\(\int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx\) [157]

Optimal result

Integrand size = 32, antiderivative size = 140 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=-\frac {c \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 b c-7 a d) \sqrt {a+b x^2}}{35 a^2 x^5}-\frac {\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^3}+\frac {\left (48 b^3 c-56 a b^2 d+70 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{105 a^4 x} \]

[Out]

-1/7*c*(b*x^2+a)^(1/2)/a/x^7+1/35*(-7*a*d+6*b*c)*(b*x^2+a)^(1/2)/a^2/x^5-1/105*(35*a^2*e-28*a*b*d+24*b^2*c)*(b
*x^2+a)^(1/2)/a^3/x^3+1/105*(-105*a^3*f+70*a^2*b*e-56*a*b^2*d+48*b^3*c)*(b*x^2+a)^(1/2)/a^4/x

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1817, 12, 270} \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} (6 b c-7 a d)}{35 a^2 x^5}-\frac {\sqrt {a+b x^2} \left (35 a^2 e-28 a b d+24 b^2 c\right )}{105 a^3 x^3}+\frac {\sqrt {a+b x^2} \left (-105 a^3 f+70 a^2 b e-56 a b^2 d+48 b^3 c\right )}{105 a^4 x}-\frac {c \sqrt {a+b x^2}}{7 a x^7} \]

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*Sqrt[a + b*x^2]),x]

[Out]

-1/7*(c*Sqrt[a + b*x^2])/(a*x^7) + ((6*b*c - 7*a*d)*Sqrt[a + b*x^2])/(35*a^2*x^5) - ((24*b^2*c - 28*a*b*d + 35
*a^2*e)*Sqrt[a + b*x^2])/(105*a^3*x^3) + ((48*b^3*c - 56*a*b^2*d + 70*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(1
05*a^4*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

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

Rule 1817

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {c \sqrt {a+b x^2}}{7 a x^7}-\frac {\int \frac {6 b c-7 a \left (d+e x^2+f x^4\right )}{x^6 \sqrt {a+b x^2}} \, dx}{7 a} \\ & = -\frac {c \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 b c-7 a d) \sqrt {a+b x^2}}{35 a^2 x^5}+\frac {\int \frac {4 b (6 b c-7 a d)-5 a \left (-7 a e-7 a f x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx}{35 a^2} \\ & = -\frac {c \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 b c-7 a d) \sqrt {a+b x^2}}{35 a^2 x^5}-\frac {\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^3}-\frac {\int \frac {2 b \left (24 b^2 c-28 a b d+35 a^2 e\right )-105 a^3 f}{x^2 \sqrt {a+b x^2}} \, dx}{105 a^3} \\ & = -\frac {c \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 b c-7 a d) \sqrt {a+b x^2}}{35 a^2 x^5}-\frac {\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^3}-\frac {\left (48 b^3 c-56 a b^2 d+70 a^2 b e-105 a^3 f\right ) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{105 a^3} \\ & = -\frac {c \sqrt {a+b x^2}}{7 a x^7}+\frac {(6 b c-7 a d) \sqrt {a+b x^2}}{35 a^2 x^5}-\frac {\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^3}+\frac {\left (48 b^3 c-56 a b^2 d+70 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{105 a^4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.74 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (48 b^3 c x^6-8 a b^2 x^4 \left (3 c+7 d x^2\right )+2 a^2 b x^2 \left (9 c+14 d x^2+35 e x^4\right )-a^3 \left (15 c+21 d x^2+35 x^4 \left (e+3 f x^2\right )\right )\right )}{105 a^4 x^7} \]

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*Sqrt[a + b*x^2]),x]

[Out]

(Sqrt[a + b*x^2]*(48*b^3*c*x^6 - 8*a*b^2*x^4*(3*c + 7*d*x^2) + 2*a^2*b*x^2*(9*c + 14*d*x^2 + 35*e*x^4) - a^3*(
15*c + 21*d*x^2 + 35*x^4*(e + 3*f*x^2))))/(105*a^4*x^7)

Maple [A] (verified)

