3.2.58 \(\int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx\) [158]

Optimal. Leaf size=189 \[ -\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}+\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^4 x^3}-\frac {2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^5 x} \]

[Out]

-1/9*c*(b*x^2+a)^(1/2)/a/x^9+1/63*(-9*a*d+8*b*c)*(b*x^2+a)^(1/2)/a^2/x^7-1/105*(21*a^2*e-18*a*b*d+16*b^2*c)*(b
*x^2+a)^(1/2)/a^3/x^5+1/315*(-105*a^3*f+84*a^2*b*e-72*a*b^2*d+64*b^3*c)*(b*x^2+a)^(1/2)/a^4/x^3-2/315*b*(-105*
a^3*f+84*a^2*b*e-72*a*b^2*d+64*b^3*c)*(b*x^2+a)^(1/2)/a^5/x

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Rubi [A]
time = 0.18, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1817, 12, 277, 270} \begin {gather*} \frac {\sqrt {a+b x^2} (8 b c-9 a d)}{63 a^2 x^7}-\frac {\sqrt {a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{105 a^3 x^5}-\frac {2 b \sqrt {a+b x^2} \left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right )}{315 a^5 x}+\frac {\sqrt {a+b x^2} \left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right )}{315 a^4 x^3}-\frac {c \sqrt {a+b x^2}}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*Sqrt[a + b*x^2]),x]

[Out]

-1/9*(c*Sqrt[a + b*x^2])/(a*x^9) + ((8*b*c - 9*a*d)*Sqrt[a + b*x^2])/(63*a^2*x^7) - ((16*b^2*c - 18*a*b*d + 21
*a^2*e)*Sqrt[a + b*x^2])/(105*a^3*x^5) + ((64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(3
15*a^4*x^3) - (2*b*(64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(315*a^5*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 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(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*} \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}-\frac {\int \frac {8 b c-9 a \left (d+e x^2+f x^4\right )}{x^8 \sqrt {a+b x^2}} \, dx}{9 a}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}+\frac {\int \frac {6 b (8 b c-9 a d)-7 a \left (-9 a e-9 a f x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx}{63 a^2}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}-\frac {\int \frac {4 b \left (48 b^2 c-54 a b d+63 a^2 e\right )-315 a^3 f}{x^4 \sqrt {a+b x^2}} \, dx}{315 a^3}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}-\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \int \frac {1}{x^4 \sqrt {a+b x^2}} \, dx}{105 a^3}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}+\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^4 x^3}+\frac {\left (2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right )\right ) \int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx}{315 a^4}\\ &=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}+\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^4 x^3}-\frac {2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^5 x}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 134, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {a+b x^2} \left (128 b^4 c x^8-16 a b^3 x^6 \left (4 c+9 d x^2\right )+24 a^2 b^2 x^4 \left (2 c+3 d x^2+7 e x^4\right )-2 a^3 b x^2 \left (20 c+27 d x^2+42 e x^4+105 f x^6\right )+a^4 \left (35 c+45 d x^2+63 e x^4+105 f x^6\right )\right )}{315 a^5 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*Sqrt[a + b*x^2]),x]

[Out]

-1/315*(Sqrt[a + b*x^2]*(128*b^4*c*x^8 - 16*a*b^3*x^6*(4*c + 9*d*x^2) + 24*a^2*b^2*x^4*(2*c + 3*d*x^2 + 7*e*x^
4) - 2*a^3*b*x^2*(20*c + 27*d*x^2 + 42*e*x^4 + 105*f*x^6) + a^4*(35*c + 45*d*x^2 + 63*e*x^4 + 105*f*x^6)))/(a^
5*x^9)

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Maple [A]
time = 0.12, size = 298, normalized size = 1.58

method	result	size
gosper	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 c \,a^{4}\right )}{315 a^{5} x^{9}}\)	\(157\)
trager	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 c \,a^{4}\right )}{315 a^{5} x^{9}}\)	\(157\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 c \,a^{4}\right )}{315 a^{5} x^{9}}\)	\(157\)
default	\(e \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )+c \left (-\frac {\sqrt {b \,x^{2}+a}}{9 a \,x^{9}}-\frac {8 b \left (-\frac {\sqrt {b \,x^{2}+a}}{7 a \,x^{7}}-\frac {6 b \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )+d \left (-\frac {\sqrt {b \,x^{2}+a}}{7 a \,x^{7}}-\frac {6 b \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )+f \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 278, normalized size = 1.47 \begin {gather*} -\frac {128 \, \sqrt {b x^{2} + a} b^{4} c}{315 \, a^{5} x} + \frac {16 \, \sqrt {b x^{2} + a} b^{3} d}{35 \, a^{4} x} + \frac {2 \, \sqrt {b x^{2} + a} b f}{3 \, a^{2} x} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} e}{15 \, a^{3} x} + \frac {64 \, \sqrt {b x^{2} + a} b^{3} c}{315 \, a^{4} x^{3}} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} d}{35 \, a^{3} x^{3}} - \frac {\sqrt {b x^{2} + a} f}{3 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a} b e}{15 \, a^{2} x^{3}} - \frac {16 \, \sqrt {b x^{2} + a} b^{2} c}{105 \, a^{3} x^{5}} + \frac {6 \, \sqrt {b x^{2} + a} b d}{35 \, a^{2} x^{5}} - \frac {\sqrt {b x^{2} + a} e}{5 \, a x^{5}} + \frac {8 \, \sqrt {b x^{2} + a} b c}{63 \, a^{2} x^{7}} - \frac {\sqrt {b x^{2} + a} d}{7 \, a x^{7}} - \frac {\sqrt {b x^{2} + a} c}{9 \, a x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

