Integrand size = 32, antiderivative size = 297 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {a^2 \left (64 A b^3-a \left (24 b^2 B-12 a b C+7 a^2 D\right )\right ) x \sqrt {a+b x^2}}{1024 b^4}+\frac {a \left (64 A b^3-a \left (24 b^2 B-12 a b C+7 a^2 D\right )\right ) x^3 \sqrt {a+b x^2}}{512 b^3}+\frac {1}{384} \left (64 A-\frac {a \left (24 b^2 B-12 a b C+7 a^2 D\right )}{b^3}\right ) x^3 \left (a+b x^2\right )^{3/2}+\frac {\left (24 b^2 B-12 a b C+7 a^2 D\right ) x^3 \left (a+b x^2\right )^{5/2}}{192 b^3}+\frac {(12 b C-7 a D) x^5 \left (a+b x^2\right )^{5/2}}{120 b^2}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}-\frac {a^3 \left (64 A b^3-a \left (24 b^2 B-12 a b C+7 a^2 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{9/2}} \] Output:
1/1024*a^2*(64*A*b^3-a*(24*B*b^2-12*C*a*b+7*D*a^2))*x*(b*x^2+a)^(1/2)/b^4+ 1/512*a*(64*A*b^3-a*(24*B*b^2-12*C*a*b+7*D*a^2))*x^3*(b*x^2+a)^(1/2)/b^3+1 /384*(64*A-a*(24*B*b^2-12*C*a*b+7*D*a^2)/b^3)*x^3*(b*x^2+a)^(3/2)+1/192*(2 4*B*b^2-12*C*a*b+7*D*a^2)*x^3*(b*x^2+a)^(5/2)/b^3+1/120*(12*C*b-7*D*a)*x^5 *(b*x^2+a)^(5/2)/b^2+1/12*D*x^7*(b*x^2+a)^(5/2)/b-1/1024*a^3*(64*A*b^3-a*( 24*B*b^2-12*C*a*b+7*D*a^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
Time = 2.14 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^5 D+10 a^4 b \left (18 C+7 D x^2\right )-8 a^3 b^2 \left (45 B+15 C x^2+7 D x^4\right )+48 a^2 b^3 \left (20 A+5 B x^2+2 C x^4+D x^6\right )+128 b^5 x^4 \left (20 A+15 B x^2+12 C x^4+10 D x^6\right )+64 a b^4 x^2 \left (70 A+45 B x^2+33 C x^4+26 D x^6\right )\right )+120 a^3 b \left (16 A b^2+3 a^2 C\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )+30 a^4 \left (24 b^2 B+7 a^2 D\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{15360 b^{9/2}} \] Input:
Integrate[x^2*(a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^5*D + 10*a^4*b*(18*C + 7*D*x^2) - 8*a^3 *b^2*(45*B + 15*C*x^2 + 7*D*x^4) + 48*a^2*b^3*(20*A + 5*B*x^2 + 2*C*x^4 + D*x^6) + 128*b^5*x^4*(20*A + 15*B*x^2 + 12*C*x^4 + 10*D*x^6) + 64*a*b^4*x^ 2*(70*A + 45*B*x^2 + 33*C*x^4 + 26*D*x^6)) + 120*a^3*b*(16*A*b^2 + 3*a^2*C )*ArcTanh[(Sqrt[b]*x)/(Sqrt[a] - Sqrt[a + b*x^2])] + 30*a^4*(24*b^2*B + 7* a^2*D)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/(15360*b^(9/2))
Time = 0.50 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.81, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2340, 1590, 27, 363, 248, 248, 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.
