Integrand size = 26, antiderivative size = 207 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {a \left (48 b^2 c e+3 a^2 d f-8 a b (d e+c f)\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (48 b^2 c e+3 a^2 d f-8 a b (d e+c f)\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}-\frac {(3 a d f-8 b (d e+c f)) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d f x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {a^2 \left (48 b^2 c e+3 a^2 d f-8 a b (d e+c f)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:
1/128*a*(48*b^2*c*e+3*a^2*d*f-8*a*b*(c*f+d*e))*x*(b*x^2+a)^(1/2)/b^2+1/192 *(48*b^2*c*e+3*a^2*d*f-8*a*b*(c*f+d*e))*x*(b*x^2+a)^(3/2)/b^2-1/48*(3*a*d* f-8*b*(c*f+d*e))*x*(b*x^2+a)^(5/2)/b^2+1/8*d*f*x^3*(b*x^2+a)^(5/2)/b+1/128 *a^2*(48*b^2*c*e+3*a^2*d*f-8*a*b*(c*f+d*e))*arctanh(b^(1/2)*x/(b*x^2+a)^(1 /2))/b^(5/2)
Time = 0.48 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.85 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-9 a^3 d f+6 a^2 b \left (4 d e+4 c f+d f x^2\right )+16 b^3 x^2 \left (6 c e+4 d e x^2+4 c f x^2+3 d f x^4\right )+8 a b^2 \left (30 c e+14 d e x^2+14 c f x^2+9 d f x^4\right )\right )-3 a^2 \left (48 b^2 c e+3 a^2 d f-8 a b (d e+c f)\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{384 b^{5/2}} \] Input:
Integrate[(a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2),x]
Output:
(Sqrt[b]*x*Sqrt[a + b*x^2]*(-9*a^3*d*f + 6*a^2*b*(4*d*e + 4*c*f + d*f*x^2) + 16*b^3*x^2*(6*c*e + 4*d*e*x^2 + 4*c*f*x^2 + 3*d*f*x^4) + 8*a*b^2*(30*c* e + 14*d*e*x^2 + 14*c*f*x^2 + 9*d*f*x^4)) - 3*a^2*(48*b^2*c*e + 3*a^2*d*f - 8*a*b*(d*e + c*f))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(384*b^(5/2))
Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {403, 299, 211, 211, 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.
| \(\displaystyle \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \left (b x^2+a\right )^{3/2} \left ((8 b d e+2 b c f-3 a d f) x^2+c (8 b e-a f)\right )dx}{8 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d f-8 a b (c f+d e)+48 b^2 c e\right ) \int \left (b x^2+a\right )^{3/2}dx}{6 b}+\frac {x \left (a+b x^2\right )^{5/2} (-3 a d f+2 b c f+8 b d e)}{6 b}}{8 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d f-8 a b (c f+d e)+48 b^2 c e\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}+\frac {x \left (a+b x^2\right )^{5/2} (-3 a d f+2 b c f+8 b d e)}{6 b}}{8 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d f-8 a b (c f+d e)+48 b^2 c e\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}+\frac {x \left (a+b x^2\right )^{5/2} (-3 a d f+2 b c f+8 b d e)}{6 b}}{8 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\left (3 a^2 d f-8 a b (c f+d e)+48 b^2 c e\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}+\frac {x \left (a+b x^2\right )^{5/2} (-3 a d f+2 b c f+8 b d e)}{6 b}}{8 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (3 a^2 d f-8 a b (c f+d e)+48 b^2 c e\right )}{6 b}+\frac {x \left (a+b x^2\right )^{5/2} (-3 a d f+2 b c f+8 b d e)}{6 b}}{8 b}+\frac {f x \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )}{8 b}\) |
Input:
Int[(a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2),x]
Output:
(f*x*(a + b*x^2)^(5/2)*(c + d*x^2))/(8*b) + (((8*b*d*e + 2*b*c*f - 3*a*d*f )*x*(a + b*x^2)^(5/2))/(6*b) + ((48*b^2*c*e + 3*a^2*d*f - 8*a*b*(d*e + c*f ))*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sq rt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/(6*b))/(8*b)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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 = 0.58 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.77