-128/315*sqrt(b*x^2 + a)*b^4*c/(a^5*x) + 16/35*sqrt(b*x^2 + a)*b^3*d/(a^4*x) + 2/3*sqrt(b*x^2 + a)*b*f/(a^2*x)
- 8/15*sqrt(b*x^2 + a)*b^2*e/(a^3*x) + 64/315*sqrt(b*x^2 + a)*b^3*c/(a^4*x^3) - 8/35*sqrt(b*x^2 + a)*b^2*d/(a
^3*x^3) - 1/3*sqrt(b*x^2 + a)*f/(a*x^3) + 4/15*sqrt(b*x^2 + a)*b*e/(a^2*x^3) - 16/105*sqrt(b*x^2 + a)*b^2*c/(a
^3*x^5) + 6/35*sqrt(b*x^2 + a)*b*d/(a^2*x^5) - 1/5*sqrt(b*x^2 + a)*e/(a*x^5) + 8/63*sqrt(b*x^2 + a)*b*c/(a^2*x
^7) - 1/7*sqrt(b*x^2 + a)*d/(a*x^7) - 1/9*sqrt(b*x^2 + a)*c/(a*x^9)

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Fricas [A]
time = 7.51, size = 152, normalized size = 0.80 \begin {gather*} -\frac {{\left (2 \, {\left (64 \, b^{4} c - 72 \, a b^{3} d - 105 \, a^{3} b f\right )} x^{8} - {\left (64 \, a b^{3} c - 72 \, a^{2} b^{2} d - 105 \, a^{4} f\right )} x^{6} + 35 \, a^{4} c + 6 \, {\left (8 \, a^{2} b^{2} c - 9 \, a^{3} b d\right )} x^{4} - 5 \, {\left (8 \, a^{3} b c - 9 \, a^{4} d\right )} x^{2} + 21 \, {\left (8 \, a^{2} b^{2} x^{8} - 4 \, a^{3} b x^{6} + 3 \, a^{4} x^{4}\right )} e\right )} \sqrt {b x^{2} + a}}{315 \, a^{5} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

-1/315*(2*(64*b^4*c - 72*a*b^3*d - 105*a^3*b*f)*x^8 - (64*a*b^3*c - 72*a^2*b^2*d - 105*a^4*f)*x^6 + 35*a^4*c +
6*(8*a^2*b^2*c - 9*a^3*b*d)*x^4 - 5*(8*a^3*b*c - 9*a^4*d)*x^2 + 21*(8*a^2*b^2*x^8 - 4*a^3*b*x^6 + 3*a^4*x^4)*
e)*sqrt(b*x^2 + a)/(a^5*x^9)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (190) = 380\).
time = 3.07, size = 1642, normalized size = 8.69 \begin {gather*} - \frac {35 a^{8} b^{\frac {33}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} - \frac {100 a^{7} b^{\frac {35}{2}} c x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} - \frac {98 a^{6} b^{\frac {37}{2}} c x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} - \frac {5 a^{6} b^{\frac {19}{2}} d \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 {28 a^{5} b^{\frac {39}{2}} c x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} - \frac {9 a^{5} b^{\frac {21}{2}} d 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 {35 a^{4} b^{\frac {41}{2}} c x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} - \frac {5 a^{4} b^{\frac {23}{2}} d 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}} e \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 {280 a^{3} b^{\frac {43}{2}} c x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} + \frac {5 a^{3} b^{\frac {25}{2}} d 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}} e 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 {560 a^{2} b^{\frac {45}{2}} c x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} + \frac {30 a^{2} b^{\frac {27}{2}} d 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}} e 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 {448 a b^{\frac {47}{2}} c x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} + \frac {40 a b^{\frac {29}{2}} d 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}} e 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 {128 b^{\frac {49}{2}} c x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{9} b^{16} x^{8} + 1260 a^{8} b^{17} x^{10} + 1890 a^{7} b^{18} x^{12} + 1260 a^{6} b^{19} x^{14} + 315 a^{5} b^{20} x^{16}} + \frac {16 b^{\frac {31}{2}} d 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}} e 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} f \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{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)/x**10/(b*x**2+a)**(1/2),x)

[Out]

