Integrand size = 29, antiderivative size = 258 \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {a^3 (A b-a B) x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^2 (A b-a B) x^2 (a+b x)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a (A b-a B) x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) x^4 (a+b x)}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^5 (a+b x)}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^4 (A b-a B) (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
-a^3*(A*b-B*a)*x*(b*x+a)/b^5/((b*x+a)^2)^(1/2)+1/2*a^2*(A*b-B*a)*x^2*(b*x+ a)/b^4/((b*x+a)^2)^(1/2)-1/3*a*(A*b-B*a)*x^3*(b*x+a)/b^3/((b*x+a)^2)^(1/2) +1/4*(A*b-B*a)*x^4*(b*x+a)/b^2/((b*x+a)^2)^(1/2)+1/5*B*x^5*(b*x+a)/b/((b*x +a)^2)^(1/2)+a^4*(A*b-B*a)*(b*x+a)*ln(b*x+a)/b^6/((b*x+a)^2)^(1/2)
Time = 1.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {x \left (60 a^4 B-30 a^3 b (2 A+B x)+10 a^2 b^2 x (3 A+2 B x)-5 a b^3 x^2 (4 A+3 B x)+3 b^4 x^3 (5 A+4 B x)\right ) \left (\sqrt {a^2} b x+a \left (\sqrt {a^2}-\sqrt {(a+b x)^2}\right )\right )}{60 b^5 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right )}+\frac {2 a^4 (-A b+a B) \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )}{b^6} \]
-1/60*(x*(60*a^4*B - 30*a^3*b*(2*A + B*x) + 10*a^2*b^2*x*(3*A + 2*B*x) - 5 *a*b^3*x^2*(4*A + 3*B*x) + 3*b^4*x^3*(5*A + 4*B*x))*(Sqrt[a^2]*b*x + a*(Sq rt[a^2] - Sqrt[(a + b*x)^2])))/(b^5*(a^2 + a*b*x - Sqrt[a^2]*Sqrt[(a + b*x )^2])) + (2*a^4*(-(A*b) + a*B)*ArcTanh[(b*x)/(Sqrt[a^2] - Sqrt[(a + b*x)^2 ])])/b^6
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1187, 27, 86, 2009}
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.
| \(\displaystyle \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b (a+b x) \int \frac {x^4 (A+B x)}{b (a+b x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {x^4 (A+B x)}{a+b x}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {(a+b x) \int \left (-\frac {(a B-A b) a^4}{b^5 (a+b x)}+\frac {(a B-A b) a^3}{b^5}-\frac {(a B-A b) x a^2}{b^4}+\frac {(a B-A b) x^2 a}{b^3}+\frac {B x^4}{b}+\frac {(A b-a B) x^3}{b^2}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (\frac {a^4 (A b-a B) \log (a+b x)}{b^6}-\frac {a^3 x (A b-a B)}{b^5}+\frac {a^2 x^2 (A b-a B)}{2 b^4}-\frac {a x^3 (A b-a B)}{3 b^3}+\frac {x^4 (A b-a B)}{4 b^2}+\frac {B x^5}{5 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-((a^3*(A*b - a*B)*x)/b^5) + (a^2*(A*b - a*B)*x^2)/(2*b^4) - ( a*(A*b - a*B)*x^3)/(3*b^3) + ((A*b - a*B)*x^4)/(4*b^2) + (B*x^5)/(5*b) + ( a^4*(A*b - a*B)*Log[a + b*x])/b^6))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.8.4.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(\frac {\left (b x +a \right ) \left (12 B \,x^{5} b^{5}+15 A \,x^{4} b^{5}-15 B \,x^{4} a \,b^{4}-20 A \,x^{3} a \,b^{4}+20 B \,x^{3} a^{2} b^{3}+30 A \,x^{2} a^{2} b^{3}-30 B \,x^{2} a^{3} b^{2}+60 A \ln \left (b x +a \right ) a^{4} b -60 A \,a^{3} b^{2} x -60 B \ln \left (b x +a \right ) a^{5}+60 B \,a^{4} b x \right )}{60 \sqrt {\left (b x +a \right )^{2}}\, b^{6}}\) | \(138\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {1}{5} b^{4} B \,x^{5}+\frac {1}{4} A \,b^{4} x^{4}-\frac {1}{4} x^{4} B \,b^{3} a -\frac {1}{3} a A \,b^{3} x^{3}+\frac {1}{3} x^{3} B \,a^{2} b^{2}+\frac {1}{2} a^{2} A \,b^{2} x^{2}-\frac {1}{2} x^{2} B \,a^{3} b -a^{3} A b x +a^{4} B x \right )}{\left (b x +a \right ) b^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{4} \left (A b -B a \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{6}}\) | \(146\) |
1/60*(b*x+a)*(12*B*x^5*b^5+15*A*x^4*b^5-15*B*x^4*a*b^4-20*A*x^3*a*b^4+20*B *x^3*a^2*b^3+30*A*x^2*a^2*b^3-30*B*x^2*a^3*b^2+60*A*ln(b*x+a)*a^4*b-60*A*a ^3*b^2*x-60*B*ln(b*x+a)*a^5+60*B*a^4*b*x)/((b*x+a)^2)^(1/2)/b^6
Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.45 \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {12 \, B b^{5} x^{5} - 15 \, {\left (B a b^{4} - A b^{5}\right )} x^{4} + 20 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 30 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 60 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} x - 60 \, {\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{60 \, b^{6}} \]
1/60*(12*B*b^5*x^5 - 15*(B*a*b^4 - A*b^5)*x^4 + 20*(B*a^2*b^3 - A*a*b^4)*x ^3 - 30*(B*a^3*b^2 - A*a^2*b^3)*x^2 + 60*(B*a^4*b - A*a^3*b^2)*x - 60*(B*a ^5 - A*a^4*b)*log(b*x + a))/b^6
Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (187) = 374\).
