Integrand size = 24, antiderivative size = 179 \[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}+\frac {5 c^2 (6 b B-7 A c) \sqrt {x}}{8 b^4 \sqrt {b x+c x^2}}-\frac {5 c^2 (6 b B-7 A c) \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{9/2}} \]
-5/8*c^2*(-7*A*c+6*B*b)*arctanh((c*x^2+b*x)^(1/2)/b^(1/2)/x^(1/2))/b^(9/2) -1/3*A/b/x^(5/2)/(c*x^2+b*x)^(1/2)+1/12*(7*A*c-6*B*b)/b^2/x^(3/2)/(c*x^2+b *x)^(1/2)+5/24*c*(-7*A*c+6*B*b)/b^3/x^(1/2)/(c*x^2+b*x)^(1/2)+5/8*c^2*(-7* A*c+6*B*b)*x^(1/2)/b^4/(c*x^2+b*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {b} \left (6 b B x \left (-2 b^2+5 b c x+15 c^2 x^2\right )-A \left (8 b^3-14 b^2 c x+35 b c^2 x^2+105 c^3 x^3\right )\right )+15 c^2 (-6 b B+7 A c) x^3 \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{24 b^{9/2} x^{5/2} \sqrt {x (b+c x)}} \]
(Sqrt[b]*(6*b*B*x*(-2*b^2 + 5*b*c*x + 15*c^2*x^2) - A*(8*b^3 - 14*b^2*c*x + 35*b*c^2*x^2 + 105*c^3*x^3)) + 15*c^2*(-6*b*B + 7*A*c)*x^3*Sqrt[b + c*x] *ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(24*b^(9/2)*x^(5/2)*Sqrt[x*(b + c*x)])
Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1220, 1135, 1135, 1132, 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.
| \(\displaystyle \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(6 b B-7 A c) \int \frac {1}{x^{3/2} \left (c x^2+b x\right )^{3/2}}dx}{6 b}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(6 b B-7 A c) \left (-\frac {5 c \int \frac {1}{\sqrt {x} \left (c x^2+b x\right )^{3/2}}dx}{4 b}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}\right )}{6 b}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(6 b B-7 A c) \left (-\frac {5 c \left (-\frac {3 c \int \frac {\sqrt {x}}{\left (c x^2+b x\right )^{3/2}}dx}{2 b}-\frac {1}{b \sqrt {x} \sqrt {b x+c x^2}}\right )}{4 b}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}\right )}{6 b}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(6 b B-7 A c) \left (-\frac {5 c \left (-\frac {3 c \left (\frac {\int \frac {1}{\sqrt {x} \sqrt {c x^2+b x}}dx}{b}+\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}\right )}{2 b}-\frac {1}{b \sqrt {x} \sqrt {b x+c x^2}}\right )}{4 b}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}\right )}{6 b}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(6 b B-7 A c) \left (-\frac {5 c \left (-\frac {3 c \left (\frac {2 \int \frac {1}{\frac {c x^2+b x}{x}-b}d\frac {\sqrt {c x^2+b x}}{\sqrt {x}}}{b}+\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}\right )}{2 b}-\frac {1}{b \sqrt {x} \sqrt {b x+c x^2}}\right )}{4 b}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}\right )}{6 b}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {(6 b B-7 A c) \left (-\frac {5 c \left (-\frac {3 c \left (\frac {2 \sqrt {x}}{b \sqrt {b x+c x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}\right )}{2 b}-\frac {1}{b \sqrt {x} \sqrt {b x+c x^2}}\right )}{4 b}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}\right )}{6 b}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}\) |
-1/3*A/(b*x^(5/2)*Sqrt[b*x + c*x^2]) + ((6*b*B - 7*A*c)*(-1/2*1/(b*x^(3/2) *Sqrt[b*x + c*x^2]) - (5*c*(-(1/(b*Sqrt[x]*Sqrt[b*x + c*x^2])) - (3*c*((2* Sqrt[x])/(b*Sqrt[b*x + c*x^2]) - (2*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqr t[x])])/b^(3/2)))/(2*b)))/(4*b)))/(6*b)
3.3.40.3.1 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[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ [0, m, 1] && IntegerQ[2*p]
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 = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (c x +b \right ) \left (57 A \,c^{2} x^{2}-42 B b c \,x^{2}-22 A b c x +12 b^{2} B x +8 A \,b^{2}\right )}{24 b^{4} x^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}}-\frac {c^{2} \left (-\frac {2 \left (35 A c -30 B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \left (-16 A c +16 B b \right )}{\sqrt {c x +b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{16 b^{4} \sqrt {x \left (c x +b \right )}}\) | \(133\) |
| default | \(\frac {\sqrt {x \left (c x +b \right )}\, \left (105 A \sqrt {c x +b}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{3} x^{3}-90 B \sqrt {c x +b}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) b \,c^{2} x^{3}-12 B \,b^{\frac {7}{2}} x +30 B \,b^{\frac {5}{2}} c \,x^{2}+90 B \,b^{\frac {3}{2}} c^{2} x^{3}-8 A \,b^{\frac {7}{2}}+14 A \,b^{\frac {5}{2}} c x -35 A \,b^{\frac {3}{2}} c^{2} x^{2}-105 A \sqrt {b}\, c^{3} x^{3}\right )}{24 x^{\frac {7}{2}} \left (c x +b \right ) b^{\frac {9}{2}}}\) | \(150\) |
-1/24*(c*x+b)*(57*A*c^2*x^2-42*B*b*c*x^2-22*A*b*c*x+12*B*b^2*x+8*A*b^2)/b^ 4/x^(5/2)/(x*(c*x+b))^(1/2)-1/16*c^2/b^4*(-2*(35*A*c-30*B*b)/b^(1/2)*arcta nh((c*x+b)^(1/2)/b^(1/2))-2*(-16*A*c+16*B*b)/(c*x+b)^(1/2))*(c*x+b)^(1/2)* x^(1/2)/(x*(c*x+b))^(1/2)
Time = 0.29 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (6 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4}\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (8 \, A b^{4} - 15 \, {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 5 \, {\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}, \frac {15 \, {\left ({\left (6 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (8 \, A b^{4} - 15 \, {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 5 \, {\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}\right ] \]
[-1/48*(15*((6*B*b*c^3 - 7*A*c^4)*x^5 + (6*B*b^2*c^2 - 7*A*b*c^3)*x^4)*sqr t(b)*log(-(c*x^2 + 2*b*x + 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 2*( 8*A*b^4 - 15*(6*B*b^2*c^2 - 7*A*b*c^3)*x^3 - 5*(6*B*b^3*c - 7*A*b^2*c^2)*x ^2 + 2*(6*B*b^4 - 7*A*b^3*c)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^5*c*x^5 + b^ 6*x^4), 1/24*(15*((6*B*b*c^3 - 7*A*c^4)*x^5 + (6*B*b^2*c^2 - 7*A*b*c^3)*x^ 4)*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (8*A*b^4 - 15*(6* B*b^2*c^2 - 7*A*b*c^3)*x^3 - 5*(6*B*b^3*c - 7*A*b^2*c^2)*x^2 + 2*(6*B*b^4 - 7*A*b^3*c)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^5*c*x^5 + b^6*x^4)]
\[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{x^{\frac {5}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{\frac {5}{2}}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {5 \, {\left (6 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{4}} + \frac {2 \, {\left (B b c^{2} - A c^{3}\right )}}{\sqrt {c x + b} b^{4}} + \frac {42 \, {\left (c x + b\right )}^{\frac {5}{2}} B b c^{2} - 96 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} c^{2} + 54 \, \sqrt {c x + b} B b^{3} c^{2} - 57 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{3} + 136 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{3} - 87 \, \sqrt {c x + b} A b^{2} c^{3}}{24 \, b^{4} c^{3} x^{3}} \]
5/8*(6*B*b*c^2 - 7*A*c^3)*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^4) + 2*(B*b*c^2 - A*c^3)/(sqrt(c*x + b)*b^4) + 1/24*(42*(c*x + b)^(5/2)*B*b*c^2 - 96*(c*x + b)^(3/2)*B*b^2*c^2 + 54*sqrt(c*x + b)*B*b^3*c^2 - 57*(c*x + b )^(5/2)*A*c^3 + 136*(c*x + b)^(3/2)*A*b*c^3 - 87*sqrt(c*x + b)*A*b^2*c^3)/ (b^4*c^3*x^3)
Timed out. \[ \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x^{5/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]