Integrand size = 24, antiderivative size = 77 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {(a B-A b x) (c+d x)^2}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {2 (A b c+a B d) (a d-b c x)}{3 a^2 b^2 \sqrt {a+b x^2}} \] Output:
-1/3*(-A*b*x+B*a)*(d*x+c)^2/a/b/(b*x^2+a)^(3/2)-2/3*(A*b*c+B*a*d)*(-b*c*x+ a*d)/a^2/b^2/(b*x^2+a)^(1/2)
Time = 1.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-2 a^3 B d^2+2 A b^3 c^2 x^3+a b^2 x \left (3 A c^2+2 B c d x^2+A d^2 x^2\right )-a^2 b \left (2 A c d+B \left (c^2+3 d^2 x^2\right )\right )}{3 a^2 b^2 \left (a+b x^2\right )^{3/2}} \] Input:
Integrate[((A + B*x)*(c + d*x)^2)/(a + b*x^2)^(5/2),x]
Output:
(-2*a^3*B*d^2 + 2*A*b^3*c^2*x^3 + a*b^2*x*(3*A*c^2 + 2*B*c*d*x^2 + A*d^2*x ^2) - a^2*b*(2*A*c*d + B*(c^2 + 3*d^2*x^2)))/(3*a^2*b^2*(a + b*x^2)^(3/2))
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {678, 453}
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.
\(\displaystyle \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 678 |
\(\displaystyle \frac {2 (a B d+A b c) \int \frac {c+d x}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}-\frac {(c+d x)^2 (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 453 |
\(\displaystyle -\frac {2 (a d-b c x) (a B d+A b c)}{3 a^2 b^2 \sqrt {a+b x^2}}-\frac {(c+d x)^2 (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((A + B*x)*(c + d*x)^2)/(a + b*x^2)^(5/2),x]
Output:
-1/3*((a*B - A*b*x)*(c + d*x)^2)/(a*b*(a + b*x^2)^(3/2)) - (2*(A*b*c + a*B *d)*(a*d - b*c*x))/(3*a^2*b^2*Sqrt[a + b*x^2])
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(d + e*x)^m*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c *(p + 1))), x] - Simp[m*((c*d*f + a*e*g)/(2*a*c*(p + 1))) Int[(d + e*x)^( m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[S implify[m + 2*p + 3], 0] && LtQ[p, -1]
Time = 1.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39
method | result | size |
gosper | \(-\frac {-A a \,b^{2} d^{2} x^{3}-2 A \,b^{3} c^{2} x^{3}-2 B a \,b^{2} c d \,x^{3}+3 B \,a^{2} b \,d^{2} x^{2}-3 A a \,b^{2} c^{2} x +2 A \,a^{2} b c d +2 B \,a^{3} d^{2}+B \,a^{2} b \,c^{2}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b^{2}}\) | \(107\) |
trager | \(-\frac {-A a \,b^{2} d^{2} x^{3}-2 A \,b^{3} c^{2} x^{3}-2 B a \,b^{2} c d \,x^{3}+3 B \,a^{2} b \,d^{2} x^{2}-3 A a \,b^{2} c^{2} x +2 A \,a^{2} b c d +2 B \,a^{3} d^{2}+B \,a^{2} b \,c^{2}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b^{2}}\) | \(107\) |
orering | \(-\frac {-A a \,b^{2} d^{2} x^{3}-2 A \,b^{3} c^{2} x^{3}-2 B a \,b^{2} c d \,x^{3}+3 B \,a^{2} b \,d^{2} x^{2}-3 A a \,b^{2} c^{2} x +2 A \,a^{2} b c d +2 B \,a^{3} d^{2}+B \,a^{2} b \,c^{2}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b^{2}}\) | \(107\) |
default | \(A \,c^{2} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d \left (A d +2 B c \right ) \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )-\frac {c \left (2 A d +B c \right )}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+B \,d^{2} \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )\) | \(162\) |
Input:
int((B*x+A)*(d*x+c)^2/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(-A*a*b^2*d^2*x^3-2*A*b^3*c^2*x^3-2*B*a*b^2*c*d*x^3+3*B*a^2*b*d^2*x^2 -3*A*a*b^2*c^2*x+2*A*a^2*b*c*d+2*B*a^3*d^2+B*a^2*b*c^2)/(b*x^2+a)^(3/2)/a^ 2/b^2
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {{\left (3 \, B a^{2} b d^{2} x^{2} - 3 \, A a b^{2} c^{2} x + B a^{2} b c^{2} + 2 \, A a^{2} b c d + 2 \, B a^{3} d^{2} - {\left (2 \, A b^{3} c^{2} + 2 \, B a b^{2} c d + A a b^{2} d^{2}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \] Input:
integrate((B*x+A)*(d*x+c)^2/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/3*(3*B*a^2*b*d^2*x^2 - 3*A*a*b^2*c^2*x + B*a^2*b*c^2 + 2*A*a^2*b*c*d + 2*B*a^3*d^2 - (2*A*b^3*c^2 + 2*B*a*b^2*c*d + A*a*b^2*d^2)*x^3)*sqrt(b*x^2 + a)/(a^2*b^4*x^4 + 2*a^3*b^3*x^2 + a^4*b^2)
\[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (c + d x\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((B*x+A)*(d*x+c)**2/(b*x**2+a)**(5/2),x)
Output:
Integral((A + B*x)*(c + d*x)**2/(a + b*x**2)**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (69) = 138\).