-35*a**8*b**(33/2)*c*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12
+ 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 100*a**7*b**(35/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(315*a**9*b*
*16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 98*
a**6*b**(37/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**1
2 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 5*a**6*b**(19/2)*d*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) - 28*a**5*b**(39/2)*c*x**6*sqrt(a/(b*x**2
) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5
*b**20*x**16) - 9*a**5*b**(21/2)*d*x**2*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) - 35*a**4*b**(41/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1
260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 5*a**4*b**(23/2
)*d*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)*e*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) -
280*a**3*b**(43/2)*c*x**10*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18
*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) + 5*a**3*b**(25/2)*d*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)*e*x**2*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) - 560*a**2*b**(45/2)*c*x**12*sqrt(
a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 +
315*a**5*b**20*x**16) + 30*a**2*b**(27/2)*d*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)*e*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) - 448*a*b**(47/2)*c*x**14*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x*
*8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) + 40*a*b**(
29/2)*d*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)*e*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) - 128*b**(49/2)*c*x**16*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b
**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) + 16*b**(31/2)*d*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)*e*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)*f*sqrt(a/(b*x**2) + 1)/(3*a*
x**2) + 2*b**(3/2)*f*sqrt(a/(b*x**2) + 1)/(3*a**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (172) = 344\).
time = 1.22, size = 667, normalized size = 3.53 \begin {gather*} \frac {4 \, {\left (315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} b^{\frac {3}{2}} f - 1995 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a b^{\frac {3}{2}} f + 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} b^{\frac {5}{2}} e + 2520 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {7}{2}} d + 5355 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {3}{2}} f - 3780 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a b^{\frac {5}{2}} e + 8064 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c - 6552 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} d - 7875 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} f + 6804 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} e - 5376 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c + 6048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} d + 6825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} f - 6216 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} e + 2304 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c - 2592 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} d - 3465 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} f + 3024 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} e - 576 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {9}{2}} c + 648 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} d + 945 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} f - 756 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} e + 64 \, a^{4} b^{\frac {9}{2}} c - 72 \, a^{5} b^{\frac {7}{2}} d - 105 \, a^{7} b^{\frac {3}{2}} f + 84 \, a^{6} b^{\frac {5}{2}} e\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

4/315*(315*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(3/2)*f - 1995*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(3/2)*f + 84
0*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(5/2)*e + 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*d + 5355*(sqrt(b)
*x - sqrt(b*x^2 + a))^10*a^2*b^(3/2)*f - 3780*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*e + 8064*(sqrt(b)*x -
sqrt(b*x^2 + a))^8*b^(9/2)*c - 6552*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d - 7875*(sqrt(b)*x - sqrt(b*x^
2 + a))^8*a^3*b^(3/2)*f + 6804*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*e - 5376*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*a*b^(9/2)*c + 6048*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(7/2)*d + 6825*(sqrt(b)*x - sqrt(b*x^2 + a))^6*
a^4*b^(3/2)*f - 6216*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(5/2)*e + 2304*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*
b^(9/2)*c - 2592*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(7/2)*d - 3465*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(3
/2)*f + 3024*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*e - 576*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^3*b^(9/2)*c
+ 648*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(7/2)*d + 945*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^6*b^(3/2)*f - 756
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*e + 64*a^4*b^(9/2)*c - 72*a^5*b^(7/2)*d - 105*a^7*b^(3/2)*f + 84*
a^6*b^(5/2)*e)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9

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Mupad [B]
time = 1.28, size = 171, normalized size = 0.90 \begin {gather*} \frac {\sqrt {b\,x^2+a}\,\left (-105\,f\,a^3+84\,e\,a^2\,b-72\,d\,a\,b^2+64\,c\,b^3\right )}{315\,a^4\,x^3}-\frac {\sqrt {b\,x^2+a}\,\left (9\,a\,d-8\,b\,c\right )}{63\,a^2\,x^7}-\frac {\sqrt {b\,x^2+a}\,\left (21\,e\,a^2-18\,d\,a\,b+16\,c\,b^2\right )}{105\,a^3\,x^5}-\frac {\sqrt {b\,x^2+a}\,\left (-210\,f\,a^3\,b+168\,e\,a^2\,b^2-144\,d\,a\,b^3+128\,c\,b^4\right )}{315\,a^5\,x}-\frac {c\,\sqrt {b\,x^2+a}}{9\,a\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^(1/2)),x)

[Out]

((a + b*x^2)^(1/2)*(64*b^3*c - 105*a^3*f - 72*a*b^2*d + 84*a^2*b*e))/(315*a^4*x^3) - ((a + b*x^2)^(1/2)*(9*a*d
- 8*b*c))/(63*a^2*x^7) - ((a + b*x^2)^(1/2)*(16*b^2*c + 21*a^2*e - 18*a*b*d))/(105*a^3*x^5) - ((a + b*x^2)^(1
/2)*(128*b^4*c + 168*a^2*b^2*e - 144*a*b^3*d - 210*a^3*b*f))/(315*a^5*x) - (c*(a + b*x^2)^(1/2))/(9*a*x^9)

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