Time = 3.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(-\frac {\left (\left (7 f \,x^{6}+\frac {7}{3} e \,x^{4}+\frac {7}{5} d \,x^{2}+c \right ) a^{3}-\frac {6 b \,x^{2} \left (\frac {35}{9} e \,x^{4}+\frac {14}{9} d \,x^{2}+c \right ) a^{2}}{5}+\frac {8 b^{2} \left (\frac {7 d \,x^{2}}{3}+c \right ) x^{4} a}{5}-\frac {16 b^{3} c \,x^{6}}{5}\right ) \sqrt {b \,x^{2}+a}}{7 x^{7} a^{4}}\)	\(92\)
gosper	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (105 a^{3} f \,x^{6}-70 a^{2} b e \,x^{6}+56 a \,b^{2} d \,x^{6}-48 b^{3} c \,x^{6}+35 a^{3} e \,x^{4}-28 a^{2} b d \,x^{4}+24 a \,b^{2} c \,x^{4}+21 a^{3} d \,x^{2}-18 a^{2} b c \,x^{2}+15 c \,a^{3}\right )}{105 x^{7} a^{4}}\)	\(111\)
trager	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (105 a^{3} f \,x^{6}-70 a^{2} b e \,x^{6}+56 a \,b^{2} d \,x^{6}-48 b^{3} c \,x^{6}+35 a^{3} e \,x^{4}-28 a^{2} b d \,x^{4}+24 a \,b^{2} c \,x^{4}+21 a^{3} d \,x^{2}-18 a^{2} b c \,x^{2}+15 c \,a^{3}\right )}{105 x^{7} a^{4}}\)	\(111\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (105 a^{3} f \,x^{6}-70 a^{2} b e \,x^{6}+56 a \,b^{2} d \,x^{6}-48 b^{3} c \,x^{6}+35 a^{3} e \,x^{4}-28 a^{2} b d \,x^{4}+24 a \,b^{2} c \,x^{4}+21 a^{3} d \,x^{2}-18 a^{2} b c \,x^{2}+15 c \,a^{3}\right )}{105 x^{7} a^{4}}\)	\(111\)
default	\(c \left (-\frac {\sqrt {b \,x^{2}+a}}{7 a \,x^{7}}-\frac {6 b \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )+e \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )-\frac {f \sqrt {b \,x^{2}+a}}{a x}+d \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )\)	\(206\)

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/7*((7*f*x^6+7/3*e*x^4+7/5*d*x^2+c)*a^3-6/5*b*x^2*(35/9*e*x^4+14/9*d*x^2+c)*a^2+8/5*b^2*(7/3*d*x^2+c)*x^4*a-
16/5*b^3*c*x^6)*(b*x^2+a)^(1/2)/x^7/a^4

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=\frac {{\left ({\left (48 \, b^{3} c - 56 \, a b^{2} d + 70 \, a^{2} b e - 105 \, a^{3} f\right )} x^{6} - {\left (24 \, a b^{2} c - 28 \, a^{2} b d + 35 \, a^{3} e\right )} x^{4} - 15 \, a^{3} c + 3 \, {\left (6 \, a^{2} b c - 7 \, a^{3} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, a^{4} x^{7}} \]

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

1/105*((48*b^3*c - 56*a*b^2*d + 70*a^2*b*e - 105*a^3*f)*x^6 - (24*a*b^2*c - 28*a^2*b*d + 35*a^3*e)*x^4 - 15*a^
3*c + 3*(6*a^2*b*c - 7*a^3*d)*x^2)*sqrt(b*x^2 + a)/(a^4*x^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (136) = 272\).

Time = 2.12 (sec) , antiderivative size = 891, normalized size of antiderivative = 6.36 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=- \frac {5 a^{6} b^{\frac {19}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {9 a^{5} b^{\frac {21}{2}} c x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {5 a^{4} b^{\frac {23}{2}} c x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {3 a^{4} b^{\frac {9}{2}} d \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} + \frac {5 a^{3} b^{\frac {25}{2}} c x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {2 a^{3} b^{\frac {11}{2}} d x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} + \frac {30 a^{2} b^{\frac {27}{2}} c x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {3 a^{2} b^{\frac {13}{2}} d x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} + \frac {40 a b^{\frac {29}{2}} c x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {12 a b^{\frac {15}{2}} d x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} + \frac {16 b^{\frac {31}{2}} c x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac {8 b^{\frac {17}{2}} d x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{2}} + 1}}{a} + \frac {2 b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \]

[In]

integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a)**(1/2),x)

[Out]

-5*a**6*b**(19/2)*c*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*
a**4*b**12*x**12) - 9*a**5*b**(21/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 10
5*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 5*a**4*b**(23/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 +
105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 3*a**4*b**(9/2)*d*sqrt(a/(b*x**2) + 1)/(15
*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) + 5*a**3*b**(25/2)*c*x**6*sqrt(a/(b*x**2) + 1)/(35*a*
*7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 2*a**3*b**(11/2)*d*x**2*sqr
t(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) + 30*a**2*b**(27/2)*c*x**8*sqrt(
a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 3*a**
2*b**(13/2)*d*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) + 40*a*b**
(29/2)*c*x**10*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*
b**12*x**12) - 12*a*b**(15/2)*d*x**6*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**
6*x**8) + 16*b**(31/2)*c*x**12*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*
x**10 + 35*a**4*b**12*x**12) - 8*b**(17/2)*d*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6
+ 15*a**3*b**6*x**8) - sqrt(b)*e*sqrt(a/(b*x**2) + 1)/(3*a*x**2) - sqrt(b)*f*sqrt(a/(b*x**2) + 1)/a + 2*b**(3/
2)*e*sqrt(a/(b*x**2) + 1)/(3*a**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.38 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=\frac {16 \, \sqrt {b x^{2} + a} b^{3} c}{35 \, a^{4} x} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} d}{15 \, a^{3} x} + \frac {2 \, \sqrt {b x^{2} + a} b e}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a} f}{a x} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} c}{35 \, a^{3} x^{3}} + \frac {4 \, \sqrt {b x^{2} + a} b d}{15 \, a^{2} x^{3}} - \frac {\sqrt {b x^{2} + a} e}{3 \, a x^{3}} + \frac {6 \, \sqrt {b x^{2} + a} b c}{35 \, a^{2} x^{5}} - \frac {\sqrt {b x^{2} + a} d}{5 \, a x^{5}} - \frac {\sqrt {b x^{2} + a} c}{7 \, a x^{7}} \]

