Integrand size = 28, antiderivative size = 300 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=-\frac {\left (35 a^3 d f^3+144 a b^2 e f (d e+c f)-40 a^2 b f^2 (3 d e+c f)-64 b^3 e^2 (d e+3 c f)\right ) x \sqrt {a+b x^2}}{128 b^4}+\frac {f \left (35 a^2 d f^2+144 b^2 e (d e+c f)-40 a b f (3 d e+c f)\right ) x^3 \sqrt {a+b x^2}}{192 b^3}-\frac {f^2 (7 a d f-8 b (3 d e+c f)) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {d f^3 x^7 \sqrt {a+b x^2}}{8 b}+\frac {\left (128 b^4 c e^3+35 a^4 d f^3+144 a^2 b^2 e f (d e+c f)-40 a^3 b f^2 (3 d e+c f)-64 a b^3 e^2 (d e+3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}} \] Output:
-1/128*(35*a^3*d*f^3+144*a*b^2*e*f*(c*f+d*e)-40*a^2*b*f^2*(c*f+3*d*e)-64*b ^3*e^2*(3*c*f+d*e))*x*(b*x^2+a)^(1/2)/b^4+1/192*f*(35*a^2*d*f^2+144*b^2*e* (c*f+d*e)-40*a*b*f*(c*f+3*d*e))*x^3*(b*x^2+a)^(1/2)/b^3-1/48*f^2*(7*a*d*f- 8*b*(c*f+3*d*e))*x^5*(b*x^2+a)^(1/2)/b^2+1/8*d*f^3*x^7*(b*x^2+a)^(1/2)/b+1 /128*(128*b^4*c*e^3+35*a^4*d*f^3+144*a^2*b^2*e*f*(c*f+d*e)-40*a^3*b*f^2*(c *f+3*d*e)-64*a*b^3*e^2*(3*c*f+d*e))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^( 9/2)
Time = 0.69 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^3 d f^3+10 a^2 b f^2 \left (36 d e+12 c f+7 d f x^2\right )-8 a b^2 f \left (2 c f \left (27 e+5 f x^2\right )+d \left (54 e^2+30 e f x^2+7 f^2 x^4\right )\right )+16 b^3 \left (2 c f \left (18 e^2+9 e f x^2+2 f^2 x^4\right )+3 d \left (4 e^3+6 e^2 f x^2+4 e f^2 x^4+f^3 x^6\right )\right )\right )-3 \left (128 b^4 c e^3+35 a^4 d f^3+144 a^2 b^2 e f (d e+c f)-40 a^3 b f^2 (3 d e+c f)-64 a b^3 e^2 (d e+3 c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{9/2}} \] Input:
Integrate[((c + d*x^2)*(e + f*x^2)^3)/Sqrt[a + b*x^2],x]
Output:
(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^3*d*f^3 + 10*a^2*b*f^2*(36*d*e + 12*c*f + 7*d*f*x^2) - 8*a*b^2*f*(2*c*f*(27*e + 5*f*x^2) + d*(54*e^2 + 30*e*f*x^2 + 7*f^2*x^4)) + 16*b^3*(2*c*f*(18*e^2 + 9*e*f*x^2 + 2*f^2*x^4) + 3*d*(4*e ^3 + 6*e^2*f*x^2 + 4*e*f^2*x^4 + f^3*x^6))) - 3*(128*b^4*c*e^3 + 35*a^4*d* f^3 + 144*a^2*b^2*e*f*(d*e + c*f) - 40*a^3*b*f^2*(3*d*e + c*f) - 64*a*b^3* e^2*(d*e + 3*c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(384*b^(9/2))
Time = 0.56 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {403, 403, 403, 299, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \frac {\left (f x^2+e\right )^2 \left ((6 b d e+8 b c f-7 a d f) x^2+(8 b c-a d) e\right )}{\sqrt {b x^2+a}}dx}{8 b}+\frac {d x \sqrt {a+b x^2} \left (e+f x^2\right )^3}{8 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {\left (f x^2+e\right ) \left (\left (8 e (3 d e+10 c f) b^2-8 a f (8 d e+5 c f) b+35 a^2 d f^2\right ) x^2+e \left (7 d f a^2-12 b d e a-8 b c f a+48 b^2 c e\right )\right )}{\sqrt {b x^2+a}}dx}{6 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (-7 a d f+8 b c f+6 b d e)}{6 b}}{8 b}+\frac {d x \sqrt {a+b x^2} \left (e+f x^2\right )^3}{8 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {e \left (-35 d f^2 a^3+4 b f (23 d e+10 c f) a^2-8 b^2 e (9 d e+14 c f) a+192 b^3 c e^2\right )-\left (-16 e^2 (3 d e+22 c f) b^3+8 a e f (31 d e+44 c f) b^2-10 a^2 f^2 (29 d e+12 c f) b+105 a^3 d f^3\right ) x^2}{\sqrt {b x^2+a}}dx}{4 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f^2-8 a b f (5 c f+8 d e)+8 b^2 e (10 c f+3 d e)\right )}{4 b}}{6 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (-7 a d f+8 b c f+6 b d e)}{6 b}}{8 b}+\frac {d x \sqrt {a+b x^2} \left (e+f x^2\right )^3}{8 b}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (35 a^4 d f^3-40 a^3 b f^2 (c f+3 d e)+144 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}-\frac {x \sqrt {a+b x^2} \left (105 a^3 d f^3-10 a^2 b f^2 (12 c f+29 d e)+8 a b^2 e f (44 c f+31 d e)-16 b^3 e^2 (22 c f+3 d e)\right )}{2 b}}{4 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f^2-8 a b f (5 c f+8 d e)+8 b^2 e (10 c f+3 d e)\right )}{4 b}}{6 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (-7 a d f+8 b c f+6 b d e)}{6 b}}{8 b}+\frac {d x \sqrt {a+b x^2} \left (e+f x^2\right )^3}{8 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (35 a^4 d f^3-40 a^3 b f^2 (c f+3 d e)+144 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}-\frac {x \sqrt {a+b x^2} \left (105 a^3 d f^3-10 a^2 b f^2 (12 c f+29 d e)+8 a b^2 e f (44 c f+31 d e)-16 b^3 e^2 (22 c f+3 d e)\right )}{2 b}}{4 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f^2-8 a b f (5 c f+8 d e)+8 b^2 e (10 c f+3 d e)\right )}{4 b}}{6 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (-7 a d f+8 b c f+6 b d e)}{6 b}}{8 b}+\frac {d x \sqrt {a+b x^2} \left (e+f x^2\right )^3}{8 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {x \sqrt {a+b x^2} \left (e+f x^2\right ) \left (35 a^2 d f^2-8 a b f (5 c f+8 d e)+8 b^2 e (10 c f+3 d e)\right )}{4 b}+\frac {\frac {3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^4 d f^3-40 a^3 b f^2 (c f+3 d e)+144 a^2 b^2 e f (c f+d e)-64 a b^3 e^2 (3 c f+d e)+128 b^4 c e^3\right )}{2 b^{3/2}}-\frac {x \sqrt {a+b x^2} \left (105 a^3 d f^3-10 a^2 b f^2 (12 c f+29 d e)+8 a b^2 e f (44 c f+31 d e)-16 b^3 e^2 (22 c f+3 d e)\right )}{2 b}}{4 b}}{6 b}+\frac {x \sqrt {a+b x^2} \left (e+f x^2\right )^2 (-7 a d f+8 b c f+6 b d e)}{6 b}}{8 b}+\frac {d x \sqrt {a+b x^2} \left (e+f x^2\right )^3}{8 b}\) |
Input:
Int[((c + d*x^2)*(e + f*x^2)^3)/Sqrt[a + b*x^2],x]
Output:
(d*x*Sqrt[a + b*x^2]*(e + f*x^2)^3)/(8*b) + (((6*b*d*e + 8*b*c*f - 7*a*d*f )*x*Sqrt[a + b*x^2]*(e + f*x^2)^2)/(6*b) + (((35*a^2*d*f^2 - 8*a*b*f*(8*d* e + 5*c*f) + 8*b^2*e*(3*d*e + 10*c*f))*x*Sqrt[a + b*x^2]*(e + f*x^2))/(4*b ) + (-1/2*((105*a^3*d*f^3 - 10*a^2*b*f^2*(29*d*e + 12*c*f) - 16*b^3*e^2*(3 *d*e + 22*c*f) + 8*a*b^2*e*f*(31*d*e + 44*c*f))*x*Sqrt[a + b*x^2])/b + (3* (128*b^4*c*e^3 + 35*a^4*d*f^3 + 144*a^2*b^2*e*f*(d*e + c*f) - 40*a^3*b*f^2 *(3*d*e + c*f) - 64*a*b^3*e^2*(d*e + 3*c*f))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)))/(4*b))/(6*b))/(8*b)
