Integrand size = 13, antiderivative size = 95 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=-\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
-35/12/a^3/x^(3/2)+1/2/a/x^(3/2)/(b*x+a)^2+7/4/a^2/x^(3/2)/(b*x+a)+35/4*b^ (3/2)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(9/2)+35/4*b/a^4/x^(1/2)
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {-8 a^3+56 a^2 b x+175 a b^2 x^2+105 b^3 x^3}{12 a^4 x^{3/2} (a+b x)^2}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
(-8*a^3 + 56*a^2*b*x + 175*a*b^2*x^2 + 105*b^3*x^3)/(12*a^4*x^(3/2)*(a + b *x)^2) + (35*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {52, 52, 61, 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.
| \(\displaystyle \int \frac {1}{x^{5/2} (a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 \int \frac {1}{x^{5/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 \left (\frac {5 \int \frac {1}{x^{5/2} (a+b x)}dx}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\) |
1/(2*a*x^(3/2)*(a + b*x)^2) + (7*(1/(a*x^(3/2)*(a + b*x)) + (5*(-2/(3*a*x^ (3/2)) - (b*(-2/(a*Sqrt[x]) - (2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]] )/a^(3/2)))/a))/(2*a)))/(4*a)
3.5.68.3.1 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 + 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_)^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]
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {2 \left (-9 b x +a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{4}+\frac {13 a \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{4}}\) | \(64\) |
| derivativedivides | \(\frac {2 b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}-\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}\) | \(67\) |
| default | \(\frac {2 b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}-\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}\) | \(67\) |
-2/3*(-9*b*x+a)/a^4/x^(3/2)+1/a^4*b^2*(2*(11/8*b*x^(3/2)+13/8*a*x^(1/2))/( b*x+a)^2+35/4/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.24 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.63 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\left [\frac {105 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, -\frac {105 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
[1/24*(105*(b^3*x^4 + 2*a*b^2*x^3 + a^2*b*x^2)*sqrt(-b/a)*log((b*x + 2*a*s qrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(105*b^3*x^3 + 175*a*b^2*x^2 + 56*a^ 2*b*x - 8*a^3)*sqrt(x))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2), -1/12*(105* (b^3*x^4 + 2*a*b^2*x^3 + a^2*b*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x ))) - (105*b^3*x^3 + 175*a*b^2*x^2 + 56*a^2*b*x - 8*a^3)*sqrt(x))/(a^4*b^2 *x^4 + 2*a^5*b*x^3 + a^6*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (88) = 176\).
Time = 65.48 (sec) , antiderivative size = 869, normalized size of antiderivative = 9.15 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{3} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{9 b^{3} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {16 a^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {112 a^{2} b x \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {350 a b^{2} x^{2} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {210 b^{3} x^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*a**3*x**(3/2)), Eq(b , 0)), (-2/(9*b**3*x**(9/2)), Eq(a, 0)), (-16*a**3*sqrt(-a/b)/(24*a**6*x** (3/2)*sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*s qrt(-a/b)) + 105*a**2*b*x**(3/2)*log(sqrt(x) - sqrt(-a/b))/(24*a**6*x**(3/ 2)*sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*sqrt (-a/b)) - 105*a**2*b*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(24*a**6*x**(3/2)* sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*sqrt(-a /b)) + 112*a**2*b*x*sqrt(-a/b)/(24*a**6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x* *(5/2)*sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*sqrt(-a/b)) + 210*a*b**2*x**(5/2 )*log(sqrt(x) - sqrt(-a/b))/(24*a**6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x**(5 /2)*sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*sqrt(-a/b)) - 210*a*b**2*x**(5/2)*l og(sqrt(x) + sqrt(-a/b))/(24*a**6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x**(5/2) *sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*sqrt(-a/b)) + 350*a*b**2*x**2*sqrt(-a/ b)/(24*a**6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4* b**2*x**(7/2)*sqrt(-a/b)) + 105*b**3*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(2 4*a**6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4*b**2* x**(7/2)*sqrt(-a/b)) - 105*b**3*x**(7/2)*log(sqrt(x) + sqrt(-a/b))/(24*a** 6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4*b**2*x**(7 /2)*sqrt(-a/b)) + 210*b**3*x**3*sqrt(-a/b)/(24*a**6*x**(3/2)*sqrt(-a/b) + 48*a**5*b*x**(5/2)*sqrt(-a/b) + 24*a**4*b**2*x**(7/2)*sqrt(-a/b)), True...
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} + 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} \]
1/12*(105*b^3*x^3 + 175*a*b^2*x^2 + 56*a^2*b*x - 8*a^3)/(a^4*b^2*x^(7/2) + 2*a^5*b*x^(5/2) + a^6*x^(3/2)) + 35/4*b^2*arctan(b*sqrt(x)/sqrt(a*b))/(sq rt(a*b)*a^4)
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} + \frac {2 \, {\left (9 \, b x - a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} + \frac {11 \, b^{3} x^{\frac {3}{2}} + 13 \, a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4}} \]
35/4*b^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + 2/3*(9*b*x - a)/(a^ 4*x^(3/2)) + 1/4*(11*b^3*x^(3/2) + 13*a*b^2*sqrt(x))/((b*x + a)^2*a^4)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {\frac {175\,b^2\,x^2}{12\,a^3}-\frac {2}{3\,a}+\frac {35\,b^3\,x^3}{4\,a^4}+\frac {14\,b\,x}{3\,a^2}}{a^2\,x^{3/2}+b^2\,x^{7/2}+2\,a\,b\,x^{5/2}}+\frac {35\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \]