Integrand size = 30, antiderivative size = 243 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx=\frac {3 b (b B c+A b d+4 a C d) \sqrt {a+b x^2}}{8 a}-\frac {b (c C+B d) \sqrt {a+b x^2}}{x}-\frac {(c C+B d) \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {(b B c+A b d+4 a C d) \left (a+b x^2\right )^{3/2}}{8 a x^2}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}-\frac {(B c+A d) \left (a+b x^2\right )^{5/2}}{4 a x^4}+b^{3/2} (c C+B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {3 b (b B c+A b d+4 a C d) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}} \] Output:
3/8*b*(A*b*d+B*b*c+4*C*a*d)*(b*x^2+a)^(1/2)/a-b*(B*d+C*c)*(b*x^2+a)^(1/2)/ x-1/3*(B*d+C*c)*(b*x^2+a)^(3/2)/x^3-1/8*(A*b*d+B*b*c+4*C*a*d)*(b*x^2+a)^(3 /2)/a/x^2-1/5*A*c*(b*x^2+a)^(5/2)/a/x^5-1/4*(A*d+B*c)*(b*x^2+a)^(5/2)/a/x^ 4+b^(3/2)*(B*d+C*c)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))-3/8*b*(A*b*d+B*b*c+ 4*C*a*d)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)
Time = 2.06 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx=-\frac {\sqrt {a+b x^2} \left (24 A b^2 c x^4+a^2 \left (6 A (4 c+5 d x)+10 x \left (3 B c+4 c C x+4 B d x+6 C d x^2\right )\right )+a b x^2 (A (48 c+75 d x)+5 x (8 C x (4 c-3 d x)+B (15 c+32 d x)))\right )}{120 a x^5}+3 \sqrt {a} b C d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {3 b^2 (B c+A d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}}-b^{3/2} (c C+B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:
Integrate[((c + d*x)*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^6,x]
Output:
-1/120*(Sqrt[a + b*x^2]*(24*A*b^2*c*x^4 + a^2*(6*A*(4*c + 5*d*x) + 10*x*(3 *B*c + 4*c*C*x + 4*B*d*x + 6*C*d*x^2)) + a*b*x^2*(A*(48*c + 75*d*x) + 5*x* (8*C*x*(4*c - 3*d*x) + B*(15*c + 32*d*x)))))/(a*x^5) + 3*Sqrt[a]*b*C*d*Arc Tanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] - (3*b^2*(B*c + A*d)*ArcTanh[( -(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/(4*Sqrt[a]) - b^(3/2)*(c*C + B*d )*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
Time = 1.05 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2338, 27, 2338, 25, 27, 537, 25, 536, 538, 224, 219, 243, 73, 221}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} (c+d x) \left (A+B x+C x^2\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {\int -\frac {5 \left (b x^2+a\right )^{3/2} \left (a C d x^2+a (c C+B d) x+a (B c+A d)\right )}{x^5}dx}{5 a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (a C d x^2+a (c C+B d) x+a (B c+A d)\right )}{x^5}dx}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {a (4 a (c C+B d)+(b B c+A b d+4 a C d) x) \left (b x^2+a\right )^{3/2}}{x^4}dx}{4 a}-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a (4 a (c C+B d)+(b B c+A b d+4 a C d) x) \left (b x^2+a\right )^{3/2}}{x^4}dx}{4 a}-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {(4 a (c C+B d)+(b B c+A b d+4 a C d) x) \left (b x^2+a\right )^{3/2}}{x^4}dx-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {\frac {1}{4} \left (-\frac {1}{2} b \int -\frac {(8 a (c C+B d)+3 (b B c+A b d+4 a C d) x) \sqrt {b x^2+a}}{x^2}dx-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \int \frac {(8 a (c C+B d)+3 (b B c+A b d+4 a C d) x) \sqrt {b x^2+a}}{x^2}dx-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (\int \frac {3 a (b B c+A b d+4 a C d)+8 a b (c C+B d) x}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (3 a (4 a C d+A b d+b B c) \int \frac {1}{x \sqrt {b x^2+a}}dx+8 a b (B d+c C) \int \frac {1}{\sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (3 a (4 a C d+A b d+b B c) \int \frac {1}{x \sqrt {b x^2+a}}dx+8 a b (B d+c C) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (3 a (4 a C d+A b d+b B c) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}+8 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (B d+c C)\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (\frac {3}{2} a (4 a C d+A b d+b B c) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}+8 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (B d+c C)\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (\frac {3 a (4 a C d+A b d+b B c) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}+8 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (B d+c C)\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} b \left (-3 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (4 a C d+A b d+b B c)-\frac {\sqrt {a+b x^2} (8 a (B d+c C)-3 x (4 a C d+A b d+b B c))}{x}+8 a \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (B d+c C)\right )-\frac {\left (a+b x^2\right )^{3/2} (3 x (4 a C d+A b d+b B c)+8 a (B d+c C))}{6 x^3}\right )-\frac {\left (a+b x^2\right )^{5/2} (A d+B c)}{4 x^4}}{a}-\frac {A c \left (a+b x^2\right )^{5/2}}{5 a x^5}\) |
Input:
Int[((c + d*x)*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^6,x]
Output:
-1/5*(A*c*(a + b*x^2)^(5/2))/(a*x^5) + (-1/4*((B*c + A*d)*(a + b*x^2)^(5/2 ))/x^4 + (-1/6*((8*a*(c*C + B*d) + 3*(b*B*c + A*b*d + 4*a*C*d)*x)*(a + b*x ^2)^(3/2))/x^3 + (b*(-(((8*a*(c*C + B*d) - 3*(b*B*c + A*b*d + 4*a*C*d)*x)* Sqrt[a + b*x^2])/x) + 8*a*Sqrt[b]*(c*C + B*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] - 3*Sqrt[a]*(b*B*c + A*b*d + 4*a*C*d)*ArcTanh[Sqrt[a + b*x^2]/Sqr t[a]]))/2)/4)/a
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.37 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (24 A \,b^{2} c \,x^{4}+160 B a b d \,x^{4}+160 C a b c \,x^{4}+75 A b d \,x^{3} a +75 B b c \,x^{3} a +60 C \,a^{2} d \,x^{3}+48 A a b c \,x^{2}+40 B \,a^{2} d \,x^{2}+40 C \,a^{2} c \,x^{2}+30 A \,a^{2} d x +30 B \,a^{2} c x +24 A \,a^{2} c \right )}{120 x^{5} a}+\frac {b \left (-\frac {\left (3 A b d +3 B b c +12 a C d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+8 B \sqrt {b}\, d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+8 C \sqrt {b}\, c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+8 d C \sqrt {b \,x^{2}+a}\right )}{8}\) | \(235\) |
| default | \(\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )-\frac {A c \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}+d C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )\) | \(315\) |
Input:
int((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/120*(b*x^2+a)^(1/2)*(24*A*b^2*c*x^4+160*B*a*b*d*x^4+160*C*a*b*c*x^4+75* A*a*b*d*x^3+75*B*a*b*c*x^3+60*C*a^2*d*x^3+48*A*a*b*c*x^2+40*B*a^2*d*x^2+40 *C*a^2*c*x^2+30*A*a^2*d*x+30*B*a^2*c*x+24*A*a^2*c)/x^5/a+1/8*b*(-(3*A*b*d+ 3*B*b*c+12*C*a*d)/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+8*B*b^(1/2 )*d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+8*C*b^(1/2)*c*ln(b^(1/2)*x+(b*x^2+a)^(1/ 2))+8*d*C*(b*x^2+a)^(1/2))