| \(\displaystyle \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {\int x^2 \left (b x^2+a\right )^{3/2} \left ((12 b C-7 a D) x^4+12 b B x^2+12 A b\right )dx}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 1590 |
| \(\displaystyle \frac {\frac {\int 5 x^2 \left (b x^2+a\right )^{3/2} \left (24 A b^2+\left (7 D a^2-12 b C a+24 b^2 B\right ) x^2\right )dx}{10 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int x^2 \left (b x^2+a\right )^{3/2} \left (24 A b^2+\left (7 D a^2-12 b C a+24 b^2 B\right ) x^2\right )dx}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {\frac {3 \left (64 A b^3-a \left (7 a^2 D-12 a b C+24 b^2 B\right )\right ) \int x^2 \left (b x^2+a\right )^{3/2}dx}{8 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} \left (7 a^2 D-12 a b C+24 b^2 B\right )}{8 b}}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {\frac {3 \left (64 A b^3-a \left (7 a^2 D-12 a b C+24 b^2 B\right )\right ) \left (\frac {1}{2} a \int x^2 \sqrt {b x^2+a}dx+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )}{8 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} \left (7 a^2 D-12 a b C+24 b^2 B\right )}{8 b}}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {\frac {3 \left (64 A b^3-a \left (7 a^2 D-12 a b C+24 b^2 B\right )\right ) \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {x^2}{\sqrt {b x^2+a}}dx+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )}{8 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} \left (7 a^2 D-12 a b C+24 b^2 B\right )}{8 b}}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {\frac {3 \left (64 A b^3-a \left (7 a^2 D-12 a b C+24 b^2 B\right )\right ) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )}{8 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} \left (7 a^2 D-12 a b C+24 b^2 B\right )}{8 b}}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {3 \left (64 A b^3-a \left (7 a^2 D-12 a b C+24 b^2 B\right )\right ) \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )}{8 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} \left (7 a^2 D-12 a b C+24 b^2 B\right )}{8 b}}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right )+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right ) \left (64 A b^3-a \left (7 a^2 D-12 a b C+24 b^2 B\right )\right )}{8 b}+\frac {x^3 \left (a+b x^2\right )^{5/2} \left (7 a^2 D-12 a b C+24 b^2 B\right )}{8 b}}{2 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 b C-7 a D)}{10 b}}{12 b}+\frac {D x^7 \left (a+b x^2\right )^{5/2}}{12 b}\) |
Input:
Int[x^2*(a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + D*x^6),x]
Output:
(D*x^7*(a + b*x^2)^(5/2))/(12*b) + (((12*b*C - 7*a*D)*x^5*(a + b*x^2)^(5/2 ))/(10*b) + (((24*b^2*B - 12*a*b*C + 7*a^2*D)*x^3*(a + b*x^2)^(5/2))/(8*b) + (3*(64*A*b^3 - a*(24*b^2*B - 12*a*b*C + 7*a^2*D))*((x^3*(a + b*x^2)^(3/ 2))/6 + (a*((x^3*Sqrt[a + b*x^2])/4 + (a*((x*Sqrt[a + b*x^2])/(2*b) - (a*A rcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2))))/4))/2))/(8*b))/(2*b))/( 12*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 0.61 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(\frac {-\left (b^{3} A -\frac {3}{8} a \,b^{2} B +\frac {3}{16} a^{2} b