| method | result | size |
| pseudoelliptic | \(\frac {\frac {3 a^{2} \left (a^{2} d f -\frac {8 a b \left (c f +d e \right )}{3}+16 c e \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{128}-\frac {3 \sqrt {b \,x^{2}+a}\, x \left (-\frac {80 a \left (\frac {3 d f \,x^{4}}{10}+\frac {7 \left (c f +d e \right ) x^{2}}{15}+c e \right ) b^{\frac {5}{2}}}{3}-\frac {32 x^{2} \left (\frac {d f \,x^{4}}{2}+\frac {2 \left (c f +d e \right ) x^{2}}{3}+c e \right ) b^{\frac {7}{2}}}{3}+a^{2} \left (\frac {2 \left (-d f \,x^{2}-4 c f -4 d e \right ) b^{\frac {3}{2}}}{3}+a d f \sqrt {b}\right )\right )}{128}}{b^{\frac {5}{2}}}\) | \(159\) |
| risch | \(-\frac {x \left (-48 b^{3} d f \,x^{6}-72 a d f \,b^{2} x^{4}-64 b^{3} c f \,x^{4}-64 d e \,b^{3} x^{4}-6 a^{2} b d f \,x^{2}-112 a c f \,b^{2} x^{2}-112 a d e \,b^{2} x^{2}-96 b^{3} c e \,x^{2}+9 a^{3} d f -24 a^{2} c f b -24 a^{2} b e d -240 a c e \,b^{2}\right ) \sqrt {b \,x^{2}+a}}{384 b^{2}}+\frac {a^{2} \left (3 a^{2} d f -8 a b c f -8 a b d e +48 c e \,b^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}\) | \(184\) |
| default | \(c e \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\left (c f +d e \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+d f \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )\) | \(237\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)*(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
3/128*(a^2*(a^2*d*f-8/3*a*b*(c*f+d*e)+16*c*e*b^2)*arctanh((b*x^2+a)^(1/2)/ x/b^(1/2))-(b*x^2+a)^(1/2)*x*(-80/3*a*(3/10*d*f*x^4+7/15*(c*f+d*e)*x^2+c*e )*b^(5/2)-32/3*x^2*(1/2*d*f*x^4+2/3*(c*f+d*e)*x^2+c*e)*b^(7/2)+a^2*(2/3*(- d*f*x^2-4*c*f-4*d*e)*b^(3/2)+a*d*f*b^(1/2))))/b^(5/2)
Time = 0.15 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.00 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\left [\frac {3 \, {\left (8 \, {\left (6 \, a^{2} b^{2} c - a^{3} b d\right )} e - {\left (8 \, a^{3} b c - 3 \, a^{4} d\right )} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (48 \, b^{4} d f x^{7} + 8 \, {\left (8 \, b^{4} d e + {\left (8 \, b^{4} c + 9 \, a b^{3} d\right )} f\right )} x^{5} + 2 \, {\left (8 \, {\left (6 \, b^{4} c + 7 \, a b^{3} d\right )} e + {\left (56 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} f\right )} x^{3} + 3 \, {\left (8 \, {\left (10 \, a b^{3} c + a^{2} b^{2} d\right )} e + {\left (8 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{3}}, -\frac {3 \, {\left (8 \, {\left (6 \, a^{2} b^{2} c - a^{3} b d\right )} e - {\left (8 \, a^{3} b c - 3 \, a^{4} d\right )} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (48 \, b^{4} d f x^{7} + 8 \, {\left (8 \, b^{4} d e + {\left (8 \, b^{4} c + 9 \, a b^{3} d\right )} f\right )} x^{5} + 2 \, {\left (8 \, {\left (6 \, b^{4} c + 7 \, a b^{3} d\right )} e + {\left (56 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} f\right )} x^{3} + 3 \, {\left (8 \, {\left (10 \, a b^{3} c + a^{2} b^{2} d\right )} e + {\left (8 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} f\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{3}}\right ] \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)*(f*x^2+e),x, algorithm="fricas")
Output:
[1/768*(3*(8*(6*a^2*b^2*c - a^3*b*d)*e - (8*a^3*b*c - 3*a^4*d)*f)*sqrt(b)* log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(48*b^4*d*f*x^7 + 8*(8 *b^4*d*e + (8*b^4*c + 9*a*b^3*d)*f)*x^5 + 2*(8*(6*b^4*c + 7*a*b^3*d)*e + ( 56*a*b^3*c + 3*a^2*b^2*d)*f)*x^3 + 3*(8*(10*a*b^3*c + a^2*b^2*d)*e + (8*a^ 2*b^2*c - 3*a^3*b*d)*f)*x)*sqrt(b*x^2 + a))/b^3, -1/384*(3*(8*(6*a^2*b^2*c - a^3*b*d)*e - (8*a^3*b*c - 3*a^4*d)*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b *x^2 + a)) - (48*b^4*d*f*x^7 + 8*(8*b^4*d*e + (8*b^4*c + 9*a*b^3*d)*f)*x^5 + 2*(8*(6*b^4*c + 7*a*b^3*d)*e + (56*a*b^3*c + 3*a^2*b^2*d)*f)*x^3 + 3*(8 *(10*a*b^3*c + a^2*b^2*d)*e + (8*a^2*b^2*c - 3*a^3*b*d)*f)*x)*sqrt(b*x^2 + a))/b^3]
Time = 0.47 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.87 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {b d f x^{7}}{8} + \frac {x^{5} \cdot \left (\frac {9 a b d f}{8} + b^{2} c f + b^{2} d e\right )}{6 b} + \frac {x^{3} \left (a^{2} d f + 2 a b c f + 2 a b d e - \frac {5 a \left (\frac {9 a b d f}{8} + b^{2} c f + b^{2} d e\right )}{6 b} + b^{2} c e\right )}{4 b} + \frac {x \left (a^{2} c f + a^{2} d e + 2 a b c e - \frac {3 a \left (a^{2} d f + 2 a b c f + 2 a b d e - \frac {5 a \left (\frac {9 a b d f}{8} + b^{2} c f + b^{2} d e\right )}{6 b} + b^{2} c e\right )}{4 b}\right )}{2 b}\right ) + \left (a^{2} c e - \frac {a \left (a^{2} c f + a^{2} d e + 2 a b c e - \frac {3 a \left (a^{2} d f + 2 a b c f + 2 a b d e - \frac {5 a \left (\frac {9 a b d f}{8} + b^{2} c f + b^{2} d e\right )}{6 b} + b^{2} c e\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (c e x + \frac {d f x^{5}}{5} + \frac {x^{3} \left (c f + d e\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)*(f*x**2+e),x)
Output:
Piecewise((sqrt(a + b*x**2)*(b*d*f*x**7/8 + x**5*(9*a*b*d*f/8 + b**2*c*f + b**2*d*e)/(6*b) + x**3*(a**2*d*f + 2*a*b*c*f + 2*a*b*d*e - 5*a*(9*a*b*d*f /8 + b**2*c*f + b**2*d*e)/(6*b) + b**2*c*e)/(4*b) + x*(a**2*c*f + a**2*d*e + 2*a*b*c*e - 3*a*(a**2*d*f + 2*a*b*c*f + 2*a*b*d*e - 5*a*(9*a*b*d*f/8 + b**2*c*f + b**2*d*e)/(6*b) + b**2*c*e)/(4*b))/(2*b)) + (a**2*c*e - a*(a**2 *c*f + a**2*d*e + 2*a*b*c*e - 3*a*(a**2*d*f + 2*a*b*c*f + 2*a*b*d*e - 5*a* (9*a*b*d*f/8 + b**2*c*f + b**2*d*e)/(6*b) + b**2*c*e)/(4*b))/(2*b))*Piecew ise((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)), (a**(3/2)*(c*e*x + d*f*x**5/5 + x**3*(c* f + d*e)/3), True))
Time = 0.04 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.15 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d f x^{3}}{8 \, b} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c e x + \frac {3}{8} \, \sqrt {b x^{2} + a} a c e x - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d f x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d f x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{3} d f x}{128 \, b^{2}} + \frac {3 \, a^{2} c e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {3 \, a^{4} d f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d e + c f\right )} x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d e + c f\right )} a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} {\left (d e + c f\right )} a^{2} x}{16 \, b} - \frac {{\left (d e + c f\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)*(f*x^2+e),x, algorithm="maxima")
Output:
1/8*(b*x^2 + a)^(5/2)*d*f*x^3/b + 1/4*(b*x^2 + a)^(3/2)*c*e*x + 3/8*sqrt(b *x^2 + a)*a*c*e*x - 1/16*(b*x^2 + a)^(5/2)*a*d*f*x/b^2 + 1/64*(b*x^2 + a)^ (3/2)*a^2*d*f*x/b^2 + 3/128*sqrt(b*x^2 + a)*a^3*d*f*x/b^2 + 3/8*a^2*c*e*ar csinh(b*x/sqrt(a*b))/sqrt(b) + 3/128*a^4*d*f*arcsinh(b*x/sqrt(a*b))/b^(5/2 ) + 1/6*(b*x^2 + a)^(5/2)*(d*e + c*f)*x/b - 1/24*(b*x^2 + a)^(3/2)*(d*e + c*f)*a*x/b - 1/16*sqrt(b*x^2 + a)*(d*e + c*f)*a^2*x/b - 1/16*(d*e + c*f)*a ^3*arcsinh(b*x/sqrt(a*b))/b^(3/2)
Time = 0.14 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b d f x^{2} + \frac {8 \, b^{7} d e + 8 \, b^{7} c f + 9 \, a b^{6} d f}{b^{6}}\right )} x^{2} + \frac {48 \, b^{7} c e + 56 \, a b^{6} d e + 56 \, a b^{6} c f + 3 \, a^{2} b^{5} d f}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (80 \, a b^{6} c e + 8 \, a^{2} b^{5} d e + 8 \, a^{2} b^{5} c f - 3 \, a^{3} b^{4} d f\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (48 \, a^{2} b^{2} c e - 8 \, a^{3} b d e - 8 \, a^{3} b c f + 3 \, a^{4} d f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)*(f*x^2+e),x, algorithm="giac")
Output:
1/384*(2*(4*(6*b*d*f*x^2 + (8*b^7*d*e + 8*b^7*c*f + 9*a*b^6*d*f)/b^6)*x^2 + (48*b^7*c*e + 56*a*b^6*d*e + 56*a*b^6*c*f + 3*a^2*b^5*d*f)/b^6)*x^2 + 3* (80*a*b^6*c*e + 8*a^2*b^5*d*e + 8*a^2*b^5*c*f - 3*a^3*b^4*d*f)/b^6)*sqrt(b *x^2 + a)*x - 1/128*(48*a^2*b^2*c*e - 8*a^3*b*d*e - 8*a^3*b*c*f + 3*a^4*d* f)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
Timed out. \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\int {\left (b\,x^2+a\right )}^{3/2}\,\left (d\,x^2+c\right )\,\left (f\,x^2+e\right ) \,d x \] Input:
int((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2),x)
Output:
int((a + b*x^2)^(3/2)*(c + d*x^2)*(e + f*x^2), x)
Time = 0.17 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.67 \[ \int \left (a+b x^2\right )^{3/2} \left (c+d x^2\right ) \left (e+f x^2\right ) \, dx=\frac {-9 \sqrt {b \,x^{2}+a}\, a^{3} b d f x +24 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c f x +24 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d e x +6 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d f \,x^{3}+240 \sqrt {b \,x^{2}+a}\, a \,b^{3} c e x +112 \sqrt {b \,x^{2}+a}\, a \,b^{3} c f \,x^{3}+112 \sqrt {b \,x^{2}+a}\, a \,b^{3} d e \,x^{3}+72 \sqrt {b \,x^{2}+a}\, a \,b^{3} d f \,x^{5}+96 \sqrt {b \,x^{2}+a}\, b^{4} c e \,x^{3}+64 \sqrt {b \,x^{2}+a}\, b^{4} c f \,x^{5}+64 \sqrt {b \,x^{2}+a}\, b^{4} d e \,x^{5}+48 \sqrt {b \,x^{2}+a}\, b^{4} d f \,x^{7}+9 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d f -24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c f -24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b d e +144 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c e}{384 b^{3}} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)*(f*x^2+e),x)
Output:
( - 9*sqrt(a + b*x**2)*a**3*b*d*f*x + 24*sqrt(a + b*x**2)*a**2*b**2*c*f*x + 24*sqrt(a + b*x**2)*a**2*b**2*d*e*x + 6*sqrt(a + b*x**2)*a**2*b**2*d*f*x **3 + 240*sqrt(a + b*x**2)*a*b**3*c*e*x + 112*sqrt(a + b*x**2)*a*b**3*c*f* x**3 + 112*sqrt(a + b*x**2)*a*b**3*d*e*x**3 + 72*sqrt(a + b*x**2)*a*b**3*d *f*x**5 + 96*sqrt(a + b*x**2)*b**4*c*e*x**3 + 64*sqrt(a + b*x**2)*b**4*c*f *x**5 + 64*sqrt(a + b*x**2)*b**4*d*e*x**5 + 48*sqrt(a + b*x**2)*b**4*d*f*x **7 + 9*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d*f - 24* sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*c*f - 24*sqrt(b )*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*d*e + 144*sqrt(b)*log ((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*b**2*c*e)/(384*b**3)