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Time = 0.67 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.80
| method | result | size |
| pseudoelliptic | \(\frac {\frac {35 \left (a^{3} \left (a d -\frac {8 b c}{7}\right ) f^{3}-\frac {24 a^{2} \left (a d -\frac {6 b c}{5}\right ) b e \,f^{2}}{7}+\frac {144 a \left (a d -\frac {4 b c}{3}\right ) b^{2} e^{2} f}{35}-\frac {64 b^{3} e^{3} \left (a d -2 b c \right )}{35}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{128}-\frac {35 \sqrt {b \,x^{2}+a}\, x \left (\left (-\frac {64 \left (\frac {3 x^{2} d}{4}+c \right ) x^{4} f^{3}}{105}-\frac {96 \left (\frac {2 x^{2} d}{3}+c \right ) x^{2} e \,f^{2}}{35}-\frac {192 e^{2} \left (\frac {x^{2} d}{2}+c \right ) f}{35}-\frac {64 d \,e^{3}}{35}\right ) b^{\frac {7}{2}}+a \left (\left (\frac {16 \left (\frac {7 x^{2} d}{10}+c \right ) x^{2} f^{2}}{21}+\frac {144 \left (\frac {5 x^{2} d}{9}+c \right ) e f}{35}+\frac {144 d \,e^{2}}{35}\right ) b^{\frac {5}{2}}+a \left (\left (\left (-\frac {2 x^{2} d}{3}-\frac {8 c}{7}\right ) f -\frac {24 d e}{7}\right ) b^{\frac {3}{2}}+a d f \sqrt {b}\right ) f \right ) f \right )}{128}}{b^{\frac {9}{2}}}\) | \(239\) |
| risch | \(-\frac {x \left (-48 d \,f^{3} b^{3} x^{6}+56 a \,b^{2} d \,f^{3} x^{4}-64 b^{3} c \,f^{3} x^{4}-192 b^{3} d e \,f^{2} x^{4}-70 a^{2} f^{3} d b \,x^{2}+80 a \,b^{2} c \,f^{3} x^{2}+240 a \,b^{2} d e \,f^{2} x^{2}-288 b^{3} c e \,f^{2} x^{2}-288 b^{3} d \,e^{2} f \,x^{2}+105 a^{3} d \,f^{3}-120 a^{2} b c \,f^{3}-360 a^{2} b d e \,f^{2}+432 a \,b^{2} c e \,f^{2}+432 a \,b^{2} d \,e^{2} f -576 b^{3} c \,e^{2} f -192 b^{3} d \,e^{3}\right ) \sqrt {b \,x^{2}+a}}{384 b^{4}}+\frac {\left (35 a^{4} d \,f^{3}-40 a^{3} b c \,f^{3}-120 a^{3} b d e \,f^{2}+144 a^{2} b^{2} c e \,f^{2}+144 a^{2} b^{2} d \,e^{2} f -192 a \,b^{3} c \,e^{2} f -64 a \,b^{3} d \,e^{3}+128 b^{4} c \,e^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {9}{2}}}\) | \(313\) |
| default | \(\frac {c \,e^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+f^{2} \left (c f +3 d e \right ) \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+3 e f \left (c f +d e \right ) \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+e^{2} \left (3 c f +d e \right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+d \,f^{3} \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )\) | \(362\) |
Input:
int((d*x^2+c)*(f*x^2+e)^3/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
35/128/b^(9/2)*((a^3*(a*d-8/7*b*c)*f^3-24/7*a^2*(a*d-6/5*b*c)*b*e*f^2+144/ 35*a*(a*d-4/3*b*c)*b^2*e^2*f-64/35*b^3*e^3*(a*d-2*b*c))*arctanh((b*x^2+a)^ (1/2)/x/b^(1/2))-(b*x^2+a)^(1/2)*x*((-64/105*(3/4*x^2*d+c)*x^4*f^3-96/35*( 2/3*x^2*d+c)*x^2*e*f^2-192/35*e^2*(1/2*x^2*d+c)*f-64/35*d*e^3)*b^(7/2)+a*( (16/21*(7/10*x^2*d+c)*x^2*f^2+144/35*(5/9*x^2*d+c)*e*f+144/35*d*e^2)*b^(5/ 2)+a*(((-2/3*x^2*d-8/7*c)*f-24/7*d*e)*b^(3/2)+a*d*f*b^(1/2))*f)*f))
