Integrand size = 29, antiderivative size = 157 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {5 (7 A b-a B)}{8 a^4 b \sqrt {x}}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}+\frac {7 A b-a B}{12 a^2 b \sqrt {x} (a+b x)^2}+\frac {5 (7 A b-a B)}{24 a^3 b \sqrt {x} (a+b x)}-\frac {5 (7 A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{9/2} \sqrt {b}} \]
-5/8*(7*A*b-B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(9/2)/b^(1/2)-5/8*(7*A* b-B*a)/a^4/b/x^(1/2)+1/3*(A*b-B*a)/a/b/(b*x+a)^3/x^(1/2)+1/12*(7*A*b-B*a)/ a^2/b/(b*x+a)^2/x^(1/2)+5/24*(7*A*b-B*a)/a^3/b/(b*x+a)/x^(1/2)
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-105 A b^3 x^3+5 a b^2 x^2 (-56 A+3 B x)+a^3 (-48 A+33 B x)+a^2 b x (-231 A+40 B x)}{24 a^4 \sqrt {x} (a+b x)^3}+\frac {5 (-7 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{8 a^{9/2} \sqrt {b}} \]
(-105*A*b^3*x^3 + 5*a*b^2*x^2*(-56*A + 3*B*x) + a^3*(-48*A + 33*B*x) + a^2 *b*x*(-231*A + 40*B*x))/(24*a^4*Sqrt[x]*(a + b*x)^3) + (5*(-7*A*b + a*B)*A rcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(8*a^(9/2)*Sqrt[b])
Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1184, 27, 87, 52, 52, 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.
\(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^4 \int \frac {A+B x}{b^4 x^{3/2} (a+b x)^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {A+B x}{x^{3/2} (a+b x)^4}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(7 A b-a B) \int \frac {1}{x^{3/2} (a+b x)^3}dx}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {(7 A b-a B) \left (\frac {5 \int \frac {1}{x^{3/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {(7 A b-a B) \left (\frac {5 \left (\frac {3 \int \frac {1}{x^{3/2} (a+b x)}dx}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {(7 A b-a B) \left (\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(7 A b-a B) \left (\frac {5 \left (\frac {3 \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(7 A b-a B) \left (\frac {5 \left (\frac {3 \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a b}+\frac {A b-a B}{3 a b \sqrt {x} (a+b x)^3}\) |
(A*b - a*B)/(3*a*b*Sqrt[x]*(a + b*x)^3) + ((7*A*b - a*B)*(1/(2*a*Sqrt[x]*( a + b*x)^2) + (5*(1/(a*Sqrt[x]*(a + b*x)) + (3*(-2/(a*Sqrt[x]) - (2*Sqrt[b ]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/(2*a)))/(4*a)))/(6*a*b)
3.8.71.3.1 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[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {19}{16} A \,b^{3}-\frac {5}{16} B a \,b^{2}\right ) x^{\frac {5}{2}}+\frac {b a \left (17 A b -5 B a \right ) x^{\frac {3}{2}}}{6}+\left (\frac {29}{16} A \,a^{2} b -\frac {11}{16} B \,a^{3}\right ) \sqrt {x}}{\left (b x +a \right )^{3}}+\frac {5 \left (7 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{16 \sqrt {b a}}\right )}{a^{4}}-\frac {2 A}{a^{4} \sqrt {x}}\) | \(105\) |
