Integrand size = 22, antiderivative size = 105 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a^2 d^4 x (d x)^{-4+m}}{c^2 (4-m) \sqrt {c x^2}}-\frac {2 a b d^3 x (d x)^{-3+m}}{c^2 (3-m) \sqrt {c x^2}}-\frac {b^2 d^2 x (d x)^{-2+m}}{c^2 (2-m) \sqrt {c x^2}} \]
-a^2*d^4*x*(d*x)^(-4+m)/c^2/(4-m)/(c*x^2)^(1/2)-2*a*b*d^3*x*(d*x)^(-3+m)/c ^2/(3-m)/(c*x^2)^(1/2)-b^2*d^2*x*(d*x)^(-2+m)/c^2/(2-m)/(c*x^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {x (d x)^m \left (a^2 \left (6-5 m+m^2\right )+2 a b \left (8-6 m+m^2\right ) x+b^2 \left (12-7 m+m^2\right ) x^2\right )}{(-4+m) (-3+m) (-2+m) \left (c x^2\right )^{5/2}} \]
(x*(d*x)^m*(a^2*(6 - 5*m + m^2) + 2*a*b*(8 - 6*m + m^2)*x + b^2*(12 - 7*m + m^2)*x^2))/((-4 + m)*(-3 + m)*(-2 + m)*(c*x^2)^(5/2))
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {30, 53, 2009}
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.
\(\displaystyle \int \frac {(a+b x)^2 (d x)^m}{\left (c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle \frac {d^5 x \int (d x)^{m-5} (a+b x)^2dx}{c^2 \sqrt {c x^2}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {d^5 x \int \left (a^2 (d x)^{m-5}+\frac {2 a b (d x)^{m-4}}{d}+\frac {b^2 (d x)^{m-3}}{d^2}\right )dx}{c^2 \sqrt {c x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d^5 x \left (-\frac {a^2 (d x)^{m-4}}{d (4-m)}-\frac {2 a b (d x)^{m-3}}{d^2 (3-m)}-\frac {b^2 (d x)^{m-2}}{d^3 (2-m)}\right )}{c^2 \sqrt {c x^2}}\) |
(d^5*x*(-((a^2*(d*x)^(-4 + m))/(d*(4 - m))) - (2*a*b*(d*x)^(-3 + m))/(d^2* (3 - m)) - (b^2*(d*x)^(-2 + m))/(d^3*(2 - m))))/(c^2*Sqrt[c*x^2])
3.10.80.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x -7 m \,x^{2} b^{2}+a^{2} m^{2}-12 a b m x +12 b^{2} x^{2}-5 a^{2} m +16 a b x +6 a^{2}\right ) \left (d x \right )^{m}}{\left (-2+m \right ) \left (-3+m \right ) \left (-4+m \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(95\) |
risch | \(\frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x -7 m \,x^{2} b^{2}+a^{2} m^{2}-12 a b m x +12 b^{2} x^{2}-5 a^{2} m +16 a b x +6 a^{2}\right ) \left (d x \right )^{m}}{c^{2} x^{3} \sqrt {c \,x^{2}}\, \left (-2+m \right ) \left (-3+m \right ) \left (-4+m \right )}\) | \(100\) |
x*(b^2*m^2*x^2+2*a*b*m^2*x-7*b^2*m*x^2+a^2*m^2-12*a*b*m*x+12*b^2*x^2-5*a^2 *m+16*a*b*x+6*a^2)*(d*x)^m/(-2+m)/(-3+m)/(-4+m)/(c*x^2)^(5/2)
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {{\left (a^{2} m^{2} - 5 \, a^{2} m + {\left (b^{2} m^{2} - 7 \, b^{2} m + 12 \, b^{2}\right )} x^{2} + 6 \, a^{2} + 2 \, {\left (a b m^{2} - 6 \, a b m + 8 \, a b\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{{\left (c^{3} m^{3} - 9 \, c^{3} m^{2} + 26 \, c^{3} m - 24 \, c^{3}\right )} x^{5}} \]
(a^2*m^2 - 5*a^2*m + (b^2*m^2 - 7*b^2*m + 12*b^2)*x^2 + 6*a^2 + 2*(a*b*m^2 - 6*a*b*m + 8*a*b)*x)*sqrt(c*x^2)*(d*x)^m/((c^3*m^3 - 9*c^3*m^2 + 26*c^3* m - 24*c^3)*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 719 vs. \(2 (94) = 188\).