Time = 0.22 (sec) , antiderivative size = 906, normalized size of antiderivative = 3.73 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6,x, algorithm="fricas")
Output:
[1/240*(120*(C*a*b*c + B*a*b*d)*sqrt(b)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 45*(B*b^2*c + (4*C*a*b + A*b^2)*d)*sqrt(a)*x^5*log(-(b *x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(120*C*a*b*d*x^5 - 8*(20* B*a*b*d + (20*C*a*b + 3*A*b^2)*c)*x^4 - 24*A*a^2*c - 15*(5*B*a*b*c + (4*C* a^2 + 5*A*a*b)*d)*x^3 - 8*(5*B*a^2*d + (5*C*a^2 + 6*A*a*b)*c)*x^2 - 30*(B* a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + a))/(a*x^5), -1/240*(240*(C*a*b*c + B*a*b *d)*sqrt(-b)*x^5*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 45*(B*b^2*c + (4*C*a *b + A*b^2)*d)*sqrt(a)*x^5*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/ x^2) - 2*(120*C*a*b*d*x^5 - 8*(20*B*a*b*d + (20*C*a*b + 3*A*b^2)*c)*x^4 - 24*A*a^2*c - 15*(5*B*a*b*c + (4*C*a^2 + 5*A*a*b)*d)*x^3 - 8*(5*B*a^2*d + ( 5*C*a^2 + 6*A*a*b)*c)*x^2 - 30*(B*a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + a))/(a* x^5), 1/120*(45*(B*b^2*c + (4*C*a*b + A*b^2)*d)*sqrt(-a)*x^5*arctan(sqrt(b *x^2 + a)*sqrt(-a)/a) + 60*(C*a*b*c + B*a*b*d)*sqrt(b)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (120*C*a*b*d*x^5 - 8*(20*B*a*b*d + (20* C*a*b + 3*A*b^2)*c)*x^4 - 24*A*a^2*c - 15*(5*B*a*b*c + (4*C*a^2 + 5*A*a*b) *d)*x^3 - 8*(5*B*a^2*d + (5*C*a^2 + 6*A*a*b)*c)*x^2 - 30*(B*a^2*c + A*a^2* d)*x)*sqrt(b*x^2 + a))/(a*x^5), -1/120*(120*(C*a*b*c + B*a*b*d)*sqrt(-b)*x ^5*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 45*(B*b^2*c + (4*C*a*b + A*b^2)*d) *sqrt(-a)*x^5*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (120*C*a*b*d*x^5 - 8*(2 0*B*a*b*d + (20*C*a*b + 3*A*b^2)*c)*x^4 - 24*A*a^2*c - 15*(5*B*a*b*c + ...
Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (233) = 466\).
Time = 9.79 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.82 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx=- \frac {A a^{2} d}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A a \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {3 A a \sqrt {b} d}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 A b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {A b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {A b^{\frac {3}{2}} d}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {3 A b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} - \frac {B \sqrt {a} b d}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{2} c}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B a \sqrt {b} c}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B a \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {B b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {B b^{\frac {3}{2}} c}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{2}} + 1}}{3} + B b^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {3 B b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} - \frac {B b^{2} d x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {C \sqrt {a} b c}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 C \sqrt {a} b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {C a \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {C a \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {C a \sqrt {b} d}{x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {C b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{3} + C b^{\frac {3}{2}} c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + \frac {C b^{\frac {3}{2}} d x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {C b^{2} c x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:
integrate((d*x+c)*(b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**6,x)
Output:
-A*a**2*d/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - A*a*sqrt(b)*c*sqrt(a/(b* x**2) + 1)/(5*x**4) - 3*A*a*sqrt(b)*d/(8*x**3*sqrt(a/(b*x**2) + 1)) - 2*A* b**(3/2)*c*sqrt(a/(b*x**2) + 1)/(5*x**2) - A*b**(3/2)*d*sqrt(a/(b*x**2) + 1)/(2*x) - A*b**(3/2)*d/(8*x*sqrt(a/(b*x**2) + 1)) - A*b**(5/2)*c*sqrt(a/( b*x**2) + 1)/(5*a) - 3*A*b**2*d*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a)) - B *sqrt(a)*b*d/(x*sqrt(1 + b*x**2/a)) - B*a**2*c/(4*sqrt(b)*x**5*sqrt(a/(b*x **2) + 1)) - 3*B*a*sqrt(b)*c/(8*x**3*sqrt(a/(b*x**2) + 1)) - B*a*sqrt(b)*d *sqrt(a/(b*x**2) + 1)/(3*x**2) - B*b**(3/2)*c*sqrt(a/(b*x**2) + 1)/(2*x) - B*b**(3/2)*c/(8*x*sqrt(a/(b*x**2) + 1)) - B*b**(3/2)*d*sqrt(a/(b*x**2) + 1)/3 + B*b**(3/2)*d*asinh(sqrt(b)*x/sqrt(a)) - 3*B*b**2*c*asinh(sqrt(a)/(s qrt(b)*x))/(8*sqrt(a)) - B*b**2*d*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - C*sqrt( a)*b*c/(x*sqrt(1 + b*x**2/a)) - 3*C*sqrt(a)*b*d*asinh(sqrt(a)/(sqrt(b)*x)) /2 - C*a*sqrt(b)*c*sqrt(a/(b*x**2) + 1)/(3*x**2) - C*a*sqrt(b)*d*sqrt(a/(b *x**2) + 1)/(2*x) + C*a*sqrt(b)*d/(x*sqrt(a/(b*x**2) + 1)) - C*b**(3/2)*c* sqrt(a/(b*x**2) + 1)/3 + C*b**(3/2)*c*asinh(sqrt(b)*x/sqrt(a)) + C*b**(3/2 )*d*x/sqrt(a/(b*x**2) + 1) - C*b**2*c*x/(sqrt(a)*sqrt(1 + b*x**2/a))
Time = 0.04 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx=-\frac {3}{2} \, C \sqrt {a} b d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {3}{2} \, \sqrt {b x^{2} + a} C b d + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C b d}{2 \, a} + \frac {\sqrt {b x^{2} + a} {\left (C c + B d\right )} b^{2} x}{a} + {\left (C c + B d\right )} b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {3 \, {\left (B c + A d\right )} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} {\left (B c + A d\right )} b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d}{2 \, a x^{2}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )}}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )}}{4 \, a x^{4}} \] Input:
integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6,x, algorithm="maxima")
Output:
-3/2*C*sqrt(a)*b*d*arcsinh(a/(sqrt(a*b)*abs(x))) + 3/2*sqrt(b*x^2 + a)*C*b *d + 1/2*(b*x^2 + a)^(3/2)*C*b*d/a + sqrt(b*x^2 + a)*(C*c + B*d)*b^2*x/a + (C*c + B*d)*b^(3/2)*arcsinh(b*x/sqrt(a*b)) - 3/8*(B*c + A*d)*b^2*arcsinh( a/(sqrt(a*b)*abs(x)))/sqrt(a) + 1/8*(b*x^2 + a)^(3/2)*(B*c + A*d)*b^2/a^2 + 3/8*sqrt(b*x^2 + a)*(B*c + A*d)*b^2/a - 1/2*(b*x^2 + a)^(5/2)*C*d/(a*x^2 ) - 2/3*(b*x^2 + a)^(3/2)*(C*c + B*d)*b/(a*x) - 1/8*(b*x^2 + a)^(5/2)*(B*c + A*d)*b/(a^2*x^2) - 1/3*(b*x^2 + a)^(5/2)*(C*c + B*d)/(a*x^3) - 1/5*(b*x ^2 + a)^(5/2)*A*c/(a*x^5) - 1/4*(b*x^2 + a)^(5/2)*(B*c + A*d)/(a*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (207) = 414\).
