Integrand size = 20, antiderivative size = 159 \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\frac {5 a^2 (8 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{64 b^4}-\frac {5 a (8 A b-7 a B) x^{3/2} \sqrt {a+b x}}{96 b^3}+\frac {(8 A b-7 a B) x^{5/2} \sqrt {a+b x}}{24 b^2}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}-\frac {5 a^3 (8 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{9/2}} \]
-5/64*a^3*(8*A*b-7*B*a)*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(9/2)-5/9 6*a*(8*A*b-7*B*a)*x^(3/2)*(b*x+a)^(1/2)/b^3+1/24*(8*A*b-7*B*a)*x^(5/2)*(b* x+a)^(1/2)/b^2+1/4*B*x^(7/2)*(b*x+a)^(1/2)/b+5/64*a^2*(8*A*b-7*B*a)*x^(1/2 )*(b*x+a)^(1/2)/b^4
Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.79 \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (-105 a^3 B+16 b^3 x^2 (4 A+3 B x)-8 a b^2 x (10 A+7 B x)+10 a^2 b (12 A+7 B x)\right )}{192 b^4}+\frac {5 a^3 (-8 A b+7 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{32 b^{9/2}} \]
(Sqrt[x]*Sqrt[a + b*x]*(-105*a^3*B + 16*b^3*x^2*(4*A + 3*B*x) - 8*a*b^2*x* (10*A + 7*B*x) + 10*a^2*b*(12*A + 7*B*x)))/(192*b^4) + (5*a^3*(-8*A*b + 7* a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/(32*b^(9/2))
Time = 0.22 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {90, 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.
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {(8 A b-7 a B) \int \frac {x^{5/2}}{\sqrt {a+b x}}dx}{8 b}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(8 A b-7 a B) \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \int \frac {x^{3/2}}{\sqrt {a+b x}}dx}{6 b}\right )}{8 b}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(8 A b-7 a B) \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \int \frac {\sqrt {x}}{\sqrt {a+b x}}dx}{4 b}\right )}{6 b}\right )}{8 b}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(8 A b-7 a B) \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{2 b}\right )}{4 b}\right )}{6 b}\right )}{8 b}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {(8 A b-7 a B) \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}\right )}{4 b}\right )}{6 b}\right )}{8 b}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(8 A b-7 a B) \left (\frac {x^{5/2} \sqrt {a+b x}}{3 b}-\frac {5 a \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b}-\frac {3 a \left (\frac {\sqrt {x} \sqrt {a+b x}}{b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}\right )}{4 b}\right )}{6 b}\right )}{8 b}+\frac {B x^{7/2} \sqrt {a+b x}}{4 b}\) |
(B*x^(7/2)*Sqrt[a + b*x])/(4*b) + ((8*A*b - 7*a*B)*((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*b)
3.6.15.3.1 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 = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(\frac {\left (48 b^{3} B \,x^{3}+64 A \,b^{3} x^{2}-56 B a \,b^{2} x^{2}-80 a \,b^{2} A x +70 a^{2} b B x +120 a^{2} b A -105 a^{3} B \right ) \sqrt {x}\, \sqrt {b x +a}}{192 b^{4}}-\frac {5 a^{3} \left (8 A b -7 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{128 b^{\frac {9}{2}} \sqrt {x}\, \sqrt {b x +a}}\) | \(135\) |
| default | \(-\frac {\sqrt {x}\, \sqrt {b x +a}\, \left (-96 B \,b^{\frac {7}{2}} x^{3} \sqrt {x \left (b x +a \right )}-128 A \,b^{\frac {7}{2}} x^{2} \sqrt {x \left (b x +a \right )}+112 B a \,b^{\frac {5}{2}} x^{2} \sqrt {x \left (b x +a \right )}+160 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} a x -140 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2} x +120 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3} b -240 A \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a^{2}-105 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{4}+210 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{3}\right )}{384 b^{\frac {9}{2}} \sqrt {x \left (b x +a \right )}}\) | \(218\) |
1/192*(48*B*b^3*x^3+64*A*b^3*x^2-56*B*a*b^2*x^2-80*A*a*b^2*x+70*B*a^2*b*x+ 120*A*a^2*b-105*B*a^3)*x^(1/2)*(b*x+a)^(1/2)/b^4-5/128*a^3*(8*A*b-7*B*a)/b ^(9/2)*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 = 0.24 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.57 \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, B b^{4} x^{3} - 105 \, B a^{3} b + 120 \, A a^{2} b^{2} - 8 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, b^{5}}, -\frac {15 \, {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (48 \, B b^{4} x^{3} - 105 \, B a^{3} b + 120 \, A a^{2} b^{2} - 8 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, b^{5}}\right ] \]