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

16/35*sqrt(b*x^2 + a)*b^3*c/(a^4*x) - 8/15*sqrt(b*x^2 + a)*b^2*d/(a^3*x) + 2/3*sqrt(b*x^2 + a)*b*e/(a^2*x) - s
qrt(b*x^2 + a)*f/(a*x) - 8/35*sqrt(b*x^2 + a)*b^2*c/(a^3*x^3) + 4/15*sqrt(b*x^2 + a)*b*d/(a^2*x^3) - 1/3*sqrt(
b*x^2 + a)*e/(a*x^3) + 6/35*sqrt(b*x^2 + a)*b*c/(a^2*x^5) - 1/5*sqrt(b*x^2 + a)*d/(a*x^5) - 1/7*sqrt(b*x^2 + a
)*c/(a*x^7)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (124) = 248\).

Time = 0.32 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.91 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=\frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} \sqrt {b} f + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {3}{2}} e - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a \sqrt {b} f + 560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {5}{2}} d - 910 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {3}{2}} e + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} \sqrt {b} f + 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {7}{2}} c - 1400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {5}{2}} d + 1540 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {3}{2}} e - 2100 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} \sqrt {b} f - 1008 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {7}{2}} c + 1176 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} d - 1260 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {3}{2}} e + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} \sqrt {b} f + 336 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {7}{2}} c - 392 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {5}{2}} d + 490 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {3}{2}} e - 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} \sqrt {b} f - 48 \, a^{3} b^{\frac {7}{2}} c + 56 \, a^{4} b^{\frac {5}{2}} d - 70 \, a^{5} b^{\frac {3}{2}} e + 105 \, a^{6} \sqrt {b} f\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*sqrt(b)*f + 210*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(3/2)*e - 630*(
sqrt(b)*x - sqrt(b*x^2 + a))^10*a*sqrt(b)*f + 560*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(5/2)*d - 910*(sqrt(b)*x -
sqrt(b*x^2 + a))^8*a*b^(3/2)*e + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*sqrt(b)*f + 1680*(sqrt(b)*x - sqrt(
b*x^2 + a))^6*b^(7/2)*c - 1400*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(5/2)*d + 1540*(sqrt(b)*x - sqrt(b*x^2 + a)
)^6*a^2*b^(3/2)*e - 2100*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*sqrt(b)*f - 1008*(sqrt(b)*x - sqrt(b*x^2 + a))^4*
a*b^(7/2)*c + 1176*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2)*d - 1260*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^
(3/2)*e + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*sqrt(b)*f + 336*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(7/2)
*c - 392*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(5/2)*d + 490*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(3/2)*e - 6
30*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*sqrt(b)*f - 48*a^3*b^(7/2)*c + 56*a^4*b^(5/2)*d - 70*a^5*b^(3/2)*e + 10
5*a^6*sqrt(b)*f)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7

Mupad [B] (verification not implemented)

Time = 6.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^8 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b\,x^2+a}\,\left (-105\,f\,a^3+70\,e\,a^2\,b-56\,d\,a\,b^2+48\,c\,b^3\right )}{105\,a^4\,x}-\frac {\sqrt {b\,x^2+a}\,\left (7\,a\,d-6\,b\,c\right )}{35\,a^2\,x^5}-\frac {\sqrt {b\,x^2+a}\,\left (35\,e\,a^2-28\,d\,a\,b+24\,c\,b^2\right )}{105\,a^3\,x^3}-\frac {c\,\sqrt {b\,x^2+a}}{7\,a\,x^7} \]

[In]

int((c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^(1/2)),x)

[Out]

((a + b*x^2)^(1/2)*(48*b^3*c - 105*a^3*f - 56*a*b^2*d + 70*a^2*b*e))/(105*a^4*x) - ((a + b*x^2)^(1/2)*(7*a*d -
6*b*c))/(35*a^2*x^5) - ((a + b*x^2)^(1/2)*(24*b^2*c + 35*a^2*e - 28*a*b*d))/(105*a^3*x^3) - (c*(a + b*x^2)^(1
/2))/(7*a*x^7)