Integrand size = 20, antiderivative size = 225 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {a^4 (12 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{512 b^4}-\frac {a^3 (12 A b-7 a B) x^{3/2} \sqrt {a+b x}}{768 b^3}+\frac {a^2 (12 A b-7 a B) x^{5/2} \sqrt {a+b x}}{960 b^2}+\frac {a (12 A b-7 a B) x^{7/2} \sqrt {a+b x}}{160 b}+\frac {(12 A b-7 a B) x^{7/2} (a+b x)^{3/2}}{60 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}-\frac {a^5 (12 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{512 b^{9/2}} \]
1/60*(12*A*b-7*B*a)*x^(7/2)*(b*x+a)^(3/2)/b+1/6*B*x^(7/2)*(b*x+a)^(5/2)/b- 1/512*a^5*(12*A*b-7*B*a)*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(9/2)-1/ 768*a^3*(12*A*b-7*B*a)*x^(3/2)*(b*x+a)^(1/2)/b^3+1/960*a^2*(12*A*b-7*B*a)* x^(5/2)*(b*x+a)^(1/2)/b^2+1/160*a*(12*A*b-7*B*a)*x^(7/2)*(b*x+a)^(1/2)/b+1 /512*a^4*(12*A*b-7*B*a)*x^(1/2)*(b*x+a)^(1/2)/b^4
Time = 0.80 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.86 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-105 a^5 B+48 a^2 b^3 x^2 (2 A+B x)+256 b^5 x^4 (6 A+5 B x)-8 a^3 b^2 x (15 A+7 B x)+10 a^4 b (18 A+7 B x)+64 a b^4 x^3 (33 A+26 B x)\right )+360 a^5 A b \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+210 a^6 B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{7680 b^{9/2}} \]
(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-105*a^5*B + 48*a^2*b^3*x^2*(2*A + B*x) + 256*b^5*x^4*(6*A + 5*B*x) - 8*a^3*b^2*x*(15*A + 7*B*x) + 10*a^4*b*(18*A + 7*B*x) + 64*a*b^4*x^3*(33*A + 26*B*x)) + 360*a^5*A*b*ArcTanh[(Sqrt[b]*Sqrt [x])/(Sqrt[a] - Sqrt[a + b*x])] + 210*a^6*B*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sq rt[a] + Sqrt[a + b*x])])/(7680*b^(9/2))
Time = 0.24 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {90, 60, 60, 60, 60, 60, 65, 219}
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.
| \(\displaystyle \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {(12 A b-7 a B) \int x^{5/2} (a+b x)^{3/2}dx}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \int x^{5/2} \sqrt {a+b x}dx+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \left (\frac {1}{8} a \int \frac {x^{5/2}}{\sqrt {a+b x}}dx+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \int \frac {x^{3/2}}{\sqrt {a+b x}}dx}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a+b x}}dx}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{2 b}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(12 A b-7 a B) \left (\frac {3}{10} a \left (\frac {1}{8} a \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\right )}{4 b}\right )}{6 b}\right )+\frac {1}{4} x^{7/2} \sqrt {a+b x}\right )+\frac {1}{5} x^{7/2} (a+b x)^{3/2}\right )}{12 b}+\frac {B x^{7/2} (a+b x)^{5/2}}{6 b}\) |
(B*x^(7/2)*(a + b*x)^(5/2))/(6*b) + ((12*A*b - 7*a*B)*((x^(7/2)*(a + b*x)^ (3/2))/5 + (3*a*((x^(7/2)*Sqrt[a + b*x])/4 + (a*((x^(5/2)*Sqrt[a + b*x])/( 3*b) - (5*a*((x^(3/2)*Sqrt[a + b*x])/(2*b) - (3*a*((Sqrt[x]*Sqrt[a + b*x]) /b - (a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/2)))/(4*b)))/(6*b)) )/8))/10))/(12*b)
3.5.90.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 1.42 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.81
| method | result | size |
| risch | \(\frac {\left (1280 b^{5} B \,x^{5}+1536 A \,b^{5} x^{4}+1664 B a \,b^{4} x^{4}+2112 A a \,b^{4} x^{3}+48 B \,a^{2} b^{3} x^{3}+96 A \,a^{2} b^{3} x^{2}-56 B \,a^{3} b^{2} x^{2}-120 a^{3} b^{2} A x +70 a^{4} b B x +180 a^{4} b A -105 a^{5} B \right ) \sqrt {x}\, \sqrt {b x +a}}{7680 b^{4}}-\frac {a^{5} \left (12 A b -7 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{1024 b^{\frac {9}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(183\) |
