Integrand size = 32, antiderivative size = 194 \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) x \sqrt {a+b x^2}}{128 b^4}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}} \]
-1/128*a*(-35*a^3*f+40*a^2*b*e-48*a*b^2*d+64*b^3*c)*arctanh(x*b^(1/2)/(b*x ^2+a)^(1/2))/b^(9/2)+1/128*(-35*a^3*f+40*a^2*b*e-48*a*b^2*d+64*b^3*c)*x*(b *x^2+a)^(1/2)/b^4+1/192*(35*a^2*f-40*a*b*e+48*b^2*d)*x^3*(b*x^2+a)^(1/2)/b ^3+1/48*(-7*a*f+8*b*e)*x^5*(b*x^2+a)^(1/2)/b^2+1/8*f*x^7*(b*x^2+a)^(1/2)/b
Time = 0.63 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^3 f+10 a^2 b \left (12 e+7 f x^2\right )-8 a b^2 \left (18 d+10 e x^2+7 f x^4\right )+16 b^3 \left (12 c+6 d x^2+4 e x^4+3 f x^6\right )\right )+6 a \left (-64 b^3 c+48 a b^2 d-40 a^2 b e+35 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{384 b^{9/2}} \]
(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^3*f + 10*a^2*b*(12*e + 7*f*x^2) - 8*a*b ^2*(18*d + 10*e*x^2 + 7*f*x^4) + 16*b^3*(12*c + 6*d*x^2 + 4*e*x^4 + 3*f*x^ 6)) + 6*a*(-64*b^3*c + 48*a*b^2*d - 40*a^2*b*e + 35*a^3*f)*ArcTanh[(Sqrt[b ]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/(384*b^(9/2))
Time = 0.41 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2340, 1590, 363, 262, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {\int \frac {x^2 \left ((8 b e-7 a f) x^4+8 b d x^2+8 b c\right )}{\sqrt {b x^2+a}}dx}{8 b}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {\frac {\int \frac {x^2 \left (48 c b^2+\left (35 f a^2-40 b e a+48 b^2 d\right ) x^2\right )}{\sqrt {b x^2+a}}dx}{6 b}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{6 b}}{8 b}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right ) \int \frac {x^2}{\sqrt {b x^2+a}}dx}{4 b}+\frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{4 b}}{6 b}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{6 b}}{8 b}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right ) \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )}{4 b}+\frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{4 b}}{6 b}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{6 b}}{8 b}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right ) \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )}{4 b}+\frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{4 b}}{6 b}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{6 b}}{8 b}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{4 b}+\frac {3 \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right ) \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{4 b}}{6 b}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{6 b}}{8 b}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}\) |
(f*x^7*Sqrt[a + b*x^2])/(8*b) + (((8*b*e - 7*a*f)*x^5*Sqrt[a + b*x^2])/(6* b) + (((48*b^2*d - 40*a*b*e + 35*a^2*f)*x^3*Sqrt[a + b*x^2])/(4*b) + (3*(6 4*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*((x*Sqrt[a + b*x^2])/(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/(4*b))/(6*b))/(8*b )
3.2.52.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ (q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q + 1)) Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Time = 3.65 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.71
