac {f x^5 \sqrt {a+b x^2}}{6 b} \]

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Rubi [A]  time = 0.12, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1815, 1159, 388, 217, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (6 a^2 b e-5 a^3 f-8 a b^2 d+16 b^3 c\right )}{16 b^{7/2}}+\frac {x \sqrt {a+b x^2} \left (5 a^2 f-6 a b e+8 b^2 d\right )}{16 b^3}+\frac {x^3 \sqrt {a+b x^2} (6 b e-5 a f)}{24 b^2}+\frac {f x^5 \sqrt {a+b x^2}}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/Sqrt[a + b*x^2],x]

[Out]

((8*b^2*d - 6*a*b*e + 5*a^2*f)*x*Sqrt[a + b*x^2])/(16*b^3) + ((6*b*e - 5*a*f)*x^3*Sqrt[a + b*x^2])/(24*b^2) +
(f*x^5*Sqrt[a + b*x^2])/(6*b) + ((16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x
^2]])/(16*b^(7/2))

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]

Rule 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{\sqrt {a+b x^2}} \, dx &=\frac {f x^5 \sqrt {a+b x^2}}{6 b}+\frac {\int \frac {6 b c+6 b d x^2+(6 b e-5 a f) x^4}{\sqrt {a+b x^2}} \, dx}{6 b}\\ &=\frac {(6 b e-5 a f) x^3 \sqrt {a+b x^2}}{24 b^2}+\frac {f x^5 \sqrt {a+b x^2}}{6 b}+\frac {\int \frac {24 b^2 c+3 \left (8 b^2 d-6 a b e+5 a^2 f\right ) x^2}{\sqrt {a+b x^2}} \, dx}{24 b^2}\\ &=\frac {\left (8 b^2 d-6 a b e+5 a^2 f\right ) x \sqrt {a+b x^2}}{16 b^3}+\frac {(6 b e-5 a f) x^3 \sqrt {a+b x^2}}{24 b^2}+\frac {f x^5 \sqrt {a+b x^2}}{6 b}-\frac {1}{16} \left (-16 c+\frac {a \left (8 b^2 d-6 a b e+5 a^2 f\right )}{b^3}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {\left (8 b^2 d-6 a b e+5 a^2 f\right ) x \sqrt {a+b x^2}}{16 b^3}+\frac {(6 b e-5 a f) x^3 \sqrt {a+b x^2}}{24 b^2}+\frac {f x^5 \sqrt {a+b x^2}}{6 b}-\frac {1}{16} \left (-16 c+\frac {a \left (8 b^2 d-6 a b e+5 a^2 f\right )}{b^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {\left (8 b^2 d-6 a b e+5 a^2 f\right ) x \sqrt {a+b x^2}}{16 b^3}+\frac {(6 b e-5 a f) x^3 \sqrt {a+b x^2}}{24 b^2}+\frac {f x^5 \sqrt {a+b x^2}}{6 b}+\frac {\left (16 c-\frac {a \left (8 b^2 d-6 a b e+5 a^2 f\right )}{b^3}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 118, normalized size = 0.81 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (15 a^2 f-2 a b \left (9 e+5 f x^2\right )+4 b^2 \left (6 d+3 e x^2+2 f x^4\right )\right )+3 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-5 a^3 f+6 a^2 b e-8 a b^2 d+16 b^3 c\right )}{48 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(15*a^2*f - 2*a*b*(9*e + 5*f*x^2) + 4*b^2*(6*d + 3*e*x^2 + 2*f*x^4)) + 3*(16*b^3*c
- 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(48*b^(7/2))

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IntegrateAlgebraic [A]  time = 0.18, size = 123, normalized size = 0.85 \begin {gather*} \frac {\sqrt {a+b x^2} \left (15 a^2 f x-18 a b e x-10 a b f x^3+24 b^2 d x+12 b^2 e x^3+8 b^2 f x^5\right )}{48 b^3}+\frac {\log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \left (5 a^3 f-6 a^2 b e+8 a b^2 d-16 b^3 c\right )}{16 b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c + d*x^2 + e*x^4 + f*x^6)/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(24*b^2*d*x - 18*a*b*e*x + 15*a^2*f*x + 12*b^2*e*x^3 - 10*a*b*f*x^3 + 8*b^2*f*x^5))/(48*b^3)
+ ((-16*b^3*c + 8*a*b^2*d - 6*a^2*b*e + 5*a^3*f)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(16*b^(7/2))

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fricas [A]  time = 1.01, size = 250, normalized size = 1.72 \begin {gather*} \left [-\frac {3 \, {\left (16 \, b^{3} c - 8 \, a b^{2} d + 6 \, a^{2} b e - 5 \, a^{3} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, b^{3} f x^{5} + 2 \, {\left (6 \, b^{3} e - 5 \, a b^{2} f\right )} x^{3} + 3 \, {\left (8 \, b^{3} d - 6 \, a b^{2} e + 5 \, a^{2} b f\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{4}}, -\frac {3 \, {\left (16 \, b^{3} c - 8 \, a b^{2} d + 6 \, a^{2} b e - 5 \, a^{3} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{3} f x^{5} + 2 \, {\left (6 \, b^{3} e - 5 \, a b^{2} f\right )} x^{3} + 3 \, {\left (8 \, b^{3} d - 6 \, a b^{2} e + 5 \, a^{2} b f\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/96*(3*(16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a)
- 2*(8*b^3*f*x^5 + 2*(6*b^3*e - 5*a*b^2*f)*x^3 + 3*(8*b^3*d - 6*a*b^2*e + 5*a^2*b*f)*x)*sqrt(b*x^2 + a))/b^4,
-1/48*(3*(16*b^3*c - 8*a*b^2*d + 6*a^2*b*e - 5*a^3*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8*b^3*f*
x^5 + 2*(6*b^3*e - 5*a*b^2*f)*x^3 + 3*(8*b^3*d - 6*a*b^2*e + 5*a^2*b*f)*x)*sqrt(b*x^2 + a))/b^4]

