Integrand size = 22, antiderivative size = 103 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {a^2 c^2 (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x}+\frac {2 a b c^2 (d x)^{7+m} \sqrt {c x^2}}{d^7 (7+m) x}+\frac {b^2 c^2 (d x)^{8+m} \sqrt {c x^2}}{d^8 (8+m) x} \] Output:
a^2*c^2*(d*x)^(6+m)*(c*x^2)^(1/2)/d^6/(6+m)/x+2*a*b*c^2*(d*x)^(7+m)*(c*x^2 )^(1/2)/d^7/(7+m)/x+b^2*c^2*(d*x)^(8+m)*(c*x^2)^(1/2)/d^8/(8+m)/x
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.47 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=x (d x)^m \left (c x^2\right )^{5/2} \left (\frac {a^2}{6+m}+\frac {2 a b x}{7+m}+\frac {b^2 x^2}{8+m}\right ) \] Input:
Integrate[(d*x)^m*(c*x^2)^(5/2)*(a + b*x)^2,x]
Output:
x*(d*x)^m*(c*x^2)^(5/2)*(a^2/(6 + m) + (2*a*b*x)/(7 + m) + (b^2*x^2)/(8 + m))
Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.75, 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.
\(\displaystyle \int \left (c x^2\right )^{5/2} (a+b x)^2 (d x)^m \, dx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle \frac {c^2 \sqrt {c x^2} \int (d x)^{m+5} (a+b x)^2dx}{d^5 x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {c^2 \sqrt {c x^2} \int \left (a^2 (d x)^{m+5}+\frac {2 a b (d x)^{m+6}}{d}+\frac {b^2 (d x)^{m+7}}{d^2}\right )dx}{d^5 x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^2 \sqrt {c x^2} \left (\frac {a^2 (d x)^{m+6}}{d (m+6)}+\frac {2 a b (d x)^{m+7}}{d^2 (m+7)}+\frac {b^2 (d x)^{m+8}}{d^3 (m+8)}\right )}{d^5 x}\) |
Input:
Int[(d*x)^m*(c*x^2)^(5/2)*(a + b*x)^2,x]
Output:
(c^2*Sqrt[c*x^2]*((a^2*(d*x)^(6 + m))/(d*(6 + m)) + (2*a*b*(d*x)^(7 + m))/ (d^2*(7 + m)) + (b^2*(d*x)^(8 + m))/(d^3*(8 + m))))/(d^5*x)
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 = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +13 m \,x^{2} b^{2}+a^{2} m^{2}+28 a b m x +42 b^{2} x^{2}+15 a^{2} m +96 a b x +56 a^{2}\right ) \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {5}{2}}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right )}\) | \(95\) |
orering | \(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +13 m \,x^{2} b^{2}+a^{2} m^{2}+28 a b m x +42 b^{2} x^{2}+15 a^{2} m +96 a b x +56 a^{2}\right ) \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {5}{2}}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right )}\) | \(95\) |
risch | \(\frac {c^{2} x^{5} \sqrt {c \,x^{2}}\, \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +13 m \,x^{2} b^{2}+a^{2} m^{2}+28 a b m x +42 b^{2} x^{2}+15 a^{2} m +96 a b x +56 a^{2}\right ) \left (d x \right )^{m}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right )}\) | \(100\) |
Input:
int((d*x)^m*(c*x^2)^(5/2)*(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
x*(b^2*m^2*x^2+2*a*b*m^2*x+13*b^2*m*x^2+a^2*m^2+28*a*b*m*x+42*b^2*x^2+15*a ^2*m+96*a*b*x+56*a^2)*(d*x)^m*(c*x^2)^(5/2)/(8+m)/(7+m)/(6+m)
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.19 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {{\left ({\left (b^{2} c^{2} m^{2} + 13 \, b^{2} c^{2} m + 42 \, b^{2} c^{2}\right )} x^{7} + 2 \, {\left (a b c^{2} m^{2} + 14 \, a b c^{2} m + 48 \, a b c^{2}\right )} x^{6} + {\left (a^{2} c^{2} m^{2} + 15 \, a^{2} c^{2} m + 56 \, a^{2} c^{2}\right )} x^{5}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 21 \, m^{2} + 146 \, m + 336} \] Input:
integrate((d*x)^m*(c*x^2)^(5/2)*(b*x+a)^2,x, algorithm="fricas")
Output:
((b^2*c^2*m^2 + 13*b^2*c^2*m + 42*b^2*c^2)*x^7 + 2*(a*b*c^2*m^2 + 14*a*b*c ^2*m + 48*a*b*c^2)*x^6 + (a^2*c^2*m^2 + 15*a^2*c^2*m + 56*a^2*c^2)*x^5)*sq rt(c*x^2)*(d*x)^m/(m^3 + 21*m^2 + 146*m + 336)
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (92) = 184\).