C -\frac {7}{64} a^{3} D\right ) a^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+x \sqrt {b \,x^{2}+a}\, \left (\frac {8 \left (\frac {1}{2} D x^{6}+\frac {3}{5} C \,x^{4}+\frac {3}{4} x^{2} B +A \right ) x^{4} b^{\frac {11}{2}}}{3}+\left (a \left (\frac {1}{20} D x^{6}+\frac {1}{10} C \,x^{4}+\frac {1}{4} x^{2} B +A \right ) b^{\frac {7}{2}}+\left (\frac {26}{15} D x^{8}+\frac {11}{5} C \,x^{6}+3 B \,x^{4}+\frac {14}{3} A \,x^{2}\right ) b^{\frac {9}{2}}-\frac {7 a^{2} \left (\left (\frac {8}{15} D x^{4}+\frac {8}{7} C \,x^{2}+\frac {24}{7} B \right ) b^{\frac {5}{2}}+\left (\left (-\frac {2 D x^{2}}{3}-\frac {12 C}{7}\right ) b^{\frac {3}{2}}+D a \sqrt {b}\right ) a \right )}{64}\right ) a \right )}{16 b^{\frac {9}{2}}}\) | \(201\) |
| default | \(A \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+B \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+C \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )+D \left (\frac {x^{7} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{12 b}-\frac {7 a \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )}{12 b}\right )\) | \(446\) |
Input:
int(x^2*(b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/16*(-(b^3*A-3/8*a*b^2*B+3/16*a^2*b*C-7/64*a^3*D)*a^3*arctanh((b*x^2+a)^( 1/2)/x/b^(1/2))+x*(b*x^2+a)^(1/2)*(8/3*(1/2*D*x^6+3/5*C*x^4+3/4*x^2*B+A)*x ^4*b^(11/2)+(a*(1/20*D*x^6+1/10*C*x^4+1/4*x^2*B+A)*b^(7/2)+(26/15*D*x^8+11 /5*C*x^6+3*B*x^4+14/3*A*x^2)*b^(9/2)-7/64*a^2*((8/15*D*x^4+8/7*C*x^2+24/7* B)*b^(5/2)+((-2/3*D*x^2-12/7*C)*b^(3/2)+D*a*b^(1/2))*a))*a))/b^(9/2)
Time = 0.21 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.67 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\left [-\frac {15 \, {\left (7 \, D a^{6} - 12 \, C a^{5} b + 24 \, B a^{4} b^{2} - 64 \, A a^{3} b^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (1280 \, D b^{6} x^{11} + 128 \, {\left (13 \, D a b^{5} + 12 \, C b^{6}\right )} x^{9} + 48 \, {\left (D a^{2} b^{4} + 44 \, C a b^{5} + 40 \, B b^{6}\right )} x^{7} - 8 \, {\left (7 \, D a^{3} b^{3} - 12 \, C a^{2} b^{4} - 360 \, B a b^{5} - 320 \, A b^{6}\right )} x^{5} + 10 \, {\left (7 \, D a^{4} b^{2} - 12 \, C a^{3} b^{3} + 24 \, B a^{2} b^{4} + 448 \, A a b^{5}\right )} x^{3} - 15 \, {\left (7 \, D a^{5} b - 12 \, C a^{4} b^{2} + 24 \, B a^{3} b^{3} - 64 \, A a^{2} b^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{30720 \, b^{5}}, -\frac {15 \, {\left (7 \, D a^{6} - 12 \, C a^{5} b + 24 \, B a^{4} b^{2} - 64 \, A a^{3} b^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (1280 \, D b^{6} x^{11} + 128 \, {\left (13 \, D a b^{5} + 12 \, C b^{6}\right )} x^{9} + 48 \, {\left (D a^{2} b^{4} + 44 \, C a b^{5} + 40 \, B b^{6}\right )} x^{7} - 8 \, {\left (7 \, D a^{3} b^{3} - 12 \, C a^{2} b^{4} - 360 \, B a b^{5} - 320 \, A b^{6}\right )} x^{5} + 10 \, {\left (7 \, D a^{4} b^{2} - 12 \, C a^{3} b^{3} + 24 \, B a^{2} b^{4} + 448 \, A a b^{5}\right )} x^{3} - 15 \, {\left (7 \, D a^{5} b - 12 \, C a^{4} b^{2} + 24 \, B a^{3} b^{3} - 64 \, A a^{2} b^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{15360 \, b^{5}}\right ] \] Input:
integrate(x^2*(b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A),x, algorithm="fricas")
