Integrand size = 18, antiderivative size = 86 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=2 a^2 A \sqrt {a+b x}+\frac {2}{3} a A (a+b x)^{3/2}+\frac {2}{5} A (a+b x)^{5/2}+\frac {2 B (a+b x)^{7/2}}{7 b}-2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
2/3*a*A*(b*x+a)^(3/2)+2/5*A*(b*x+a)^(5/2)+2/7*B*(b*x+a)^(7/2)/b-2*a^(5/2)* A*arctanh((b*x+a)^(1/2)/a^(1/2))+2*a^2*A*(b*x+a)^(1/2)
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=\frac {2 \left (105 a^2 A b \sqrt {a+b x}+35 a A b (a+b x)^{3/2}+21 A b (a+b x)^{5/2}+15 B (a+b x)^{7/2}\right )}{105 b}-2 a^{5/2} A \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \]
(2*(105*a^2*A*b*Sqrt[a + b*x] + 35*a*A*b*(a + b*x)^(3/2) + 21*A*b*(a + b*x )^(5/2) + 15*B*(a + b*x)^(7/2)))/(105*b) - 2*a^(5/2)*A*ArcTanh[Sqrt[a + b* x]/Sqrt[a]]
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {90, 60, 60, 60, 73, 221}
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.
\(\displaystyle \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle A \int \frac {(a+b x)^{5/2}}{x}dx+\frac {2 B (a+b x)^{7/2}}{7 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle A \left (a \int \frac {(a+b x)^{3/2}}{x}dx+\frac {2}{5} (a+b x)^{5/2}\right )+\frac {2 B (a+b x)^{7/2}}{7 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle A \left (a \left (a \int \frac {\sqrt {a+b x}}{x}dx+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )+\frac {2 B (a+b x)^{7/2}}{7 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle A \left (a \left (a \left (a \int \frac {1}{x \sqrt {a+b x}}dx+2 \sqrt {a+b x}\right )+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )+\frac {2 B (a+b x)^{7/2}}{7 b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle A \left (a \left (a \left (\frac {2 a \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b}+2 \sqrt {a+b x}\right )+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )+\frac {2 B (a+b x)^{7/2}}{7 b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle A \left (a \left (a \left (2 \sqrt {a+b x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )+\frac {2 B (a+b x)^{7/2}}{7 b}\) |
(2*B*(a + b*x)^(7/2))/(7*b) + A*((2*(a + b*x)^(5/2))/5 + a*((2*(a + b*x)^( 3/2))/3 + a*(2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])))
3.5.15.3.1 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[((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*(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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 1.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 A b \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b a \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A \,a^{2} b \sqrt {b x +a}-2 A \,a^{\frac {5}{2}} b \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b}\) | \(72\) |
default | \(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 A b \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 A b a \left (b x +a \right )^{\frac {3}{2}}}{3}+2 A \,a^{2} b \sqrt {b x +a}-2 A \,a^{\frac {5}{2}} b \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b}\) | \(72\) |
pseudoelliptic | \(\frac {-2 A \,a^{\frac {5}{2}} b \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {46 \left (\frac {3 x^{2} \left (\frac {5 B x}{7}+A \right ) b^{3}}{23}+\frac {11 \left (\frac {45 B x}{77}+A \right ) x a \,b^{2}}{23}+a^{2} \left (\frac {45 B x}{161}+A \right ) b +\frac {15 a^{3} B}{161}\right ) \sqrt {b x +a}}{15}}{b}\) | \(80\) |
2/b*(1/7*B*(b*x+a)^(7/2)+1/5*A*b*(b*x+a)^(5/2)+1/3*A*b*a*(b*x+a)^(3/2)+A*a ^2*b*(b*x+a)^(1/2)-A*a^(5/2)*b*arctanh((b*x+a)^(1/2)/a^(1/2)))
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.42 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=\left [\frac {105 \, A a^{\frac {5}{2}} b \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (15 \, B b^{3} x^{3} + 15 \, B a^{3} + 161 \, A a^{2} b + 3 \, {\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} + {\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x\right )} \sqrt {b x + a}}{105 \, b}, \frac {2 \, {\left (105 \, A \sqrt {-a} a^{2} b \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, B b^{3} x^{3} + 15 \, B a^{3} + 161 \, A a^{2} b + 3 \, {\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} + {\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x\right )} \sqrt {b x + a}\right )}}{105 \, b}\right ] \]
[1/105*(105*A*a^(5/2)*b*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*( 15*B*b^3*x^3 + 15*B*a^3 + 161*A*a^2*b + 3*(15*B*a*b^2 + 7*A*b^3)*x^2 + (45 *B*a^2*b + 77*A*a*b^2)*x)*sqrt(b*x + a))/b, 2/105*(105*A*sqrt(-a)*a^2*b*ar ctan(sqrt(b*x + a)*sqrt(-a)/a) + (15*B*b^3*x^3 + 15*B*a^3 + 161*A*a^2*b + 3*(15*B*a*b^2 + 7*A*b^3)*x^2 + (45*B*a^2*b + 77*A*a*b^2)*x)*sqrt(b*x + a)) /b]
Time = 1.58 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=\begin {cases} \frac {2 A a^{3} \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 A a^{2} \sqrt {a + b x} + \frac {2 A a \left (a + b x\right )^{\frac {3}{2}}}{3} + \frac {2 A \left (a + b x\right )^{\frac {5}{2}}}{5} + \frac {2 B \left (a + b x\right )^{\frac {7}{2}}}{7 b} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (A \log {\left (B x \right )} + B x\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*A*a**3*atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a) + 2*A*a**2*sqrt( a + b*x) + 2*A*a*(a + b*x)**(3/2)/3 + 2*A*(a + b*x)**(5/2)/5 + 2*B*(a + b* x)**(7/2)/(7*b), Ne(b, 0)), (a**(5/2)*(A*log(B*x) + B*x), True))
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=A a^{\frac {5}{2}} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} B + 21 \, {\left (b x + a\right )}^{\frac {5}{2}} A b + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b + 105 \, \sqrt {b x + a} A a^{2} b\right )}}{105 \, b} \]
A*a^(5/2)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) + 2/105 *(15*(b*x + a)^(7/2)*B + 21*(b*x + a)^(5/2)*A*b + 35*(b*x + a)^(3/2)*A*a*b + 105*sqrt(b*x + a)*A*a^2*b)/b
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=\frac {2 \, A a^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} B b^{6} + 21 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{7} + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{7} + 105 \, \sqrt {b x + a} A a^{2} b^{7}\right )}}{105 \, b^{7}} \]
2*A*a^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2/105*(15*(b*x + a)^(7/2 )*B*b^6 + 21*(b*x + a)^(5/2)*A*b^7 + 35*(b*x + a)^(3/2)*A*a*b^7 + 105*sqrt (b*x + a)*A*a^2*b^7)/b^7
Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.49 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x} \, dx=\left (\frac {2\,A\,b-2\,B\,a}{5\,b}+\frac {2\,B\,a}{5\,b}\right )\,{\left (a+b\,x\right )}^{5/2}+a^2\,\left (\frac {2\,A\,b-2\,B\,a}{b}+\frac {2\,B\,a}{b}\right )\,\sqrt {a+b\,x}+\frac {2\,B\,{\left (a+b\,x\right )}^{7/2}}{7\,b}+\frac {a\,\left (\frac {2\,A\,b-2\,B\,a}{b}+\frac {2\,B\,a}{b}\right )\,{\left (a+b\,x\right )}^{3/2}}{3}+A\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]