Integrand size = 31, antiderivative size = 255 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (7 A b-3 a B) (a+b x)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b-3 a B) (a+b x)}{4 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {b} (7 A b-3 a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
1/4*(7*A*b-3*B*a)/a^2/b/x^(3/2)/((b*x+a)^2)^(1/2)+1/2*(A*b-B*a)/a/b/x^(3/2 )/(b*x+a)/((b*x+a)^2)^(1/2)-5/12*(7*A*b-3*B*a)*(b*x+a)/a^3/b/x^(3/2)/((b*x +a)^2)^(1/2)+5/4*(7*A*b-3*B*a)*(b*x+a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))*b^( 1/2)/a^(9/2)/((b*x+a)^2)^(1/2)+5/4*(7*A*b-3*B*a)*(b*x+a)/a^4/x^(1/2)/((b*x +a)^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.51 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {(a+b x) \left (-\frac {\sqrt {a} \left (-105 A b^3 x^3+8 a^3 (A+3 B x)+5 a b^2 x^2 (-35 A+9 B x)+a^2 b x (-56 A+75 B x)\right )}{x^{3/2}}+15 \sqrt {b} (7 A b-3 a B) (a+b x)^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{12 a^{9/2} \left ((a+b x)^2\right )^{3/2}} \]
((a + b*x)*(-((Sqrt[a]*(-105*A*b^3*x^3 + 8*a^3*(A + 3*B*x) + 5*a*b^2*x^2*( -35*A + 9*B*x) + a^2*b*x*(-56*A + 75*B*x)))/x^(3/2)) + 15*Sqrt[b]*(7*A*b - 3*a*B)*(a + b*x)^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(12*a^(9/2)*((a + b*x)^2)^(3/2))
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.62, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {1187, 27, 87, 52, 61, 61, 73, 218}
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.
| \(\displaystyle \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^3 (a+b x) \int \frac {A+B x}{b^3 x^{5/2} (a+b x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {A+B x}{x^{5/2} (a+b x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(7 A b-3 a B) \int \frac {1}{x^{5/2} (a+b x)^2}dx}{4 a b}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(7 A b-3 a B) \left (\frac {5 \int \frac {1}{x^{5/2} (a+b x)}dx}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(7 A b-3 a B) \left (\frac {5 \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(7 A b-3 a B) \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(7 A b-3 a B) \left (\frac {5 \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(7 A b-3 a B) \left (\frac {5 \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{3/2} (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x)^2) + ((7*A*b - 3*a*B)*(1/ (a*x^(3/2)*(a + b*x)) + (5*(-2/(3*a*x^(3/2)) - (b*(-2/(a*Sqrt[x]) - (2*Sqr t[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/a))/(2*a)))/(4*a*b)))/Sq rt[a^2 + 2*a*b*x + b^2*x^2]
3.9.24.3.1 Defintions of rubi rules used
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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.18 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(-\frac {2 \left (-9 A b x +3 a B x +a A \right ) \sqrt {\left (b x +a \right )^{2}}}{3 a^{4} x^{\frac {3}{2}} \left (b x +a \right )}+\frac {b \left (\frac {2 \left (\frac {11}{8} A \,b^{2}-\frac {7}{8} a b B \right ) x^{\frac {3}{2}}+\frac {a \left (13 A b -9 B a \right ) \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {5 \left (7 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{4 \sqrt {b a}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{4} \left (b x +a \right )}\) | \(130\) |
| default | \(\frac {\left (105 A \,x^{3} \sqrt {b a}\, b^{3}+105 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) b^{4}-45 B \,x^{3} \sqrt {b a}\, a \,b^{2}-45 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{3}+210 A \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a \,b^{3}-90 B \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{2}+175 A \,x^{2} \sqrt {b a}\, a \,b^{2}+105 A \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{2} b^{2}-75 B \,x^{2} \sqrt {b a}\, a^{2} b -45 B \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right ) a^{3} b +56 A x \sqrt {b a}\, a^{2} b -24 B x \sqrt {b a}\, a^{3}-8 A \,a^{3} \sqrt {b a}\right ) \left (b x +a \right )}{12 x^{\frac {3}{2}} \sqrt {b a}\, a^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(253\) |