Time = 1.60 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.81 \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} \left (\frac {B x^{4}}{5 b^{2}} + \frac {x^{3} \left (A - \frac {9 B a}{5 b}\right )}{4 b^{2}} + \frac {x^{2} \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b^{2}} + \frac {x \left (- \frac {3 a^{2} \left (A - \frac {9 B a}{5 b}\right )}{4 b^{2}} - \frac {5 a \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b}\right )}{2 b^{2}} + \frac {- \frac {2 a^{2} \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b^{2}} - \frac {3 a \left (- \frac {3 a^{2} \left (A - \frac {9 B a}{5 b}\right )}{4 b^{2}} - \frac {5 a \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b}\right )}{2 b}}{b^{2}}\right ) + \frac {\left (\frac {a}{b} + x\right ) \left (- \frac {a^{2} \left (- \frac {3 a^{2} \left (A - \frac {9 B a}{5 b}\right )}{4 b^{2}} - \frac {5 a \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b}\right )}{2 b^{2}} - \frac {a \left (- \frac {2 a^{2} \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b^{2}} - \frac {3 a \left (- \frac {3 a^{2} \left (A - \frac {9 B a}{5 b}\right )}{4 b^{2}} - \frac {5 a \left (- \frac {4 B a^{2}}{5 b^{2}} - \frac {7 a \left (A - \frac {9 B a}{5 b}\right )}{4 b}\right )}{3 b}\right )}{2 b}\right )}{b}\right ) \log {\left (\frac {a}{b} + x \right )}}{\sqrt {b^{2} \left (\frac {a}{b} + x\right )^{2}}} & \text {for}\: b^{2} \neq 0 \\\frac {\frac {A \left (a^{8} \sqrt {a^{2} + 2 a b x} - \frac {4 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {6 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5} - \frac {4 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9}\right )}{8 a^{4} b^{4}} + \frac {B \left (- a^{10} \sqrt {a^{2} + 2 a b x} + \frac {5 a^{8} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} - 2 a^{6} \left (a^{2} + 2 a b x\right )^{\frac {5}{2}} + \frac {10 a^{4} \left (a^{2} + 2 a b x\right )^{\frac {7}{2}}}{7} - \frac {5 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {9}{2}}}{9} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {11}{2}}}{11}\right )}{16 a^{5} b^{5}}}{2 a b} & \text {for}\: a b \neq 0 \\\frac {\frac {A x^{5}}{5} + \frac {B x^{6}}{6}}{\sqrt {a^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(B*x**4/(5*b**2) + x**3*(A - 9 *B*a/(5*b))/(4*b**2) + x**2*(-4*B*a**2/(5*b**2) - 7*a*(A - 9*B*a/(5*b))/(4 *b))/(3*b**2) + x*(-3*a**2*(A - 9*B*a/(5*b))/(4*b**2) - 5*a*(-4*B*a**2/(5* b**2) - 7*a*(A - 9*B*a/(5*b))/(4*b))/(3*b))/(2*b**2) + (-2*a**2*(-4*B*a**2 /(5*b**2) - 7*a*(A - 9*B*a/(5*b))/(4*b))/(3*b**2) - 3*a*(-3*a**2*(A - 9*B* a/(5*b))/(4*b**2) - 5*a*(-4*B*a**2/(5*b**2) - 7*a*(A - 9*B*a/(5*b))/(4*b)) /(3*b))/(2*b))/b**2) + (a/b + x)*(-a**2*(-3*a**2*(A - 9*B*a/(5*b))/(4*b**2 ) - 5*a*(-4*B*a**2/(5*b**2) - 7*a*(A - 9*B*a/(5*b))/(4*b))/(3*b))/(2*b**2) - a*(-2*a**2*(-4*B*a**2/(5*b**2) - 7*a*(A - 9*B*a/(5*b))/(4*b))/(3*b**2) - 3*a*(-3*a**2*(A - 9*B*a/(5*b))/(4*b**2) - 5*a*(-4*B*a**2/(5*b**2) - 7*a* (A - 9*B*a/(5*b))/(4*b))/(3*b))/(2*b))/b)*log(a/b + x)/sqrt(b**2*(a/b + x) **2), Ne(b**2, 0)), ((A*(a**8*sqrt(a**2 + 2*a*b*x) - 4*a**6*(a**2 + 2*a*b* x)**(3/2)/3 + 6*a**4*(a**2 + 2*a*b*x)**(5/2)/5 - 4*a**2*(a**2 + 2*a*b*x)** (7/2)/7 + (a**2 + 2*a*b*x)**(9/2)/9)/(8*a**4*b**4) + B*(-a**10*sqrt(a**2 + 2*a*b*x) + 5*a**8*(a**2 + 2*a*b*x)**(3/2)/3 - 2*a**6*(a**2 + 2*a*b*x)**(5 /2) + 10*a**4*(a**2 + 2*a*b*x)**(7/2)/7 - 5*a**2*(a**2 + 2*a*b*x)**(9/2)/9 + (a**2 + 2*a*b*x)**(11/2)/11)/(16*a**5*b**5))/(2*a*b), Ne(a*b, 0)), ((A* x**5/5 + B*x**6/6)/sqrt(a**2), True))
