3.149 \(\int \frac{c+d x^2+e x^4+f x^6}{x^7 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \left (8 a^2 b e-16 a^3 f-6 a b^2 d+5 b^3 c\right )}{16 a^{7/2}}-\frac{\sqrt{a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{16 a^3 x^2}+\frac{\sqrt{a+b x^2} (5 b c-6 a d)}{24 a^2 x^4}-\frac{c \sqrt{a+b x^2}}{6 a x^6} \]

[Out]

-(c*Sqrt[a + b*x^2])/(6*a*x^6) + ((5*b*c - 6*a*d)*Sqrt[a + b*x^2])/(24*a^2*x^4) - ((5*b^2*c - 6*a*b*d + 8*a^2*
e)*Sqrt[a + b*x^2])/(16*a^3*x^2) + ((5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[
a]])/(16*a^(7/2))

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Rubi [A]  time = 0.276366, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1799, 1621, 897, 1157, 385, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \left (8 a^2 b e-16 a^3 f-6 a b^2 d+5 b^3 c\right )}{16 a^{7/2}}-\frac{\sqrt{a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{16 a^3 x^2}+\frac{\sqrt{a+b x^2} (5 b c-6 a d)}{24 a^2 x^4}-\frac{c \sqrt{a+b x^2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^7*Sqrt[a + b*x^2]),x]

[Out]

-(c*Sqrt[a + b*x^2])/(6*a*x^6) + ((5*b*c - 6*a*d)*Sqrt[a + b*x^2])/(24*a^2*x^4) - ((5*b^2*c - 6*a*b*d + 8*a^2*
e)*Sqrt[a + b*x^2])/(16*a^3*x^2) + ((5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[
a]])/(16*a^(7/2))

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1621

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*
(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expo
n[Px, x], 2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 385

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^7 \sqrt{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^4 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{c \sqrt{a+b x^2}}{6 a x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (5 b c-6 a d)-3 a e x-3 a f x^2}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac{c \sqrt{a+b x^2}}{6 a x^6}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\frac{1}{2} b^2 (5 b c-6 a d)+3 a^2 b e-3 a^3 f}{b^2}-\frac{\left (3 a b e-6 a^2 f\right ) x^2}{b^2}-\frac{3 a f x^4}{b^2}}{\left (-\frac{a}{b}+\frac{x^2}{b}\right )^3} \, dx,x,\sqrt{a+b x^2}\right )}{3 a b}\\ &=-\frac{c \sqrt{a+b x^2}}{6 a x^6}+\frac{(5 b c-6 a d) \sqrt{a+b x^2}}{24 a^2 x^4}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{2} \left (5 b c-6 a d+\frac{8 a^2 e}{b}-\frac{8 a^3 f}{b^2}\right )-\frac{12 a^2 f x^2}{b^2}}{\left (-\frac{a}{b}+\frac{x^2}{b}\right )^2} \, dx,x,\sqrt{a+b x^2}\right )}{12 a^2}\\ &=-\frac{c \sqrt{a+b x^2}}{6 a x^6}+\frac{(5 b c-6 a d) \sqrt{a+b x^2}}{24 a^2 x^4}-\frac{\left (5 b^2 c-6 a b d+8 a^2 e\right ) \sqrt{a+b x^2}}{16 a^3 x^2}+\frac{\left (b^2 \left (\frac{12 a^3 f}{b^3}-\frac{3 \left (5 b c-6 a d+\frac{8 a^2 e}{b}-\frac{8 a^3 f}{b^2}\right )}{2 b}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{24 a^3}\\ &=-\frac{c \sqrt{a+b x^2}}{6 a x^6}+\frac{(5 b c-6 a d) \sqrt{a+b x^2}}{24 a^2 x^4}-\frac{\left (5 b^2 c-6 a b d+8 a^2 e\right ) \sqrt{a+b x^2}}{16 a^3 x^2}+\frac{\left (5 b^3 c-6 a b^2 d+8 a^2 b e-16 a^3 f\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.962059, size = 162, normalized size = 1.11 \[ \frac{b^3 c \sqrt{a+b x^2} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b x^2}{a}+1\right )}{a^4}-\frac{b^2 d \sqrt{a+b x^2} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{b x^2}{a}+1\right )}{a^3}-\frac{b e \sqrt{a+b x^2} \left (\frac{a}{b x^2}-\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{2 a^2}-\frac{f \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^7*Sqrt[a + b*x^2]),x]

[Out]