Time = 0.25 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.09 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\left [\frac {3 \, {\left (64 \, {\left (2 \, b^{4} c - a b^{3} d\right )} e^{3} - 48 \, {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} e^{2} f + 24 \, {\left (6 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} e f^{2} - 5 \, {\left (8 \, a^{3} b c - 7 \, a^{4} d\right )} f^{3}\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} d f^{3} x^{7} + 8 \, {\left (24 \, b^{4} d e f^{2} + {\left (8 \, b^{4} c - 7 \, a b^{3} d\right )} f^{3}\right )} x^{5} + 2 \, {\left (144 \, b^{4} d e^{2} f + 24 \, {\left (6 \, b^{4} c - 5 \, a b^{3} d\right )} e f^{2} - 5 \, {\left (8 \, a b^{3} c - 7 \, a^{2} b^{2} d\right )} f^{3}\right )} x^{3} + 3 \, {\left (64 \, b^{4} d e^{3} + 48 \, {\left (4 \, b^{4} c - 3 \, a b^{3} d\right )} e^{2} f - 24 \, {\left (6 \, a b^{3} c - 5 \, a^{2} b^{2} d\right )} e f^{2} + 5 \, {\left (8 \, a^{2} b^{2} c - 7 \, a^{3} b d\right )} f^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{5}}, -\frac {3 \, {\left (64 \, {\left (2 \, b^{4} c - a b^{3} d\right )} e^{3} - 48 \, {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} e^{2} f + 24 \, {\left (6 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} e f^{2} - 5 \, {\left (8 \, a^{3} b c - 7 \, a^{4} d\right )} f^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d f^{3} x^{7} + 8 \, {\left (24 \, b^{4} d e f^{2} + {\left (8 \, b^{4} c - 7 \, a b^{3} d\right )} f^{3}\right )} x^{5} + 2 \, {\left (144 \, b^{4} d e^{2} f + 24 \, {\left (6 \, b^{4} c - 5 \, a b^{3} d\right )} e f^{2} - 5 \, {\left (8 \, a b^{3} c - 7 \, a^{2} b^{2} d\right )} f^{3}\right )} x^{3} + 3 \, {\left (64 \, b^{4} d e^{3} + 48 \, {\left (4 \, b^{4} c - 3 \, a b^{3} d\right )} e^{2} f - 24 \, {\left (6 \, a b^{3} c - 5 \, a^{2} b^{2} d\right )} e f^{2} + 5 \, {\left (8 \, a^{2} b^{2} c - 7 \, a^{3} b d\right )} f^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{5}}\right ] \] Input:
integrate((d*x^2+c)*(f*x^2+e)^3/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
[1/768*(3*(64*(2*b^4*c - a*b^3*d)*e^3 - 48*(4*a*b^3*c - 3*a^2*b^2*d)*e^2*f + 24*(6*a^2*b^2*c - 5*a^3*b*d)*e*f^2 - 5*(8*a^3*b*c - 7*a^4*d)*f^3)*sqrt( b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(48*b^4*d*f^3*x^7 + 8*(24*b^4*d*e*f^2 + (8*b^4*c - 7*a*b^3*d)*f^3)*x^5 + 2*(144*b^4*d*e^2*f + 24*(6*b^4*c - 5*a*b^3*d)*e*f^2 - 5*(8*a*b^3*c - 7*a^2*b^2*d)*f^3)*x^3 + 3 *(64*b^4*d*e^3 + 48*(4*b^4*c - 3*a*b^3*d)*e^2*f - 24*(6*a*b^3*c - 5*a^2*b^ 2*d)*e*f^2 + 5*(8*a^2*b^2*c - 7*a^3*b*d)*f^3)*x)*sqrt(b*x^2 + a))/b^5, -1/ 384*(3*(64*(2*b^4*c - a*b^3*d)*e^3 - 48*(4*a*b^3*c - 3*a^2*b^2*d)*e^2*f + 24*(6*a^2*b^2*c - 5*a^3*b*d)*e*f^2 - 5*(8*a^3*b*c - 7*a^4*d)*f^3)*sqrt(-b) *arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (48*b^4*d*f^3*x^7 + 8*(24*b^4*d*e*f^ 2 + (8*b^4*c - 7*a*b^3*d)*f^3)*x^5 + 2*(144*b^4*d*e^2*f + 24*(6*b^4*c - 5* a*b^3*d)*e*f^2 - 5*(8*a*b^3*c - 7*a^2*b^2*d)*f^3)*x^3 + 3*(64*b^4*d*e^3 + 48*(4*b^4*c - 3*a*b^3*d)*e^2*f - 24*(6*a*b^3*c - 5*a^2*b^2*d)*e*f^2 + 5*(8 *a^2*b^2*c - 7*a^3*b*d)*f^3)*x)*sqrt(b*x^2 + a))/b^5]
