Integrand size = 13, antiderivative size = 95 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=-\frac {35 a \sqrt {x}}{4 b^4}+\frac {35 x^{3/2}}{12 b^3}-\frac {x^{7/2}}{2 b (a+b x)^2}-\frac {7 x^{5/2}}{4 b^2 (a+b x)}+\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]
35/12*x^(3/2)/b^3-1/2*x^(7/2)/b/(b*x+a)^2-7/4*x^(5/2)/b^2/(b*x+a)+35/4*a^( 3/2)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/b^(9/2)-35/4*a*x^(1/2)/b^4
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {\sqrt {x} \left (-105 a^3-175 a^2 b x-56 a b^2 x^2+8 b^3 x^3\right )}{12 b^4 (a+b x)^2}+\frac {35 a^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{9/2}} \]
(Sqrt[x]*(-105*a^3 - 175*a^2*b*x - 56*a*b^2*x^2 + 8*b^3*x^3))/(12*b^4*(a + b*x)^2) + (35*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9/2))
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {51, 51, 60, 60, 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.
\(\displaystyle \int \frac {x^{7/2}}{(a+b x)^3} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {7 \int \frac {x^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {7 \left (\frac {5 \int \frac {x^{3/2}}{a+b x}dx}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \int \frac {\sqrt {x}}{a+b x}dx}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} (a+b x)}dx}{b}\right )}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{a+b x}d\sqrt {x}}{b}\right )}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}\right )}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\) |
-1/2*x^(7/2)/(b*(a + b*x)^2) + (7*(-(x^(5/2)/(b*(a + b*x))) + (5*((2*x^(3/ 2))/(3*b) - (a*((2*Sqrt[x])/b - (2*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a ]])/b^(3/2)))/b))/(2*b)))/(4*b)
3.5.62.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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_)^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 = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {2 \left (-b x +9 a \right ) \sqrt {x}}{3 b^{4}}+\frac {a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{4}-\frac {11 a \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{4}}\) | \(66\) |
derivativedivides | \(-\frac {2 \left (-\frac {b \,x^{\frac {3}{2}}}{3}+3 a \sqrt {x}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{8}-\frac {11 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(68\) |
default | \(-\frac {2 \left (-\frac {b \,x^{\frac {3}{2}}}{3}+3 a \sqrt {x}\right )}{b^{4}}+\frac {2 a^{2} \left (\frac {-\frac {13 b \,x^{\frac {3}{2}}}{8}-\frac {11 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{4}}\) | \(68\) |
-2/3*(-b*x+9*a)*x^(1/2)/b^4+a^2/b^4*(2*(-13/8*b*x^(3/2)-11/8*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 = 227, normalized size of antiderivative = 2.39 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\left [\frac {105 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac {105 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{3} x^{3} - 56 \, a b^{2} x^{2} - 175 \, a^{2} b x - 105 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]
[1/24*(105*(a*b^2*x^2 + 2*a^2*b*x + a^3)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x) *sqrt(-a/b) - a)/(b*x + a)) + 2*(8*b^3*x^3 - 56*a*b^2*x^2 - 175*a^2*b*x - 105*a^3)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/12*(105*(a*b^2*x^2 + 2*a^2*b*x + a^3)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (8*b^3*x^3 - 56 *a*b^2*x^2 - 175*a^2*b*x - 105*a^3)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^ 4)]
Leaf count of result is larger than twice the leaf count of optimal. 762 vs. \(2 (87) = 174\).
Time = 66.07 (sec) , antiderivative size = 762, normalized size of antiderivative = 8.02 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a^{3}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b^{3}} & \text {for}\: a = 0 \\\frac {105 a^{4} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {105 a^{4} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {210 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} + \frac {210 a^{3} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {210 a^{3} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {350 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} + \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {105 a^{2} b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} - \frac {112 a b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} + \frac {16 b^{4} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{24 a^{2} b^{5} \sqrt {- \frac {a}{b}} + 48 a b^{6} x \sqrt {- \frac {a}{b}} + 24 b^{7} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(9/2)/(9*a**3), Eq(b, 0)), (2*x**(3/2)/(3*b**3), Eq(a, 0)), (105*a**4*log(sqrt(x) - sqrt(-a/b)) /(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/ b)) - 105*a**4*log(sqrt(x) + sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b **6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 210*a**3*b*sqrt(x)*sqrt(-a/b )/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a /b)) + 210*a**3*b*x*log(sqrt(x) - sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 4 8*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 210*a**3*b*x*log(sqrt(x ) + sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b** 7*x**2*sqrt(-a/b)) - 350*a**2*b**2*x**(3/2)*sqrt(-a/b)/(24*a**2*b**5*sqrt( -a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 105*a**2*b**2* x**2*log(sqrt(x) - sqrt(-a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt (-a/b) + 24*b**7*x**2*sqrt(-a/b)) - 105*a**2*b**2*x**2*log(sqrt(x) + sqrt( -a/b))/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sq rt(-a/b)) - 112*a*b**3*x**(5/2)*sqrt(-a/b)/(24*a**2*b**5*sqrt(-a/b) + 48*a *b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a/b)) + 16*b**4*x**(7/2)*sqrt(-a/b )/(24*a**2*b**5*sqrt(-a/b) + 48*a*b**6*x*sqrt(-a/b) + 24*b**7*x**2*sqrt(-a /b)), True))
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=-\frac {13 \, a^{2} b x^{\frac {3}{2}} + 11 \, a^{3} \sqrt {x}}{4 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} + \frac {2 \, {\left (b x^{\frac {3}{2}} - 9 \, a \sqrt {x}\right )}}{3 \, b^{4}} \]
-1/4*(13*a^2*b*x^(3/2) + 11*a^3*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4) + 35/4*a^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + 2/3*(b*x^(3/2) - 9 *a*sqrt(x))/b^4
Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {35 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{4}} - \frac {13 \, a^{2} b x^{\frac {3}{2}} + 11 \, a^{3} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{4}} + \frac {2 \, {\left (b^{6} x^{\frac {3}{2}} - 9 \, a b^{5} \sqrt {x}\right )}}{3 \, b^{9}} \]
35/4*a^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) - 1/4*(13*a^2*b*x^(3/ 2) + 11*a^3*sqrt(x))/((b*x + a)^2*b^4) + 2/3*(b^6*x^(3/2) - 9*a*b^5*sqrt(x ))/b^9
Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {x^{7/2}}{(a+b x)^3} \, dx=\frac {2\,x^{3/2}}{3\,b^3}-\frac {\frac {11\,a^3\,\sqrt {x}}{4}+\frac {13\,a^2\,b\,x^{3/2}}{4}}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}-\frac {6\,a\,\sqrt {x}}{b^4}+\frac {35\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{9/2}} \]