| method | result | size |
| pseudoelliptic | \(\frac {\frac {35 a \left (f \,a^{3}-\frac {8}{7} a^{2} b e +\frac {48}{35} a \,b^{2} d -\frac {64}{35} b^{3} c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{128}-\frac {35 \left (\left (-\frac {16}{35} f \,x^{6}-\frac {64}{105} e \,x^{4}-\frac {32}{35} d \,x^{2}-\frac {64}{35} c \right ) b^{\frac {7}{2}}+\left (\left (\frac {8}{15} f \,x^{4}+\frac {16}{21} e \,x^{2}+\frac {48}{35} d \right ) b^{\frac {5}{2}}+a \left (\left (-\frac {2 f \,x^{2}}{3}-\frac {8 e}{7}\right ) b^{\frac {3}{2}}+a f \sqrt {b}\right )\right ) a \right ) x \sqrt {b \,x^{2}+a}}{128}}{b^{\frac {9}{2}}}\) | \(137\) |
| risch | \(-\frac {x \left (-48 f \,x^{6} b^{3}+56 a \,b^{2} f \,x^{4}-64 b^{3} e \,x^{4}-70 a^{2} b f \,x^{2}+80 a \,b^{2} e \,x^{2}-96 b^{3} d \,x^{2}+105 f \,a^{3}-120 a^{2} b e +144 a \,b^{2} d -192 b^{3} c \right ) \sqrt {b \,x^{2}+a}}{384 b^{4}}+\frac {a \left (35 f \,a^{3}-40 a^{2} b e +48 a \,b^{2} d -64 b^{3} c \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {9}{2}}}\) | \(150\) |
| default | \(f \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )+e \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+d \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+c \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\) | \(306\) |
35/128*(a*(f*a^3-8/7*a^2*b*e+48/35*a*b^2*d-64/35*b^3*c)*arctanh((b*x^2+a)^ (1/2)/x/b^(1/2))-((-16/35*f*x^6-64/105*e*x^4-32/35*d*x^2-64/35*c)*b^(7/2)+ ((8/15*f*x^4+16/21*e*x^2+48/35*d)*b^(5/2)+a*((-2/3*f*x^2-8/7*e)*b^(3/2)+a* f*b^(1/2)))*a)*x*(b*x^2+a)^(1/2))/b^(9/2)
Time = 0.31 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.70 \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\left [-\frac {3 \, {\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, b^{4} f x^{7} + 8 \, {\left (8 \, b^{4} e - 7 \, a b^{3} f\right )} x^{5} + 2 \, {\left (48 \, b^{4} d - 40 \, a b^{3} e + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \, {\left (64 \, b^{4} c - 48 \, a b^{3} d + 40 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{5}}, \frac {3 \, {\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, b^{4} f x^{7} + 8 \, {\left (8 \, b^{4} e - 7 \, a b^{3} f\right )} x^{5} + 2 \, {\left (48 \, b^{4} d - 40 \, a b^{3} e + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \, {\left (64 \, b^{4} c - 48 \, a b^{3} d + 40 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{5}}\right ] \]
[-1/768*(3*(64*a*b^3*c - 48*a^2*b^2*d + 40*a^3*b*e - 35*a^4*f)*sqrt(b)*log (-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(48*b^4*f*x^7 + 8*(8*b^4* e - 7*a*b^3*f)*x^5 + 2*(48*b^4*d - 40*a*b^3*e + 35*a^2*b^2*f)*x^3 + 3*(64* b^4*c - 48*a*b^3*d + 40*a^2*b^2*e - 35*a^3*b*f)*x)*sqrt(b*x^2 + a))/b^5, 1 /384*(3*(64*a*b^3*c - 48*a^2*b^2*d + 40*a^3*b*e - 35*a^4*f)*sqrt(-b)*arcta n(sqrt(-b)*x/sqrt(b*x^2 + a)) + (48*b^4*f*x^7 + 8*(8*b^4*e - 7*a*b^3*f)*x^ 5 + 2*(48*b^4*d - 40*a*b^3*e + 35*a^2*b^2*f)*x^3 + 3*(64*b^4*c - 48*a*b^3* d + 40*a^2*b^2*e - 35*a^3*b*f)*x)*sqrt(b*x^2 + a))/b^5]