Time = 0.54 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.29 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {d f^{3} x^{7}}{8 b} + \frac {x^{5} \left (- \frac {7 a d f^{3}}{8 b} + c f^{3} + 3 d e f^{2}\right )}{6 b} + \frac {x^{3} \left (- \frac {5 a \left (- \frac {7 a d f^{3}}{8 b} + c f^{3} + 3 d e f^{2}\right )}{6 b} + 3 c e f^{2} + 3 d e^{2} f\right )}{4 b} + \frac {x \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a d f^{3}}{8 b} + c f^{3} + 3 d e f^{2}\right )}{6 b} + 3 c e f^{2} + 3 d e^{2} f\right )}{4 b} + 3 c e^{2} f + d e^{3}\right )}{2 b}\right ) + \left (- \frac {a \left (- \frac {3 a \left (- \frac {5 a \left (- \frac {7 a d f^{3}}{8 b} + c f^{3} + 3 d e f^{2}\right )}{6 b} + 3 c e f^{2} + 3 d e^{2} f\right )}{4 b} + 3 c e^{2} f + d e^{3}\right )}{2 b} + c e^{3}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {c e^{3} x + \frac {d f^{3} x^{9}}{9} + \frac {x^{7} \left (c f^{3} + 3 d e f^{2}\right )}{7} + \frac {x^{5} \cdot \left (3 c e f^{2} + 3 d e^{2} f\right )}{5} + \frac {x^{3} \cdot \left (3 c e^{2} f + d e^{3}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**2+c)*(f*x**2+e)**3/(b*x**2+a)**(1/2),x)
Output:
Piecewise((sqrt(a + b*x**2)*(d*f**3*x**7/(8*b) + x**5*(-7*a*d*f**3/(8*b) + c*f**3 + 3*d*e*f**2)/(6*b) + x**3*(-5*a*(-7*a*d*f**3/(8*b) + c*f**3 + 3*d *e*f**2)/(6*b) + 3*c*e*f**2 + 3*d*e**2*f)/(4*b) + x*(-3*a*(-5*a*(-7*a*d*f* *3/(8*b) + c*f**3 + 3*d*e*f**2)/(6*b) + 3*c*e*f**2 + 3*d*e**2*f)/(4*b) + 3 *c*e**2*f + d*e**3)/(2*b)) + (-a*(-3*a*(-5*a*(-7*a*d*f**3/(8*b) + c*f**3 + 3*d*e*f**2)/(6*b) + 3*c*e*f**2 + 3*d*e**2*f)/(4*b) + 3*c*e**2*f + d*e**3) /(2*b) + c*e**3)*Piecewise((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)), Ne(b, 0)), ((c*e**3*x + d*f* *3*x**9/9 + x**7*(c*f**3 + 3*d*e*f**2)/7 + x**5*(3*c*e*f**2 + 3*d*e**2*f)/ 5 + x**3*(3*c*e**2*f + d*e**3)/3)/sqrt(a), True))
Time = 0.05 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} d f^{3} x^{7}}{8 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a d f^{3} x^{5}}{48 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} d f^{3} x^{3}}{192 \, b^{3}} + \frac {{\left (3 \, d e f^{2} + c f^{3}\right )} \sqrt {b x^{2} + a} x^{5}}{6 \, b} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} d f^{3} x}{128 \, b^{4}} + \frac {c e^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} + \frac {35 \, a^{4} d f^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{2}}} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} \sqrt {b x^{2} + a} a x^{3}}{24 \, b^{2}} + \frac {3 \, {\left (d e^{2} f + c e f^{2}\right )} \sqrt {b x^{2} + a} x^{3}}{4 \, b} + \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b^{3}} - \frac {9 \, {\left (d e^{2} f + c e f^{2}\right )} \sqrt {b x^{2} + a} a x}{8 \, b^{2}} + \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} \sqrt {b x^{2} + a} x}{2 \, b} - \frac {5 \, {\left (3 \, d e f^{2} + c f^{3}\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} + \frac {9 \, {\left (d e^{2} f + c e f^{2}\right )} a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {{\left (d e^{3} + 3 \, c e^{2} f\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} \] Input:
integrate((d*x^2+c)*(f*x^2+e)^3/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
1/8*sqrt(b*x^2 + a)*d*f^3*x^7/b - 7/48*sqrt(b*x^2 + a)*a*d*f^3*x^5/b^2 + 3 5/192*sqrt(b*x^2 + a)*a^2*d*f^3*x^3/b^3 + 1/6*(3*d*e*f^2 + c*f^3)*sqrt(b*x ^2 + a)*x^5/b - 35/128*sqrt(b*x^2 + a)*a^3*d*f^3*x/b^4 + c*e^3*arcsinh(b*x /sqrt(a*b))/sqrt(b) + 35/128*a^4*d*f^3*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 5/ 24*(3*d*e*f^2 + c*f^3)*sqrt(b*x^2 + a)*a*x^3/b^2 + 3/4*(d*e^2*f + c*e*f^2) *sqrt(b*x^2 + a)*x^3/b + 5/16*(3*d*e*f^2 + c*f^3)*sqrt(b*x^2 + a)*a^2*x/b^ 3 - 9/8*(d*e^2*f + c*e*f^2)*sqrt(b*x^2 + a)*a*x/b^2 + 1/2*(d*e^3 + 3*c*e^2 *f)*sqrt(b*x^2 + a)*x/b - 5/16*(3*d*e*f^2 + c*f^3)*a^3*arcsinh(b*x/sqrt(a* b))/b^(7/2) + 9/8*(d*e^2*f + c*e*f^2)*a^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 1/2*(d*e^3 + 3*c*e^2*f)*a*arcsinh(b*x/sqrt(a*b))/b^(3/2)
Time = 0.14 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, d f^{3} x^{2}}{b} + \frac {24 \, b^{6} d e f^{2} + 8 \, b^{6} c f^{3} - 7 \, a b^{5} d f^{3}}{b^{7}}\right )} x^{2} + \frac {144 \, b^{6} d e^{2} f + 144 \, b^{6} c e f^{2} - 120 \, a b^{5} d e f^{2} - 40 \, a b^{5} c f^{3} + 35 \, a^{2} b^{4} d f^{3}}{b^{7}}\right )} x^{2} + \frac {3 \, {\left (64 \, b^{6} d e^{3} + 192 \, b^{6} c e^{2} f - 144 \, a b^{5} d e^{2} f - 144 \, a b^{5} c e f^{2} + 120 \, a^{2} b^{4} d e f^{2} + 40 \, a^{2} b^{4} c f^{3} - 35 \, a^{3} b^{3} d f^{3}\right )}}{b^{7}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (128 \, b^{4} c e^{3} - 64 \, a b^{3} d e^{3} - 192 \, a b^{3} c e^{2} f + 144 \, a^{2} b^{2} d e^{2} f + 144 \, a^{2} b^{2} c e f^{2} - 120 \, a^{3} b d e f^{2} - 40 \, a^{3} b c f^{3} + 35 \, a^{4} d f^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {9}{2}}} \] Input:
integrate((d*x^2+c)*(f*x^2+e)^3/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
1/384*(2*(4*(6*d*f^3*x^2/b + (24*b^6*d*e*f^2 + 8*b^6*c*f^3 - 7*a*b^5*d*f^3 )/b^7)*x^2 + (144*b^6*d*e^2*f + 144*b^6*c*e*f^2 - 120*a*b^5*d*e*f^2 - 40*a *b^5*c*f^3 + 35*a^2*b^4*d*f^3)/b^7)*x^2 + 3*(64*b^6*d*e^3 + 192*b^6*c*e^2* f - 144*a*b^5*d*e^2*f - 144*a*b^5*c*e*f^2 + 120*a^2*b^4*d*e*f^2 + 40*a^2*b ^4*c*f^3 - 35*a^3*b^3*d*f^3)/b^7)*sqrt(b*x^2 + a)*x - 1/128*(128*b^4*c*e^3 - 64*a*b^3*d*e^3 - 192*a*b^3*c*e^2*f + 144*a^2*b^2*d*e^2*f + 144*a^2*b^2* c*e*f^2 - 120*a^3*b*d*e*f^2 - 40*a^3*b*c*f^3 + 35*a^4*d*f^3)*log(abs(-sqrt (b)*x + sqrt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (d\,x^2+c\right )\,{\left (f\,x^2+e\right )}^3}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((c + d*x^2)*(e + f*x^2)^3)/(a + b*x^2)^(1/2),x)
Output:
int(((c + d*x^2)*(e + f*x^2)^3)/(a + b*x^2)^(1/2), x)