Output:
[-1/30720*(15*(7*D*a^6 - 12*C*a^5*b + 24*B*a^4*b^2 - 64*A*a^3*b^3)*sqrt(b) *log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(1280*D*b^6*x^11 + 12 8*(13*D*a*b^5 + 12*C*b^6)*x^9 + 48*(D*a^2*b^4 + 44*C*a*b^5 + 40*B*b^6)*x^7 - 8*(7*D*a^3*b^3 - 12*C*a^2*b^4 - 360*B*a*b^5 - 320*A*b^6)*x^5 + 10*(7*D* a^4*b^2 - 12*C*a^3*b^3 + 24*B*a^2*b^4 + 448*A*a*b^5)*x^3 - 15*(7*D*a^5*b - 12*C*a^4*b^2 + 24*B*a^3*b^3 - 64*A*a^2*b^4)*x)*sqrt(b*x^2 + a))/b^5, -1/1 5360*(15*(7*D*a^6 - 12*C*a^5*b + 24*B*a^4*b^2 - 64*A*a^3*b^3)*sqrt(-b)*arc tan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (1280*D*b^6*x^11 + 128*(13*D*a*b^5 + 12* C*b^6)*x^9 + 48*(D*a^2*b^4 + 44*C*a*b^5 + 40*B*b^6)*x^7 - 8*(7*D*a^3*b^3 - 12*C*a^2*b^4 - 360*B*a*b^5 - 320*A*b^6)*x^5 + 10*(7*D*a^4*b^2 - 12*C*a^3* b^3 + 24*B*a^2*b^4 + 448*A*a*b^5)*x^3 - 15*(7*D*a^5*b - 12*C*a^4*b^2 + 24* B*a^3*b^3 - 64*A*a^2*b^4)*x)*sqrt(b*x^2 + a))/b^5]
Time = 0.70 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.70 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\begin {cases} - \frac {a \left (A a^{2} - \frac {3 a \left (2 A a b + B a^{2} - \frac {5 a \left (A b^{2} + 2 B a b + C a^{2} - \frac {7 a \left (B b^{2} + 2 C a b + D a^{2} - \frac {9 a \left (C b^{2} + \frac {13 D a b}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b}\right )}{4 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a + b x^{2}} \left (\frac {D b x^{11}}{12} + \frac {x^{9} \left (C b^{2} + \frac {13 D a b}{12}\right )}{10 b} + \frac {x^{7} \left (B b^{2} + 2 C a b + D a^{2} - \frac {9 a \left (C b^{2} + \frac {13 D a b}{12}\right )}{10 b}\right )}{8 b} + \frac {x^{5} \left (A b^{2} + 2 B a b + C a^{2} - \frac {7 a \left (B b^{2} + 2 C a b + D a^{2} - \frac {9 a \left (C b^{2} + \frac {13 D a b}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b} + \frac {x^{3} \cdot \left (2 A a b + B a^{2} - \frac {5 a \left (A b^{2} + 2 B a b + C a^{2} - \frac {7 a \left (B b^{2} + 2 C a b + D a^{2} - \frac {9 a \left (C b^{2} + \frac {13 D a b}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b}\right )}{4 b} + \frac {x \left (A a^{2} - \frac {3 a \left (2 A a b + B a^{2} - \frac {5 a \left (A b^{2} + 2 B a b + C a^{2} - \frac {7 a \left (B b^{2} + 2 C a b + D a^{2} - \frac {9 a \left (C b^{2} + \frac {13 D a b}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{5}}{5} + \frac {C x^{7}}{7} + \frac {D x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(b*x**2+a)**(3/2)*(D*x**6+C*x**4+B*x**2+A),x)
Output:
Piecewise((-a*(A*a**2 - 3*a*(2*A*a*b + B*a**2 - 5*a*(A*b**2 + 2*B*a*b + C* a**2 - 7*a*(B*b**2 + 2*C*a*b + D*a**2 - 9*a*(C*b**2 + 13*D*a*b/12)/(10*b)) /(8*b))/(6*b))/(4*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/s qrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/(2*b) + sqrt(a + b*x**2) *(D*b*x**11/12 + x**9*(C*b**2 + 13*D*a*b/12)/(10*b) + x**7*(B*b**2 + 2*C*a *b + D*a**2 - 9*a*(C*b**2 + 13*D*a*b/12)/(10*b))/(8*b) + x**5*(A*b**2 + 2* B*a*b + C*a**2 - 7*a*(B*b**2 + 2*C*a*b + D*a**2 - 9*a*(C*b**2 + 13*D*a*b/1 2)/(10*b))/(8*b))/(6*b) + x**3*(2*A*a*b + B*a**2 - 5*a*(A*b**2 + 2*B*a*b + C*a**2 - 7*a*(B*b**2 + 2*C*a*b + D*a**2 - 9*a*(C*b**2 + 13*D*a*b/12)/(10* b))/(8*b))/(6*b))/(4*b) + x*(A*a**2 - 3*a*(2*A*a*b + B*a**2 - 5*a*(A*b**2 + 2*B*a*b + C*a**2 - 7*a*(B*b**2 + 2*C*a*b + D*a**2 - 9*a*(C*b**2 + 13*D*a *b/12)/(10*b))/(8*b))/(6*b))/(4*b))/(2*b)), Ne(b, 0)), (a**(3/2)*(A*x**3/3 + B*x**5/5 + C*x**7/7 + D*x**9/9), True))