Time = 0.57 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03 \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\begin {cases} - \frac {a \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a f}{8 b} + e\right )}{6 b} + d\right )}{4 b} + c\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a + b x^{2}} \left (\frac {f x^{7}}{8 b} + \frac {x^{5} \left (- \frac {7 a f}{8 b} + e\right )}{6 b} + \frac {x^{3} \left (- \frac {5 a \left (- \frac {7 a f}{8 b} + e\right )}{6 b} + d\right )}{4 b} + \frac {x \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a f}{8 b} + e\right )}{6 b} + d\right )}{4 b} + c\right )}{2 b}\right ) & \text {for}\: b \neq 0 \\\frac {\frac {c x^{3}}{3} + \frac {d x^{5}}{5} + \frac {e x^{7}}{7} + \frac {f x^{9}}{9}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Piecewise((-a*(-3*a*(-5*a*(-7*a*f/(8*b) + e)/(6*b) + d)/(4*b) + c)*Piecewi se((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/ sqrt(b*x**2), True))/(2*b) + sqrt(a + b*x**2)*(f*x**7/(8*b) + x**5*(-7*a*f /(8*b) + e)/(6*b) + x**3*(-5*a*(-7*a*f/(8*b) + e)/(6*b) + d)/(4*b) + x*(-3 *a*(-5*a*(-7*a*f/(8*b) + e)/(6*b) + d)/(4*b) + c)/(2*b)), Ne(b, 0)), ((c*x **3/3 + d*x**5/5 + e*x**7/7 + f*x**9/9)/sqrt(a), True))
Time = 0.20 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.31 \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} f x^{7}}{8 \, b} + \frac {\sqrt {b x^{2} + a} e x^{5}}{6 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a f x^{5}}{48 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a e x^{3}}{24 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} f x^{3}}{192 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x}{2 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a d x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} e x}{16 \, b^{3}} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} f x}{128 \, b^{4}} - \frac {a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {3 \, a^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} + \frac {35 \, a^{4} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{2}}} \]
1/8*sqrt(b*x^2 + a)*f*x^7/b + 1/6*sqrt(b*x^2 + a)*e*x^5/b - 7/48*sqrt(b*x^ 2 + a)*a*f*x^5/b^2 + 1/4*sqrt(b*x^2 + a)*d*x^3/b - 5/24*sqrt(b*x^2 + a)*a* e*x^3/b^2 + 35/192*sqrt(b*x^2 + a)*a^2*f*x^3/b^3 + 1/2*sqrt(b*x^2 + a)*c*x /b - 3/8*sqrt(b*x^2 + a)*a*d*x/b^2 + 5/16*sqrt(b*x^2 + a)*a^2*e*x/b^3 - 35 /128*sqrt(b*x^2 + a)*a^3*f*x/b^4 - 1/2*a*c*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 3/8*a^2*d*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 5/16*a^3*e*arcsinh(b*x/sqrt(a *b))/b^(7/2) + 35/128*a^4*f*arcsinh(b*x/sqrt(a*b))/b^(9/2)
Time = 0.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, f x^{2}}{b} + \frac {8 \, b^{6} e - 7 \, a b^{5} f}{b^{7}}\right )} x^{2} + \frac {48 \, b^{6} d - 40 \, a b^{5} e + 35 \, a^{2} b^{4} f}{b^{7}}\right )} x^{2} + \frac {3 \, {\left (64 \, b^{6} c - 48 \, a b^{5} d + 40 \, a^{2} b^{4} e - 35 \, a^{3} b^{3} f\right )}}{b^{7}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {9}{2}}} \]
1/384*(2*(4*(6*f*x^2/b + (8*b^6*e - 7*a*b^5*f)/b^7)*x^2 + (48*b^6*d - 40*a *b^5*e + 35*a^2*b^4*f)/b^7)*x^2 + 3*(64*b^6*c - 48*a*b^5*d + 40*a^2*b^4*e - 35*a^3*b^3*f)/b^7)*sqrt(b*x^2 + a)*x + 1/128*(64*a*b^3*c - 48*a^2*b^2*d + 40*a^3*b*e - 35*a^4*f)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {x^2\,\left (f\,x^6+e\,x^4+d\,x^2+c\right )}{\sqrt {b\,x^2+a}} \,d x \]