Time = 0.24 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.96 \[ \int \frac {\left (c+d x^2\right ) \left (e+f x^2\right )^3}{\sqrt {a+b x^2}} \, dx=\frac {-105 \sqrt {b \,x^{2}+a}\, a^{3} b d \,f^{3} x +120 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,f^{3} x +360 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d e \,f^{2} x +70 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,f^{3} x^{3}-432 \sqrt {b \,x^{2}+a}\, a \,b^{3} c e \,f^{2} x -80 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,f^{3} x^{3}-432 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,e^{2} f x -240 \sqrt {b \,x^{2}+a}\, a \,b^{3} d e \,f^{2} x^{3}-56 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,f^{3} x^{5}+576 \sqrt {b \,x^{2}+a}\, b^{4} c \,e^{2} f x +288 \sqrt {b \,x^{2}+a}\, b^{4} c e \,f^{2} x^{3}+64 \sqrt {b \,x^{2}+a}\, b^{4} c \,f^{3} x^{5}+192 \sqrt {b \,x^{2}+a}\, b^{4} d \,e^{3} x +288 \sqrt {b \,x^{2}+a}\, b^{4} d \,e^{2} f \,x^{3}+192 \sqrt {b \,x^{2}+a}\, b^{4} d e \,f^{2} x^{5}+48 \sqrt {b \,x^{2}+a}\, b^{4} d \,f^{3} x^{7}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d \,f^{3}-120 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c \,f^{3}-360 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d e \,f^{2}+432 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c e \,f^{2}+432 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} d \,e^{2} f -576 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c \,e^{2} f -192 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} d \,e^{3}+384 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,e^{3}}{384 b^{5}} \] Input:
int((d*x^2+c)*(f*x^2+e)^3/(b*x^2+a)^(1/2),x)
Output:
( - 105*sqrt(a + b*x**2)*a**3*b*d*f**3*x + 120*sqrt(a + b*x**2)*a**2*b**2* c*f**3*x + 360*sqrt(a + b*x**2)*a**2*b**2*d*e*f**2*x + 70*sqrt(a + b*x**2) *a**2*b**2*d*f**3*x**3 - 432*sqrt(a + b*x**2)*a*b**3*c*e*f**2*x - 80*sqrt( a + b*x**2)*a*b**3*c*f**3*x**3 - 432*sqrt(a + b*x**2)*a*b**3*d*e**2*f*x - 240*sqrt(a + b*x**2)*a*b**3*d*e*f**2*x**3 - 56*sqrt(a + b*x**2)*a*b**3*d*f **3*x**5 + 576*sqrt(a + b*x**2)*b**4*c*e**2*f*x + 288*sqrt(a + b*x**2)*b** 4*c*e*f**2*x**3 + 64*sqrt(a + b*x**2)*b**4*c*f**3*x**5 + 192*sqrt(a + b*x* *2)*b**4*d*e**3*x + 288*sqrt(a + b*x**2)*b**4*d*e**2*f*x**3 + 192*sqrt(a + b*x**2)*b**4*d*e*f**2*x**5 + 48*sqrt(a + b*x**2)*b**4*d*f**3*x**7 + 105*s qrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d*f**3 - 120*sqrt( b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*c*f**3 - 360*sqrt(b) *log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*d*e*f**2 + 432*sqrt(b) *log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*e*f**2 + 432*sqrt (b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*d*e**2*f - 576*s qrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**3*c*e**2*f - 192*s qrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**3*d*e**3 + 384*sqr t(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**4*c*e**3)/(384*b**5)