Time = 0.04 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.37 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D x^{7}}{12 \, b} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a x^{5}}{120 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C x^{5}}{10 \, b} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a^{2} x^{3}}{192 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C a x^{3}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{3}}{8 \, b} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a^{3} x}{384 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} D a^{4} x}{1536 \, b^{4}} + \frac {7 \, \sqrt {b x^{2} + a} D a^{5} x}{1024 \, b^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C a^{2} x}{32 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C a^{3} x}{128 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a} C a^{4} x}{256 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{3} x}{128 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} A a^{2} x}{16 \, b} + \frac {7 \, D a^{6} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{1024 \, b^{\frac {9}{2}}} - \frac {3 \, C a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {7}{2}}} + \frac {3 \, B a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {A a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} \] Input:
integrate(x^2*(b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A),x, algorithm="maxima")
Output:
1/12*(b*x^2 + a)^(5/2)*D*x^7/b - 7/120*(b*x^2 + a)^(5/2)*D*a*x^5/b^2 + 1/1 0*(b*x^2 + a)^(5/2)*C*x^5/b + 7/192*(b*x^2 + a)^(5/2)*D*a^2*x^3/b^3 - 1/16 *(b*x^2 + a)^(5/2)*C*a*x^3/b^2 + 1/8*(b*x^2 + a)^(5/2)*B*x^3/b - 7/384*(b* x^2 + a)^(5/2)*D*a^3*x/b^4 + 7/1536*(b*x^2 + a)^(3/2)*D*a^4*x/b^4 + 7/1024 *sqrt(b*x^2 + a)*D*a^5*x/b^4 + 1/32*(b*x^2 + a)^(5/2)*C*a^2*x/b^3 - 1/128* (b*x^2 + a)^(3/2)*C*a^3*x/b^3 - 3/256*sqrt(b*x^2 + a)*C*a^4*x/b^3 - 1/16*( b*x^2 + a)^(5/2)*B*a*x/b^2 + 1/64*(b*x^2 + a)^(3/2)*B*a^2*x/b^2 + 3/128*sq rt(b*x^2 + a)*B*a^3*x/b^2 + 1/6*(b*x^2 + a)^(5/2)*A*x/b - 1/24*(b*x^2 + a) ^(3/2)*A*a*x/b - 1/16*sqrt(b*x^2 + a)*A*a^2*x/b + 7/1024*D*a^6*arcsinh(b*x /sqrt(a*b))/b^(9/2) - 3/256*C*a^5*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/128*B *a^4*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 1/16*A*a^3*arcsinh(b*x/sqrt(a*b))/b^ (3/2)
Time = 0.14 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.89 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, D b x^{2} + \frac {13 \, D a b^{10} + 12 \, C b^{11}}{b^{10}}\right )} x^{2} + \frac {3 \, {\left (D a^{2} b^{9} + 44 \, C a b^{10} + 40 \, B b^{11}\right )}}{b^{10}}\right )} x^{2} - \frac {7 \, D a^{3} b^{8} - 12 \, C a^{2} b^{9} - 360 \, B a b^{10} - 320 \, A b^{11}}{b^{10}}\right )} x^{2} + \frac {5 \, {\left (7 \, D a^{4} b^{7} - 12 \, C a^{3} b^{8} + 24 \, B a^{2} b^{9} + 448 \, A a b^{10}\right )}}{b^{10}}\right )} x^{2} - \frac {15 \, {\left (7 \, D a^{5} b^{6} - 12 \, C a^{4} b^{7} + 24 \, B a^{3} b^{8} - 64 \, A a^{2} b^{9}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (7 \, D a^{6} - 12 \, C a^{5} b + 24 \, B a^{4} b^{2} - 64 \, A a^{3} b^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {9}{2}}} \] Input:
integrate(x^2*(b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A),x, algorithm="giac")
Output:
1/15360*(2*(4*(2*(8*(10*D*b*x^2 + (13*D*a*b^10 + 12*C*b^11)/b^10)*x^2 + 3* (D*a^2*b^9 + 44*C*a*b^10 + 40*B*b^11)/b^10)*x^2 - (7*D*a^3*b^8 - 12*C*a^2* b^9 - 360*B*a*b^10 - 320*A*b^11)/b^10)*x^2 + 5*(7*D*a^4*b^7 - 12*C*a^3*b^8 + 24*B*a^2*b^9 + 448*A*a*b^10)/b^10)*x^2 - 15*(7*D*a^5*b^6 - 12*C*a^4*b^7 + 24*B*a^3*b^8 - 64*A*a^2*b^9)/b^10)*sqrt(b*x^2 + a)*x - 1/1024*(7*D*a^6 - 12*C*a^5*b + 24*B*a^4*b^2 - 64*A*a^3*b^3)*log(abs(-sqrt(b)*x + sqrt(b*x^ 2 + a)))/b^(9/2)
Timed out. \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\int x^2\,{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x^2+C\,x^4+x^6\,D\right ) \,d x \] Input:
int(x^2*(a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + x^6*D),x)
Output:
int(x^2*(a + b*x^2)^(3/2)*(A + B*x^2 + C*x^4 + x^6*D), x)
Time = 0.19 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.22 \[ \int x^2 \left (a+b x^2\right )^{3/2} \left (A+B x^2+C x^4+D x^6\right ) \, dx=\frac {-105 \sqrt {b \,x^{2}+a}\, a^{5} b d x +180 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} c x +70 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} d \,x^{3}+600 \sqrt {b \,x^{2}+a}\, a^{3} b^{4} x -120 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} c \,x^{3}-56 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} d \,x^{5}+4720 \sqrt {b \,x^{2}+a}\, a^{2} b^{5} x^{3}+96 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} c \,x^{5}+48 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} d \,x^{7}+5440 \sqrt {b \,x^{2}+a}\, a \,b^{6} x^{5}+2112 \sqrt {b \,x^{2}+a}\, a \,b^{5} c \,x^{7}+1664 \sqrt {b \,x^{2}+a}\, a \,b^{5} d \,x^{9}+1920 \sqrt {b \,x^{2}+a}\, b^{7} x^{7}+1536 \sqrt {b \,x^{2}+a}\, b^{6} c \,x^{9}+1280 \sqrt {b \,x^{2}+a}\, b^{6} d \,x^{11}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{6} d -180 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} b c -600 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b^{3}}{15360 b^{5}} \] Input:
int(x^2*(b*x^2+a)^(3/2)*(D*x^6+C*x^4+B*x^2+A),x)
Output:
( - 105*sqrt(a + b*x**2)*a**5*b*d*x + 180*sqrt(a + b*x**2)*a**4*b**2*c*x + 70*sqrt(a + b*x**2)*a**4*b**2*d*x**3 + 600*sqrt(a + b*x**2)*a**3*b**4*x - 120*sqrt(a + b*x**2)*a**3*b**3*c*x**3 - 56*sqrt(a + b*x**2)*a**3*b**3*d*x **5 + 4720*sqrt(a + b*x**2)*a**2*b**5*x**3 + 96*sqrt(a + b*x**2)*a**2*b**4 *c*x**5 + 48*sqrt(a + b*x**2)*a**2*b**4*d*x**7 + 5440*sqrt(a + b*x**2)*a*b **6*x**5 + 2112*sqrt(a + b*x**2)*a*b**5*c*x**7 + 1664*sqrt(a + b*x**2)*a*b **5*d*x**9 + 1920*sqrt(a + b*x**2)*b**7*x**7 + 1536*sqrt(a + b*x**2)*b**6* c*x**9 + 1280*sqrt(a + b*x**2)*b**6*d*x**11 + 105*sqrt(b)*log((sqrt(a + b* x**2) + sqrt(b)*x)/sqrt(a))*a**6*d - 180*sqrt(b)*log((sqrt(a + b*x**2) + s qrt(b)*x)/sqrt(a))*a**5*b*c - 600*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)* x)/sqrt(a))*a**4*b**3)/(15360*b**5)