| default | \(-\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (-2560 B \,b^{\frac {11}{2}} x^{5} \sqrt {x \left (b x +a \right )}-3072 A \,b^{\frac {11}{2}} x^{4} \sqrt {x \left (b x +a \right )}-3328 B a \,b^{\frac {9}{2}} x^{4} \sqrt {x \left (b x +a \right )}-4224 A a \,b^{\frac {9}{2}} x^{3} \sqrt {x \left (b x +a \right )}-96 B \,a^{2} b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}-192 A \,a^{2} b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+112 B \,a^{3} b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+240 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a^{3} x -140 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{4} x +180 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{5} b -360 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{4}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{6}+210 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{5}\right )}{15360 b^{\frac {9}{2}} \sqrt {x \left (b x +a \right )}}\) | \(302\) |
1/7680/b^4*(1280*B*b^5*x^5+1536*A*b^5*x^4+1664*B*a*b^4*x^4+2112*A*a*b^4*x^ 3+48*B*a^2*b^3*x^3+96*A*a^2*b^3*x^2-56*B*a^3*b^2*x^2-120*A*a^3*b^2*x+70*B* a^4*b*x+180*A*a^4*b-105*B*a^5)*x^(1/2)*(b*x+a)^(1/2)-1/1024*a^5/b^(9/2)*(1 2*A*b-7*B*a)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))*(x*(b*x+a))^(1/2)/x ^(1/2)/(b*x+a)^(1/2)
Time = 0.25 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.52 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\left [-\frac {15 \, {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{15360 \, b^{5}}, -\frac {15 \, {\left (7 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (1280 \, B b^{6} x^{5} - 105 \, B a^{5} b + 180 \, A a^{4} b^{2} + 128 \, {\left (13 \, B a b^{5} + 12 \, A b^{6}\right )} x^{4} + 48 \, {\left (B a^{2} b^{4} + 44 \, A a b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{7680 \, b^{5}}\right ] \]
[-1/15360*(15*(7*B*a^6 - 12*A*a^5*b)*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*s qrt(b)*sqrt(x) + a) - 2*(1280*B*b^6*x^5 - 105*B*a^5*b + 180*A*a^4*b^2 + 12 8*(13*B*a*b^5 + 12*A*b^6)*x^4 + 48*(B*a^2*b^4 + 44*A*a*b^5)*x^3 - 8*(7*B*a ^3*b^3 - 12*A*a^2*b^4)*x^2 + 10*(7*B*a^4*b^2 - 12*A*a^3*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^5, -1/7680*(15*(7*B*a^6 - 12*A*a^5*b)*sqrt(-b)*arctan(sqrt( b*x + a)*sqrt(-b)/(b*sqrt(x))) - (1280*B*b^6*x^5 - 105*B*a^5*b + 180*A*a^4 *b^2 + 128*(13*B*a*b^5 + 12*A*b^6)*x^4 + 48*(B*a^2*b^4 + 44*A*a*b^5)*x^3 - 8*(7*B*a^3*b^3 - 12*A*a^2*b^4)*x^2 + 10*(7*B*a^4*b^2 - 12*A*a^3*b^3)*x)*s qrt(b*x + a)*sqrt(x))/b^5]
Timed out. \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.24 \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=-\frac {7 \, \sqrt {b x^{2} + a x} B a^{4} x}{256 \, b^{3}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2} x}{96 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{3} x}{64 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B x}{6 \, b} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a x}{8 \, b} + \frac {7 \, B a^{6} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {9}{2}}} - \frac {3 \, A a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {7}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{5}}{512 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{3}}{192 \, b^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} A a^{4}}{128 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {5}{2}} B a}{60 \, b^{2}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a^{2}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{5 \, b} \]
-7/256*sqrt(b*x^2 + a*x)*B*a^4*x/b^3 + 7/96*(b*x^2 + a*x)^(3/2)*B*a^2*x/b^ 2 + 3/64*sqrt(b*x^2 + a*x)*A*a^3*x/b^2 + 1/6*(b*x^2 + a*x)^(5/2)*B*x/b - 1 /8*(b*x^2 + a*x)^(3/2)*A*a*x/b + 7/1024*B*a^6*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) - 3/256*A*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x) *sqrt(b))/b^(7/2) - 7/512*sqrt(b*x^2 + a*x)*B*a^5/b^4 + 7/192*(b*x^2 + a*x )^(3/2)*B*a^3/b^3 + 3/128*sqrt(b*x^2 + a*x)*A*a^4/b^3 - 7/60*(b*x^2 + a*x) ^(5/2)*B*a/b^2 - 1/16*(b*x^2 + a*x)^(3/2)*A*a^2/b^2 + 1/5*(b*x^2 + a*x)^(5 /2)*A/b
Timed out. \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\text {Timed out} \]
Timed out. \[ \int x^{5/2} (a+b x)^{3/2} (A+B x) \, dx=\int x^{5/2}\,\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2} \,d x \]