Time = 4.43 (sec) , antiderivative size = 719, normalized size of antiderivative = 6.85 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\begin {cases} d^{2} \left (- \frac {a^{2} x^{3}}{2 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {2 a b x^{4}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b^{2} x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}}\right ) & \text {for}\: m = 2 \\d^{3} \left (- \frac {a^{2} x^{4}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {2 a b x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b^{2} x^{6}}{\left (c x^{2}\right )^{\frac {5}{2}}}\right ) & \text {for}\: m = 3 \\d^{4} \left (\frac {a^{2} x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {2 a b x^{6}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b^{2} x^{7}}{2 \left (c x^{2}\right )^{\frac {5}{2}}}\right ) & \text {for}\: m = 4 \\\frac {a^{2} m^{2} x \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {5 a^{2} m x \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {6 a^{2} x \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {2 a b m^{2} x^{2} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {12 a b m x^{2} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {16 a b x^{2} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b^{2} m^{2} x^{3} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {7 b^{2} m x^{3} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} + \frac {12 b^{2} x^{3} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {5}{2}} - 9 m^{2} \left (c x^{2}\right )^{\frac {5}{2}} + 26 m \left (c x^{2}\right )^{\frac {5}{2}} - 24 \left (c x^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((d**2*(-a**2*x**3/(2*(c*x**2)**(5/2)) - 2*a*b*x**4/(c*x**2)**(5/ 2) + b**2*x**5*log(x)/(c*x**2)**(5/2)), Eq(m, 2)), (d**3*(-a**2*x**4/(c*x* *2)**(5/2) + 2*a*b*x**5*log(x)/(c*x**2)**(5/2) + b**2*x**6/(c*x**2)**(5/2) ), Eq(m, 3)), (d**4*(a**2*x**5*log(x)/(c*x**2)**(5/2) + 2*a*b*x**6/(c*x**2 )**(5/2) + b**2*x**7/(2*(c*x**2)**(5/2))), Eq(m, 4)), (a**2*m**2*x*(d*x)** m/(m**3*(c*x**2)**(5/2) - 9*m**2*(c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2) - 24*(c*x**2)**(5/2)) - 5*a**2*m*x*(d*x)**m/(m**3*(c*x**2)**(5/2) - 9*m**2*( c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2) - 24*(c*x**2)**(5/2)) + 6*a**2*x*(d* x)**m/(m**3*(c*x**2)**(5/2) - 9*m**2*(c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2 ) - 24*(c*x**2)**(5/2)) + 2*a*b*m**2*x**2*(d*x)**m/(m**3*(c*x**2)**(5/2) - 9*m**2*(c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2) - 24*(c*x**2)**(5/2)) - 12* a*b*m*x**2*(d*x)**m/(m**3*(c*x**2)**(5/2) - 9*m**2*(c*x**2)**(5/2) + 26*m* (c*x**2)**(5/2) - 24*(c*x**2)**(5/2)) + 16*a*b*x**2*(d*x)**m/(m**3*(c*x**2 )**(5/2) - 9*m**2*(c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2) - 24*(c*x**2)**(5 /2)) + b**2*m**2*x**3*(d*x)**m/(m**3*(c*x**2)**(5/2) - 9*m**2*(c*x**2)**(5 /2) + 26*m*(c*x**2)**(5/2) - 24*(c*x**2)**(5/2)) - 7*b**2*m*x**3*(d*x)**m/ (m**3*(c*x**2)**(5/2) - 9*m**2*(c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2) - 24 *(c*x**2)**(5/2)) + 12*b**2*x**3*(d*x)**m/(m**3*(c*x**2)**(5/2) - 9*m**2*( c*x**2)**(5/2) + 26*m*(c*x**2)**(5/2) - 24*(c*x**2)**(5/2)), True))
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.61 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {b^{2} d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 2\right )} x^{2}} + \frac {2 \, a b d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 3\right )} x^{3}} + \frac {a^{2} d^{m} x^{m}}{c^{\frac {5}{2}} {\left (m - 4\right )} x^{4}} \]
b^2*d^m*x^m/(c^(5/2)*(m - 2)*x^2) + 2*a*b*d^m*x^m/(c^(5/2)*(m - 3)*x^3) + a^2*d^m*x^m/(c^(5/2)*(m - 4)*x^4)
\[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {5}{2}}} \,d x } \]
Time = 0.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx=\frac {a^2\,{\left (d\,x\right )}^m}{c^2\,x^3\,\sqrt {c\,x^2}\,\left (m-4\right )}+\frac {b^2\,{\left (d\,x\right )}^m}{c^2\,x\,\sqrt {c\,x^2}\,\left (m-2\right )}+\frac {2\,a\,b\,{\left (d\,x\right )}^m}{c^2\,x^2\,\sqrt {c\,x^2}\,\left (m-3\right )} \]