Time = 0.19 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.05 \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B x^{4}}{5 \, b^{2}} - \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a x^{3}}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A x^{3}}{4 \, b^{2}} - \frac {77 \, B a^{3} x^{2}}{60 \, b^{4}} + \frac {13 \, A a^{2} x^{2}}{12 \, b^{3}} + \frac {47 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{2} x^{2}}{60 \, b^{4}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a x^{2}}{12 \, b^{3}} + \frac {77 \, B a^{4} x}{30 \, b^{5}} - \frac {13 \, A a^{3} x}{6 \, b^{4}} - \frac {B a^{5} \log \left (x + \frac {a}{b}\right )}{b^{6}} + \frac {A a^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {47 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{4}}{30 \, b^{6}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{3}}{6 \, b^{5}} \]
1/5*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*x^4/b^2 - 9/20*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a*x^3/b^3 + 1/4*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*x^3/b^2 - 77/60*B *a^3*x^2/b^4 + 13/12*A*a^2*x^2/b^3 + 47/60*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B *a^2*x^2/b^4 - 7/12*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*a*x^2/b^3 + 77/30*B*a^ 4*x/b^5 - 13/6*A*a^3*x/b^4 - B*a^5*log(x + a/b)/b^6 + A*a^4*log(x + a/b)/b ^5 - 47/30*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a^4/b^6 + 7/6*sqrt(b^2*x^2 + 2* a*b*x + a^2)*A*a^3/b^5
Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {12 \, B b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) - 15 \, B a b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, A b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, B a^{2} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, A a b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, B a^{3} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, A a^{2} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 60 \, B a^{4} x \mathrm {sgn}\left (b x + a\right ) - 60 \, A a^{3} b x \mathrm {sgn}\left (b x + a\right )}{60 \, b^{5}} - \frac {{\left (B a^{5} \mathrm {sgn}\left (b x + a\right ) - A a^{4} b \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \]
1/60*(12*B*b^4*x^5*sgn(b*x + a) - 15*B*a*b^3*x^4*sgn(b*x + a) + 15*A*b^4*x ^4*sgn(b*x + a) + 20*B*a^2*b^2*x^3*sgn(b*x + a) - 20*A*a*b^3*x^3*sgn(b*x + a) - 30*B*a^3*b*x^2*sgn(b*x + a) + 30*A*a^2*b^2*x^2*sgn(b*x + a) + 60*B*a ^4*x*sgn(b*x + a) - 60*A*a^3*b*x*sgn(b*x + a))/b^5 - (B*a^5*sgn(b*x + a) - A*a^4*b*sgn(b*x + a))*log(abs(b*x + a))/b^6
Timed out. \[ \int \frac {x^4 (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {x^4\,\left (A+B\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]