-2/3*(-9*A*b*x+3*B*a*x+A*a)/a^4/x^(3/2)*((b*x+a)^2)^(1/2)/(b*x+a)+1/a^4*b* (2*((11/8*A*b^2-7/8*a*b*B)*x^(3/2)+1/8*a*(13*A*b-9*B*a)*x^(1/2))/(b*x+a)^2 +5/4*(7*A*b-3*B*a)/(b*a)^(1/2)*arctan(b*x^(1/2)/(b*a)^(1/2)))*((b*x+a)^2)^ (1/2)/(b*x+a)
Time = 0.30 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.49 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
[-1/24*(15*((3*B*a*b^2 - 7*A*b^3)*x^4 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^3 + (3 *B*a^3 - 7*A*a^2*b)*x^2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a) /(b*x + a)) + 2*(8*A*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^2 + 8*(3*B*a^3 - 7*A*a^2*b)*x)*sqrt(x))/(a^4*b^2*x^4 + 2*a^5* b*x^3 + a^6*x^2), 1/12*(15*((3*B*a*b^2 - 7*A*b^3)*x^4 + 2*(3*B*a^2*b - 7*A *a*b^2)*x^3 + (3*B*a^3 - 7*A*a^2*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*s qrt(x))) - (8*A*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a *b^2)*x^2 + 8*(3*B*a^3 - 7*A*a^2*b)*x)*sqrt(x))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)]
\[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{x^{\frac {5}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.34 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {420 \, {\left (B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{\frac {5}{2}} + 5 \, {\left ({\left (B a b^{5} - 7 \, A b^{6}\right )} x^{2} + 21 \, {\left (B a^{2} b^{4} - 3 \, A a b^{5}\right )} x\right )} x^{\frac {5}{2}} - 5 \, {\left (9 \, {\left (B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{2} - 119 \, {\left (B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x\right )} \sqrt {x} - \frac {16 \, {\left (5 \, {\left (B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} - 21 \, {\left (B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )}}{\sqrt {x}} - \frac {48 \, {\left ({\left (B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} - {\left (B a^{6} - 3 \, A a^{5} b\right )} x\right )}}{x^{\frac {3}{2}}} + \frac {16 \, {\left (3 \, A a^{5} b x^{2} + A a^{6} x\right )}}{x^{\frac {5}{2}}}}{24 \, {\left (a^{6} b^{3} x^{3} + 3 \, a^{7} b^{2} x^{2} + 3 \, a^{8} b x + a^{9}\right )}} - \frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} + \frac {5 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} \sqrt {x}\right )}}{24 \, a^{6}} \]
-1/24*(420*(B*a^3*b^3 - 3*A*a^2*b^4)*x^(5/2) + 5*((B*a*b^5 - 7*A*b^6)*x^2 + 21*(B*a^2*b^4 - 3*A*a*b^5)*x)*x^(5/2) - 5*(9*(B*a^3*b^3 - 7*A*a^2*b^4)*x ^2 - 119*(B*a^4*b^2 - 3*A*a^3*b^3)*x)*sqrt(x) - 16*(5*(B*a^4*b^2 - 7*A*a^3 *b^3)*x^2 - 21*(B*a^5*b - 3*A*a^4*b^2)*x)/sqrt(x) - 48*((B*a^5*b - 7*A*a^4 *b^2)*x^2 - (B*a^6 - 3*A*a^5*b)*x)/x^(3/2) + 16*(3*A*a^5*b*x^2 + A*a^6*x)/ x^(5/2))/(a^6*b^3*x^3 + 3*a^7*b^2*x^2 + 3*a^8*b*x + a^9) - 5/4*(3*B*a*b - 7*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + 5/24*((B*a*b^2 - 7* A*b^3)*x^(3/2) + 6*(3*B*a^2*b - 7*A*a*b^2)*sqrt(x))/a^6
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, {\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right )} - \frac {7 \, B a b^{2} x^{\frac {3}{2}} - 11 \, A b^{3} x^{\frac {3}{2}} + 9 \, B a^{2} b \sqrt {x} - 13 \, A a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4} \mathrm {sgn}\left (b x + a\right )} \]
-5/4*(3*B*a*b - 7*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*sgn(b* x + a)) - 2/3*(3*B*a*x - 9*A*b*x + A*a)/(a^4*x^(3/2)*sgn(b*x + a)) - 1/4*( 7*B*a*b^2*x^(3/2) - 11*A*b^3*x^(3/2) + 9*B*a^2*b*sqrt(x) - 13*A*a*b^2*sqrt (x))/((b*x + a)^2*a^4*sgn(b*x + a))
Timed out. \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{x^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]