Time = 7.29 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.81 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\begin {cases} \frac {- \frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{2 x^{7}} - \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{6}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}} \log {\left (x \right )}}{x^{5}}}{d^{8}} & \text {for}\: m = -8 \\\frac {- \frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{6}} + \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}} \log {\left (x \right )}}{x^{5}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{4}}}{d^{7}} & \text {for}\: m = -7 \\\frac {\frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}} \log {\left (x \right )}}{x^{5}} + \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{4}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{2 x^{3}}}{d^{6}} & \text {for}\: m = -6 \\\frac {a^{2} m^{2} x \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {15 a^{2} m x \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {56 a^{2} x \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {2 a b m^{2} x^{2} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {28 a b m x^{2} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {96 a b x^{2} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {b^{2} m^{2} x^{3} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {13 b^{2} m x^{3} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} + \frac {42 b^{2} x^{3} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{3} + 21 m^{2} + 146 m + 336} & \text {otherwise} \end {cases} \] Input:
integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a)**2,x)
Output:
Piecewise(((-a**2*(c*x**2)**(5/2)/(2*x**7) - 2*a*b*(c*x**2)**(5/2)/x**6 + b**2*(c*x**2)**(5/2)*log(x)/x**5)/d**8, Eq(m, -8)), ((-a**2*(c*x**2)**(5/2 )/x**6 + 2*a*b*(c*x**2)**(5/2)*log(x)/x**5 + b**2*(c*x**2)**(5/2)/x**4)/d* *7, Eq(m, -7)), ((a**2*(c*x**2)**(5/2)*log(x)/x**5 + 2*a*b*(c*x**2)**(5/2) /x**4 + b**2*(c*x**2)**(5/2)/(2*x**3))/d**6, Eq(m, -6)), (a**2*m**2*x*(c*x **2)**(5/2)*(d*x)**m/(m**3 + 21*m**2 + 146*m + 336) + 15*a**2*m*x*(c*x**2) **(5/2)*(d*x)**m/(m**3 + 21*m**2 + 146*m + 336) + 56*a**2*x*(c*x**2)**(5/2 )*(d*x)**m/(m**3 + 21*m**2 + 146*m + 336) + 2*a*b*m**2*x**2*(c*x**2)**(5/2 )*(d*x)**m/(m**3 + 21*m**2 + 146*m + 336) + 28*a*b*m*x**2*(c*x**2)**(5/2)* (d*x)**m/(m**3 + 21*m**2 + 146*m + 336) + 96*a*b*x**2*(c*x**2)**(5/2)*(d*x )**m/(m**3 + 21*m**2 + 146*m + 336) + b**2*m**2*x**3*(c*x**2)**(5/2)*(d*x) **m/(m**3 + 21*m**2 + 146*m + 336) + 13*b**2*m*x**3*(c*x**2)**(5/2)*(d*x)* *m/(m**3 + 21*m**2 + 146*m + 336) + 42*b**2*x**3*(c*x**2)**(5/2)*(d*x)**m/ (m**3 + 21*m**2 + 146*m + 336), True))
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.62 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {b^{2} c^{\frac {5}{2}} d^{m} x^{8} x^{m}}{m + 8} + \frac {2 \, a b c^{\frac {5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac {a^{2} c^{\frac {5}{2}} d^{m} x^{6} x^{m}}{m + 6} \] Input:
integrate((d*x)^m*(c*x^2)^(5/2)*(b*x+a)^2,x, algorithm="maxima")
Output:
b^2*c^(5/2)*d^m*x^8*x^m/(m + 8) + 2*a*b*c^(5/2)*d^m*x^7*x^m/(m + 7) + a^2* c^(5/2)*d^m*x^6*x^m/(m + 6)
Exception generated. \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x)^m*(c*x^2)^(5/2)*(b*x+a)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 22.68 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.23 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {a^2\,c^2\,x^5\,\sqrt {c\,x^2}\,\left (m^2+15\,m+56\right )}{m^3+21\,m^2+146\,m+336}+\frac {b^2\,c^2\,x^7\,\sqrt {c\,x^2}\,\left (m^2+13\,m+42\right )}{m^3+21\,m^2+146\,m+336}+\frac {2\,a\,b\,c^2\,x^6\,\sqrt {c\,x^2}\,\left (m^2+14\,m+48\right )}{m^3+21\,m^2+146\,m+336}\right ) \] Input:
int((d*x)^m*(c*x^2)^(5/2)*(a + b*x)^2,x)
Output:
(d*x)^m*((a^2*c^2*x^5*(c*x^2)^(1/2)*(15*m + m^2 + 56))/(146*m + 21*m^2 + m ^3 + 336) + (b^2*c^2*x^7*(c*x^2)^(1/2)*(13*m + m^2 + 42))/(146*m + 21*m^2 + m^3 + 336) + (2*a*b*c^2*x^6*(c*x^2)^(1/2)*(14*m + m^2 + 48))/(146*m + 21 *m^2 + m^3 + 336))
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.92 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx=\frac {x^{m} d^{m} \sqrt {c}\, c^{2} x^{6} \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +13 b^{2} m \,x^{2}+a^{2} m^{2}+28 a b m x +42 b^{2} x^{2}+15 a^{2} m +96 a b x +56 a^{2}\right )}{m^{3}+21 m^{2}+146 m +336} \] Input:
int((d*x)^m*(c*x^2)^(5/2)*(b*x+a)^2,x)
Output:
(x**m*d**m*sqrt(c)*c**2*x**6*(a**2*m**2 + 15*a**2*m + 56*a**2 + 2*a*b*m**2 *x + 28*a*b*m*x + 96*a*b*x + b**2*m**2*x**2 + 13*b**2*m*x**2 + 42*b**2*x** 2))/(m**3 + 21*m**2 + 146*m + 336)