default | \(-\frac {2 \left (\frac {\left (\frac {19}{16} A \,b^{3}-\frac {5}{16} B a \,b^{2}\right ) x^{\frac {5}{2}}+\frac {b a \left (17 A b -5 B a \right ) x^{\frac {3}{2}}}{6}+\left (\frac {29}{16} A \,a^{2} b -\frac {11}{16} B \,a^{3}\right ) \sqrt {x}}{\left (b x +a \right )^{3}}+\frac {5 \left (7 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{16 \sqrt {b a}}\right )}{a^{4}}-\frac {2 A}{a^{4} \sqrt {x}}\) | \(105\) |
risch | \(-\frac {2 A}{a^{4} \sqrt {x}}-\frac {\frac {2 \left (\frac {19}{16} A \,b^{3}-\frac {5}{16} B a \,b^{2}\right ) x^{\frac {5}{2}}+\frac {b a \left (17 A b -5 B a \right ) x^{\frac {3}{2}}}{3}+2 \left (\frac {29}{16} A \,a^{2} b -\frac {11}{16} B \,a^{3}\right ) \sqrt {x}}{\left (b x +a \right )^{3}}+\frac {5 \left (7 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{8 \sqrt {b a}}}{a^{4}}\) | \(106\) |
-2/a^4*(((19/16*A*b^3-5/16*B*a*b^2)*x^(5/2)+1/6*b*a*(17*A*b-5*B*a)*x^(3/2) +(29/16*A*a^2*b-11/16*B*a^3)*x^(1/2))/(b*x+a)^3+5/16*(7*A*b-B*a)/(b*a)^(1/ 2)*arctan(b*x^(1/2)/(b*a)^(1/2)))-2*A/a^4/x^(1/2)
Time = 0.32 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.83 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [\frac {15 \, {\left ({\left (B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 3 \, {\left (B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 3 \, {\left (B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + {\left (B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (48 \, A a^{4} b - 15 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{3} - 40 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} - 33 \, {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x}}{48 \, {\left (a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{3} + 3 \, a^{7} b^{2} x^{2} + a^{8} b x\right )}}, -\frac {15 \, {\left ({\left (B a b^{3} - 7 \, A b^{4}\right )} x^{4} + 3 \, {\left (B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 3 \, {\left (B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + {\left (B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (48 \, A a^{4} b - 15 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{3} - 40 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} - 33 \, {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{3} + 3 \, a^{7} b^{2} x^{2} + a^{8} b x\right )}}\right ] \]
[1/48*(15*((B*a*b^3 - 7*A*b^4)*x^4 + 3*(B*a^2*b^2 - 7*A*a*b^3)*x^3 + 3*(B* a^3*b - 7*A*a^2*b^2)*x^2 + (B*a^4 - 7*A*a^3*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(48*A*a^4*b - 15*(B*a^2*b^3 - 7*A*a *b^4)*x^3 - 40*(B*a^3*b^2 - 7*A*a^2*b^3)*x^2 - 33*(B*a^4*b - 7*A*a^3*b^2)* x)*sqrt(x))/(a^5*b^4*x^4 + 3*a^6*b^3*x^3 + 3*a^7*b^2*x^2 + a^8*b*x), -1/24 *(15*((B*a*b^3 - 7*A*b^4)*x^4 + 3*(B*a^2*b^2 - 7*A*a*b^3)*x^3 + 3*(B*a^3*b - 7*A*a^2*b^2)*x^2 + (B*a^4 - 7*A*a^3*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b *sqrt(x))) + (48*A*a^4*b - 15*(B*a^2*b^3 - 7*A*a*b^4)*x^3 - 40*(B*a^3*b^2 - 7*A*a^2*b^3)*x^2 - 33*(B*a^4*b - 7*A*a^3*b^2)*x)*sqrt(x))/(a^5*b^4*x^4 + 3*a^6*b^3*x^3 + 3*a^7*b^2*x^2 + a^8*b*x)]
Leaf count of result is larger than twice the leaf count of optimal. 2660 vs. \(2 (146) = 292\).