[-1/384*(15*(7*B*a^4 - 8*A*a^3*b)*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt (b)*sqrt(x) + a) - 2*(48*B*b^4*x^3 - 105*B*a^3*b + 120*A*a^2*b^2 - 8*(7*B* a*b^3 - 8*A*b^4)*x^2 + 10*(7*B*a^2*b^2 - 8*A*a*b^3)*x)*sqrt(b*x + a)*sqrt( x))/b^5, -1/192*(15*(7*B*a^4 - 8*A*a^3*b)*sqrt(-b)*arctan(sqrt(b*x + a)*sq rt(-b)/(b*sqrt(x))) - (48*B*b^4*x^3 - 105*B*a^3*b + 120*A*a^2*b^2 - 8*(7*B *a*b^3 - 8*A*b^4)*x^2 + 10*(7*B*a^2*b^2 - 8*A*a*b^3)*x)*sqrt(b*x + a)*sqrt (x))/b^5]
Time = 47.22 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.91 \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\frac {5 A a^{\frac {5}{2}} \sqrt {x}}{8 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {5 A a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A \sqrt {a} x^{\frac {5}{2}}}{12 b \sqrt {1 + \frac {b x}{a}}} - \frac {5 A a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {A x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} - \frac {35 B a^{\frac {7}{2}} \sqrt {x}}{64 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {35 B a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {7 B a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B \sqrt {a} x^{\frac {7}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {35 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {9}{2}}} + \frac {B x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]
5*A*a**(5/2)*sqrt(x)/(8*b**3*sqrt(1 + b*x/a)) + 5*A*a**(3/2)*x**(3/2)/(24* b**2*sqrt(1 + b*x/a)) - A*sqrt(a)*x**(5/2)/(12*b*sqrt(1 + b*x/a)) - 5*A*a* *3*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(7/2)) + A*x**(7/2)/(3*sqrt(a)*sqr t(1 + b*x/a)) - 35*B*a**(7/2)*sqrt(x)/(64*b**4*sqrt(1 + b*x/a)) - 35*B*a** (5/2)*x**(3/2)/(192*b**3*sqrt(1 + b*x/a)) + 7*B*a**(3/2)*x**(5/2)/(96*b**2 *sqrt(1 + b*x/a)) - B*sqrt(a)*x**(7/2)/(24*b*sqrt(1 + b*x/a)) + 35*B*a**4* asinh(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(9/2)) + B*x**(9/2)/(4*sqrt(a)*sqrt( 1 + b*x/a))
Time = 0.20 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.30 \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\frac {\sqrt {b x^{2} + a x} B x^{3}}{4 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} B a x^{2}}{24 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} A x^{2}}{3 \, b} + \frac {35 \, \sqrt {b x^{2} + a x} B a^{2} x}{96 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a x} A a x}{12 \, b^{2}} + \frac {35 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {9}{2}}} - \frac {5 \, A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {7}{2}}} - \frac {35 \, \sqrt {b x^{2} + a x} B a^{3}}{64 \, b^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{8 \, b^{3}} \]
1/4*sqrt(b*x^2 + a*x)*B*x^3/b - 7/24*sqrt(b*x^2 + a*x)*B*a*x^2/b^2 + 1/3*s qrt(b*x^2 + a*x)*A*x^2/b + 35/96*sqrt(b*x^2 + a*x)*B*a^2*x/b^3 - 5/12*sqrt (b*x^2 + a*x)*A*a*x/b^2 + 35/128*B*a^4*log(2*b*x + a + 2*sqrt(b*x^2 + a*x) *sqrt(b))/b^(9/2) - 5/16*A*a^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b) )/b^(7/2) - 35/64*sqrt(b*x^2 + a*x)*B*a^3/b^4 + 5/8*sqrt(b*x^2 + a*x)*A*a^ 2/b^3
Time = 151.79 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.42 \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\frac {\frac {8 \, {\left (\frac {15 \, a^{3} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {3}{2}}} + \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} - \frac {13 \, a}{b^{2}}\right )} + \frac {33 \, a^{2}}{b^{2}}\right )}\right )} A {\left | b \right |}}{b^{2}} - \frac {{\left (\frac {105 \, a^{4} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {5}{2}}} - {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} - \frac {25 \, a}{b^{3}}\right )} + \frac {163 \, a^{2}}{b^{3}}\right )} - \frac {279 \, a^{3}}{b^{3}}\right )} \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a}\right )} B {\left | b \right |}}{b^{2}}}{192 \, b} \]
1/192*(8*(15*a^3*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b)) )/b^(3/2) + sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a )/b^2 - 13*a/b^2) + 33*a^2/b^2))*A*abs(b)/b^2 - (105*a^4*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b)))/b^(5/2) - (2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 - 25*a/b^3) + 163*a^2/b^3) - 279*a^3/b^3)*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a))*B*abs(b)/b^2)/b
Timed out. \[ \int \frac {x^{5/2} (A+B x)}{\sqrt {a+b x}} \, dx=\int \frac {x^{5/2}\,\left (A+B\,x\right )}{\sqrt {a+b\,x}} \,d x \]