-((f*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]) - (b*e*Sqrt[a + b*x^2]*(a/(b*x^2) - ArcTanh[Sqrt[1 + (b*x^2)/a
]]/Sqrt[1 + (b*x^2)/a]))/(2*a^2) - (b^2*d*Sqrt[a + b*x^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (b*x^2)/a])/a^3 +
(b^3*c*Sqrt[a + b*x^2]*Hypergeometric2F1[1/2, 4, 3/2, 1 + (b*x^2)/a])/a^4

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Maple [A]  time = 0.01, size = 238, normalized size = 1.6 \begin{align*} -{f\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{d}{4\,a{x}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{3\,bd}{8\,{a}^{2}{x}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{3\,{b}^{2}d}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{e}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{be}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{c}{6\,a{x}^{6}}\sqrt{b{x}^{2}+a}}+{\frac{5\,bc}{24\,{a}^{2}{x}^{4}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{b}^{2}c}{16\,{a}^{3}{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{b}^{3}c}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^7/(b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^7/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62761, size = 605, normalized size = 4.14 \begin{align*} \left [-\frac{3 \,{\left (5 \, b^{3} c - 6 \, a b^{2} d + 8 \, a^{2} b e - 16 \, a^{3} f\right )} \sqrt{a} x^{6} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (5 \, a b^{2} c - 6 \, a^{2} b d + 8 \, a^{3} e\right )} x^{4} + 8 \, a^{3} c - 2 \,{\left (5 \, a^{2} b c - 6 \, a^{3} d\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{96 \, a^{4} x^{6}}, -\frac{3 \,{\left (5 \, b^{3} c - 6 \, a b^{2} d + 8 \, a^{2} b e - 16 \, a^{3} f\right )} \sqrt{-a} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \,{\left (5 \, a b^{2} c - 6 \, a^{2} b d + 8 \, a^{3} e\right )} x^{4} + 8 \, a^{3} c - 2 \,{\left (5 \, a^{2} b c - 6 \, a^{3} d\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{48 \, a^{4} x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^7/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 110.921, size = 303, normalized size = 2.08 \begin{align*} - \frac{c}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{d}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b} c}{24 a x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b} d}{8 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{\sqrt{b} e \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} - \frac{5 b^{\frac{3}{2}} c}{48 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{3}{2}} d}{8 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 b^{\frac{5}{2}} c}{16 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{f \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} + \frac{b e \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} - \frac{3 b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{5}{2}}} + \frac{5 b^{3} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**7/(b*x**2+a)**(1/2),x)

[Out]

-c/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - d/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + sqrt(b)*c/(24*a*x**5*sqrt
(a/(b*x**2) + 1)) + sqrt(b)*d/(8*a*x**3*sqrt(a/(b*x**2) + 1)) - sqrt(b)*e*sqrt(a/(b*x**2) + 1)/(2*a*x) - 5*b**
(3/2)*c/(48*a**2*x**3*sqrt(a/(b*x**2) + 1)) + 3*b**(3/2)*d/(8*a**2*x*sqrt(a/(b*x**2) + 1)) - 5*b**(5/2)*c/(16*
a**3*x*sqrt(a/(b*x**2) + 1)) - f*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a) + b*e*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/
2)) - 3*b**2*d*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(5/2)) + 5*b**3*c*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(7/2))

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Giac [A]  time = 1.21274, size = 313, normalized size = 2.14 \begin{align*} -\frac{\frac{3 \,{\left (5 \, b^{4} c - 6 \, a b^{3} d - 16 \, a^{3} b f + 8 \, a^{2} b^{2} e\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{4} c - 40 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{4} c + 33 \, \sqrt{b x^{2} + a} a^{2} b^{4} c - 18 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a b^{3} d + 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b^{3} d - 30 \, \sqrt{b x^{2} + a} a^{3} b^{3} d + 24 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} b^{2} e - 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} b^{2} e + 24 \, \sqrt{b x^{2} + a} a^{4} b^{2} e}{a^{3} b^{3} x^{6}}}{48 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^7/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

-1/48*(3*(5*b^4*c - 6*a*b^3*d - 16*a^3*b*f + 8*a^2*b^2*e)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) + (1
5*(b*x^2 + a)^(5/2)*b^4*c - 40*(b*x^2 + a)^(3/2)*a*b^4*c + 33*sqrt(b*x^2 + a)*a^2*b^4*c - 18*(b*x^2 + a)^(5/2)
*a*b^3*d + 48*(b*x^2 + a)^(3/2)*a^2*b^3*d - 30*sqrt(b*x^2 + a)*a^3*b^3*d + 24*(b*x^2 + a)^(5/2)*a^2*b^2*e - 48
*(b*x^2 + a)^(3/2)*a^3*b^2*e + 24*sqrt(b*x^2 + a)*a^4*b^2*e)/(a^3*b^3*x^6))/b