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giac [A]  time = 0.56, size = 129, normalized size = 0.89 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, f x^{2}}{b} - \frac {5 \, a b^{3} f - 6 \, b^{4} e}{b^{5}}\right )} x^{2} + \frac {3 \, {\left (8 \, b^{4} d + 5 \, a^{2} b^{2} f - 6 \, a b^{3} e\right )}}{b^{5}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (16 \, b^{3} c - 8 \, a b^{2} d - 5 \, a^{3} f + 6 \, a^{2} b e\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/48*(2*(4*f*x^2/b - (5*a*b^3*f - 6*b^4*e)/b^5)*x^2 + 3*(8*b^4*d + 5*a^2*b^2*f - 6*a*b^3*e)/b^5)*sqrt(b*x^2 +
a)*x - 1/16*(16*b^3*c - 8*a*b^2*d - 5*a^3*f + 6*a^2*b*e)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)

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maple [A]  time = 0.01, size = 203, normalized size = 1.40 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, f \,x^{5}}{6 b}-\frac {5 \sqrt {b \,x^{2}+a}\, a f \,x^{3}}{24 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, e \,x^{3}}{4 b}-\frac {5 a^{3} f \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {7}{2}}}+\frac {3 a^{2} e \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}-\frac {a d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}+\frac {c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {5 \sqrt {b \,x^{2}+a}\, a^{2} f x}{16 b^{3}}-\frac {3 \sqrt {b \,x^{2}+a}\, a e x}{8 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, d x}{2 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x)

[Out]

1/6*f*x^5*(b*x^2+a)^(1/2)/b-5/24*f*a/b^2*x^3*(b*x^2+a)^(1/2)+5/16*f*a^2/b^3*x*(b*x^2+a)^(1/2)-5/16*f*a^3/b^(7/
2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+1/4*e*x^3/b*(b*x^2+a)^(1/2)-3/8*e*a/b^2*x*(b*x^2+a)^(1/2)+3/8*e*a^2/b^(5/2)*l
n(b^(1/2)*x+(b*x^2+a)^(1/2))+1/2*d*x/b*(b*x^2+a)^(1/2)-1/2*d*a/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+c*ln(b^(1
/2)*x+(b*x^2+a)^(1/2))/b^(1/2)

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maxima [A]  time = 1.35, size = 174, normalized size = 1.20 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{5}}{6 \, b} + \frac {\sqrt {b x^{2} + a} e x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a f x^{3}}{24 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x}{2 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a e x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} f x}{16 \, b^{3}} + \frac {c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {a d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {3 \, a^{2} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

1/6*sqrt(b*x^2 + a)*f*x^5/b + 1/4*sqrt(b*x^2 + a)*e*x^3/b - 5/24*sqrt(b*x^2 + a)*a*f*x^3/b^2 + 1/2*sqrt(b*x^2
+ a)*d*x/b - 3/8*sqrt(b*x^2 + a)*a*e*x/b^2 + 5/16*sqrt(b*x^2 + a)*a^2*f*x/b^3 + c*arcsinh(b*x/sqrt(a*b))/sqrt(
b) - 1/2*a*d*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 3/8*a^2*e*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 5/16*a^3*f*arcsinh(b*
x/sqrt(a*b))/b^(7/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f\,x^6+e\,x^4+d\,x^2+c}{\sqrt {b\,x^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2)^(1/2),x)

[Out]

int((c + d*x^2 + e*x^4 + f*x^6)/(a + b*x^2)^(1/2), x)

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sympy [A]  time = 13.71, size = 362, normalized size = 2.50 \begin {gather*} \frac {5 a^{\frac {5}{2}} f x}{16 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {3}{2}} e x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {3}{2}} f x^{3}}{48 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {\sqrt {a} d x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {\sqrt {a} e x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} f x^{5}}{24 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 a^{3} f \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {7}{2}}} + \frac {3 a^{2} e \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + c \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) + \frac {e x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {f x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

5*a**(5/2)*f*x/(16*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*e*x/(8*b**2*sqrt(1 + b*x**2/a)) + 5*a**(3/2)*f*x**3/(
48*b**2*sqrt(1 + b*x**2/a)) + sqrt(a)*d*x*sqrt(1 + b*x**2/a)/(2*b) - sqrt(a)*e*x**3/(8*b*sqrt(1 + b*x**2/a)) -
sqrt(a)*f*x**5/(24*b*sqrt(1 + b*x**2/a)) - 5*a**3*f*asinh(sqrt(b)*x/sqrt(a))/(16*b**(7/2)) + 3*a**2*e*asinh(s
qrt(b)*x/sqrt(a))/(8*b**(5/2)) - a*d*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2)) + c*Piecewise((sqrt(-a/b)*asin(x*sq
rt(-b/a))/sqrt(a), (a > 0) & (b < 0)), (sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), (a > 0) & (b > 0)), (sqrt(-a/b)*
acosh(x*sqrt(-b/a))/sqrt(-a), (b > 0) & (a < 0))) + e*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a)) + f*x**7/(6*sqrt(a)*
sqrt(1 + b*x**2/a))

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