Time = 0.23 (sec) , antiderivative size = 761, normalized size of antiderivative = 3.13 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6,x, algorithm="giac")
Output:
sqrt(b*x^2 + a)*C*b*d - (C*b^(3/2)*c + B*b^(3/2)*d)*log(abs(-sqrt(b)*x + s qrt(b*x^2 + a))) + 3/4*(B*b^2*c + 4*C*a*b*d + A*b^2*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) + 1/60*(75*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*b^2*c + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9*C*a*b*d + 75*(sqrt(b)* x - sqrt(b*x^2 + a))^9*A*b^2*d + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a*b ^(3/2)*c + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2)*c + 240*(sqrt(b)* x - sqrt(b*x^2 + a))^8*B*a*b^(3/2)*d - 30*(sqrt(b)*x - sqrt(b*x^2 + a))^7* B*a*b^2*c - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^2*b*d - 30*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a*b^2*d - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^2 *b^(3/2)*c - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2)*d + 880*(sq rt(b)*x - sqrt(b*x^2 + a))^4*C*a^3*b^(3/2)*c + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2)*c + 880*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/ 2)*d + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a^3*b^2*c + 120*(sqrt(b)*x - s qrt(b*x^2 + a))^3*C*a^4*b*d + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^3*b^2 *d - 560*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^4*b^(3/2)*c - 560*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2)*d - 75*(sqrt(b)*x - sqrt(b*x^2 + a))*B* a^4*b^2*c - 60*(sqrt(b)*x - sqrt(b*x^2 + a))*C*a^5*b*d - 75*(sqrt(b)*x - s qrt(b*x^2 + a))*A*a^4*b^2*d + 160*C*a^5*b^(3/2)*c + 24*A*a^4*b^(5/2)*c + 1 60*B*a^5*b^(3/2)*d)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5
Timed out. \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^6} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2))/x^6,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x)*(A + B*x + C*x^2))/x^6, x)
Time = 0.23 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.19 \[ \int \frac {(c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6} \, dx=\frac {-24 \sqrt {b \,x^{2}+a}\, a^{3} c -30 \sqrt {b \,x^{2}+a}\, a^{3} d x -48 \sqrt {b \,x^{2}+a}\, a^{2} b c \,x^{2}-30 \sqrt {b \,x^{2}+a}\, a^{2} b c x -75 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{3}-40 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{2}-40 \sqrt {b \,x^{2}+a}\, a^{2} c^{2} x^{2}-60 \sqrt {b \,x^{2}+a}\, a^{2} c d \,x^{3}-24 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{4}-75 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{3}-160 \sqrt {b \,x^{2}+a}\, a \,b^{2} d \,x^{4}-160 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{4}+120 \sqrt {b \,x^{2}+a}\, a b c d \,x^{5}+45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{5}+180 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{5}+45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{5}-180 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{5}+120 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{5}+120 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} x^{5}-24 \sqrt {b}\, a \,b^{2} c \,x^{5}+64 \sqrt {b}\, a \,b^{2} d \,x^{5}+64 \sqrt {b}\, a b \,c^{2} x^{5}}{120 a \,x^{5}} \] Input:
int((d*x+c)*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6,x)
Output:
( - 24*sqrt(a + b*x**2)*a**3*c - 30*sqrt(a + b*x**2)*a**3*d*x - 48*sqrt(a + b*x**2)*a**2*b*c*x**2 - 30*sqrt(a + b*x**2)*a**2*b*c*x - 75*sqrt(a + b*x **2)*a**2*b*d*x**3 - 40*sqrt(a + b*x**2)*a**2*b*d*x**2 - 40*sqrt(a + b*x** 2)*a**2*c**2*x**2 - 60*sqrt(a + b*x**2)*a**2*c*d*x**3 - 24*sqrt(a + b*x**2 )*a*b**2*c*x**4 - 75*sqrt(a + b*x**2)*a*b**2*c*x**3 - 160*sqrt(a + b*x**2) *a*b**2*d*x**4 - 160*sqrt(a + b*x**2)*a*b*c**2*x**4 + 120*sqrt(a + b*x**2) *a*b*c*d*x**5 + 45*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sq rt(a))*a*b**2*d*x**5 + 180*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt( b)*x)/sqrt(a))*a*b*c*d*x**5 + 45*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c*x**5 - 45*sqrt(a)*log((sqrt(a + b*x**2) + sqrt (a) + sqrt(b)*x)/sqrt(a))*a*b**2*d*x**5 - 180*sqrt(a)*log((sqrt(a + b*x**2 ) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*d*x**5 - 45*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c*x**5 + 120*sqrt(b)*log((sqr t(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2*d*x**5 + 120*sqrt(b)*log((sqrt( a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c**2*x**5 - 24*sqrt(b)*a*b**2*c*x**5 + 64*sqrt(b)*a*b**2*d*x**5 + 64*sqrt(b)*a*b*c**2*x**5)/(120*a*x**5)