Integrand size = 24, antiderivative size = 142 \[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}-\frac {(6 b B-5 A c) \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}+\frac {c (6 b B-5 A c) \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}-\frac {c^2 (6 b B-5 A c) \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}} \]
-1/8*c^2*(-5*A*c+6*B*b)*arctanh((c*x^2+b*x)^(1/2)/b^(1/2)/x^(1/2))/b^(7/2) -1/3*A*(c*x^2+b*x)^(1/2)/b/x^(7/2)-1/12*(-5*A*c+6*B*b)*(c*x^2+b*x)^(1/2)/b ^2/x^(5/2)+1/8*c*(-5*A*c+6*B*b)*(c*x^2+b*x)^(1/2)/b^3/x^(3/2)
Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {-\sqrt {b} (b+c x) \left (6 b B x (2 b-3 c x)+A \left (8 b^2-10 b c x+15 c^2 x^2\right )\right )+3 c^2 (-6 b B+5 A c) x^3 \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{24 b^{7/2} x^{5/2} \sqrt {x (b+c x)}} \]
(-(Sqrt[b]*(b + c*x)*(6*b*B*x*(2*b - 3*c*x) + A*(8*b^2 - 10*b*c*x + 15*c^2 *x^2))) + 3*c^2*(-6*b*B + 5*A*c)*x^3*Sqrt[b + c*x]*ArcTanh[Sqrt[b + c*x]/S qrt[b]])/(24*b^(7/2)*x^(5/2)*Sqrt[x*(b + c*x)])
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1220, 1135, 1135, 1136, 220}
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.
| \(\displaystyle \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(6 b B-5 A c) \int \frac {1}{x^{5/2} \sqrt {c x^2+b x}}dx}{6 b}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(6 b B-5 A c) \left (-\frac {3 c \int \frac {1}{x^{3/2} \sqrt {c x^2+b x}}dx}{4 b}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}\right )}{6 b}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(6 b B-5 A c) \left (-\frac {3 c \left (-\frac {c \int \frac {1}{\sqrt {x} \sqrt {c x^2+b x}}dx}{2 b}-\frac {\sqrt {b x+c x^2}}{b x^{3/2}}\right )}{4 b}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}\right )}{6 b}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(6 b B-5 A c) \left (-\frac {3 c \left (-\frac {c \int \frac {1}{\frac {c x^2+b x}{x}-b}d\frac {\sqrt {c x^2+b x}}{\sqrt {x}}}{b}-\frac {\sqrt {b x+c x^2}}{b x^{3/2}}\right )}{4 b}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}\right )}{6 b}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {(6 b B-5 A c) \left (-\frac {3 c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {\sqrt {b x+c x^2}}{b x^{3/2}}\right )}{4 b}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}\right )}{6 b}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}\) |
-1/3*(A*Sqrt[b*x + c*x^2])/(b*x^(7/2)) + ((6*b*B - 5*A*c)*(-1/2*Sqrt[b*x + c*x^2]/(b*x^(5/2)) - (3*c*(-(Sqrt[b*x + c*x^2]/(b*x^(3/2))) + (c*ArcTanh[ Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/b^(3/2)))/(4*b)))/(6*b)
3.3.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))) Int [(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I ntegerQ[2*p]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 0.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(-\frac {\left (c x +b \right ) \left (15 A \,c^{2} x^{2}-18 B b c \,x^{2}-10 A b c x +12 b^{2} B x +8 A \,b^{2}\right )}{24 b^{3} x^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}}+\frac {c^{2} \left (5 A c -6 B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{8 b^{\frac {7}{2}} \sqrt {x \left (c x +b \right )}}\) | \(109\) |
| default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (15 A \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{3} x^{3}-18 B \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) b \,c^{2} x^{3}-15 A \,c^{2} x^{2} \sqrt {c x +b}\, \sqrt {b}+18 B \,b^{\frac {3}{2}} c \,x^{2} \sqrt {c x +b}+10 A \,b^{\frac {3}{2}} c x \sqrt {c x +b}-12 B \,b^{\frac {5}{2}} x \sqrt {c x +b}-8 A \,b^{\frac {5}{2}} \sqrt {c x +b}\right )}{24 b^{\frac {7}{2}} x^{\frac {7}{2}} \sqrt {c x +b}}\) | \(147\) |
-1/24*(c*x+b)*(15*A*c^2*x^2-18*B*b*c*x^2-10*A*b*c*x+12*B*b^2*x+8*A*b^2)/b^ 3/x^(5/2)/(x*(c*x+b))^(1/2)+1/8*c^2*(5*A*c-6*B*b)/b^(7/2)*arctanh((c*x+b)^ (1/2)/b^(1/2))*(c*x+b)^(1/2)*x^(1/2)/(x*(c*x+b))^(1/2)
Time = 0.29 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.70 \[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\left [-\frac {3 \, {\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} \sqrt {b} x^{4} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (8 \, A b^{3} - 3 \, {\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, b^{4} x^{4}}, \frac {3 \, {\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} \sqrt {-b} x^{4} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (8 \, A b^{3} - 3 \, {\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, b^{4} x^{4}}\right ] \]
[-1/48*(3*(6*B*b*c^2 - 5*A*c^3)*sqrt(b)*x^4*log(-(c*x^2 + 2*b*x + 2*sqrt(c *x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 2*(8*A*b^3 - 3*(6*B*b^2*c - 5*A*b*c^2) *x^2 + 2*(6*B*b^3 - 5*A*b^2*c)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^4*x^4), 1/ 24*(3*(6*B*b*c^2 - 5*A*c^3)*sqrt(-b)*x^4*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^ 2 + b*x)) - (8*A*b^3 - 3*(6*B*b^2*c - 5*A*b*c^2)*x^2 + 2*(6*B*b^3 - 5*A*b^ 2*c)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^4*x^4)]
\[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{x^{\frac {7}{2}} \sqrt {x \left (b + c x\right )}}\, dx \]
\[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} x^{\frac {7}{2}}} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {\frac {3 \, {\left (6 \, B b c^{3} - 5 \, A c^{4}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} + \frac {18 \, {\left (c x + b\right )}^{\frac {5}{2}} B b c^{3} - 48 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} c^{3} + 30 \, \sqrt {c x + b} B b^{3} c^{3} - 15 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{4} + 40 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{4} - 33 \, \sqrt {c x + b} A b^{2} c^{4}}{b^{3} c^{3} x^{3}}}{24 \, c} \]
1/24*(3*(6*B*b*c^3 - 5*A*c^4)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3 ) + (18*(c*x + b)^(5/2)*B*b*c^3 - 48*(c*x + b)^(3/2)*B*b^2*c^3 + 30*sqrt(c *x + b)*B*b^3*c^3 - 15*(c*x + b)^(5/2)*A*c^4 + 40*(c*x + b)^(3/2)*A*b*c^4 - 33*sqrt(c*x + b)*A*b^2*c^4)/(b^3*c^3*x^3))/c
Timed out. \[ \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A+B\,x}{x^{7/2}\,\sqrt {c\,x^2+b\,x}} \,d x \]