Time = 80.92 (sec) , antiderivative size = 2660, normalized size of antiderivative = 16.94 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]
Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0) ), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a**4, Eq(b, 0)), ((-2*A/(9*x**(9/2)) - 2* B/(7*x**(7/2)))/b**4, Eq(a, 0)), (-105*A*a**3*b*sqrt(x)*log(sqrt(x) - sqrt (-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 105*A*a**3*b*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/ b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 96*A*a**3*b*sqrt(-a/b)/(48*a**7*b*s qrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**( 5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 315*A*a**2*b**2*x**( 3/2)*log(sqrt(x) - sqrt(-a/b))/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b* *2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4* x**(7/2)*sqrt(-a/b)) + 315*A*a**2*b**2*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/ (48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a* *5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) - 462*A*a* *2*b**2*x*sqrt(-a/b)/(48*a**7*b*sqrt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2 )*sqrt(-a/b) + 144*a**5*b**3*x**(5/2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*s qrt(-a/b)) - 315*A*a*b**3*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(48*a**7*b*sq rt(x)*sqrt(-a/b) + 144*a**6*b**2*x**(3/2)*sqrt(-a/b) + 144*a**5*b**3*x**(5 /2)*sqrt(-a/b) + 48*a**4*b**4*x**(7/2)*sqrt(-a/b)) + 315*A*a*b**3*x**(5...
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {48 \, A a^{3} - 15 \, {\left (B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 40 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{2} - 33 \, {\left (B a^{3} - 7 \, A a^{2} b\right )} x}{24 \, {\left (a^{4} b^{3} x^{\frac {7}{2}} + 3 \, a^{5} b^{2} x^{\frac {5}{2}} + 3 \, a^{6} b x^{\frac {3}{2}} + a^{7} \sqrt {x}\right )}} + \frac {5 \, {\left (B a - 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} \]
-1/24*(48*A*a^3 - 15*(B*a*b^2 - 7*A*b^3)*x^3 - 40*(B*a^2*b - 7*A*a*b^2)*x^ 2 - 33*(B*a^3 - 7*A*a^2*b)*x)/(a^4*b^3*x^(7/2) + 3*a^5*b^2*x^(5/2) + 3*a^6 *b*x^(3/2) + a^7*sqrt(x)) + 5/8*(B*a - 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/ (sqrt(a*b)*a^4)
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {5 \, {\left (B a - 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} - \frac {2 \, A}{a^{4} \sqrt {x}} + \frac {15 \, B a b^{2} x^{\frac {5}{2}} - 57 \, A b^{3} x^{\frac {5}{2}} + 40 \, B a^{2} b x^{\frac {3}{2}} - 136 \, A a b^{2} x^{\frac {3}{2}} + 33 \, B a^{3} \sqrt {x} - 87 \, A a^{2} b \sqrt {x}}{24 \, {\left (b x + a\right )}^{3} a^{4}} \]
5/8*(B*a - 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2*A/(a^4*s qrt(x)) + 1/24*(15*B*a*b^2*x^(5/2) - 57*A*b^3*x^(5/2) + 40*B*a^2*b*x^(3/2) - 136*A*a*b^2*x^(3/2) + 33*B*a^3*sqrt(x) - 87*A*a^2*b*sqrt(x))/((b*x + a) ^3*a^4)
Time = 9.96 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\frac {2\,A}{a}+\frac {11\,x\,\left (7\,A\,b-B\,a\right )}{8\,a^2}+\frac {5\,b^2\,x^3\,\left (7\,A\,b-B\,a\right )}{8\,a^4}+\frac {5\,b\,x^2\,\left (7\,A\,b-B\,a\right )}{3\,a^3}}{a^3\,\sqrt {x}+b^3\,x^{7/2}+3\,a^2\,b\,x^{3/2}+3\,a\,b^2\,x^{5/2}}-\frac {5\,\mathrm {atan}\left (\frac {5\,\sqrt {b}\,\sqrt {x}\,\left (7\,A\,b-B\,a\right )}{\sqrt {a}\,\left (35\,A\,b-5\,B\,a\right )}\right )\,\left (7\,A\,b-B\,a\right )}{8\,a^{9/2}\,\sqrt {b}} \]