Time = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.19 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {B d^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, A c^{2} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A c^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B c^{2}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {2 \, A c d}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {2 \, B a d^{2}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {{\left (2 \, B c d + A d^{2}\right )} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {{\left (2 \, B c d + A d^{2}\right )} x}{3 \, \sqrt {b x^{2} + a} a b} \] Input:
integrate((B*x+A)*(d*x+c)^2/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
-B*d^2*x^2/((b*x^2 + a)^(3/2)*b) + 2/3*A*c^2*x/(sqrt(b*x^2 + a)*a^2) + 1/3 *A*c^2*x/((b*x^2 + a)^(3/2)*a) - 1/3*B*c^2/((b*x^2 + a)^(3/2)*b) - 2/3*A*c *d/((b*x^2 + a)^(3/2)*b) - 2/3*B*a*d^2/((b*x^2 + a)^(3/2)*b^2) - 1/3*(2*B* c*d + A*d^2)*x/((b*x^2 + a)^(3/2)*b) + 1/3*(2*B*c*d + A*d^2)*x/(sqrt(b*x^2 + a)*a*b)
Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (\frac {3 \, A c^{2}}{a} - {\left (\frac {3 \, B d^{2}}{b} - \frac {{\left (2 \, A b^{3} c^{2} + 2 \, B a b^{2} c d + A a b^{2} d^{2}\right )} x}{a^{2} b^{2}}\right )} x\right )} x - \frac {B a^{2} b c^{2} + 2 \, A a^{2} b c d + 2 \, B a^{3} d^{2}}{a^{2} b^{2}}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \] Input:
integrate((B*x+A)*(d*x+c)^2/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
1/3*((3*A*c^2/a - (3*B*d^2/b - (2*A*b^3*c^2 + 2*B*a*b^2*c*d + A*a*b^2*d^2) *x/(a^2*b^2))*x)*x - (B*a^2*b*c^2 + 2*A*a^2*b*c*d + 2*B*a^3*d^2)/(a^2*b^2) )/(b*x^2 + a)^(3/2)
Time = 6.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {2\,B\,a^3\,d^2+B\,a^2\,b\,c^2+2\,A\,a^2\,b\,c\,d+3\,B\,a^2\,b\,d^2\,x^2-3\,A\,a\,b^2\,c^2\,x-2\,B\,a\,b^2\,c\,d\,x^3-A\,a\,b^2\,d^2\,x^3-2\,A\,b^3\,c^2\,x^3}{3\,a^2\,b^2\,{\left (b\,x^2+a\right )}^{3/2}} \] Input:
int(((A + B*x)*(c + d*x)^2)/(a + b*x^2)^(5/2),x)
Output:
-(2*B*a^3*d^2 + B*a^2*b*c^2 - 2*A*b^3*c^2*x^3 - A*a*b^2*d^2*x^3 + 3*B*a^2* b*d^2*x^2 + 2*A*a^2*b*c*d - 3*A*a*b^2*c^2*x - 2*B*a*b^2*c*d*x^3)/(3*a^2*b^ 2*(a + b*x^2)^(3/2))
Time = 0.26 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.66 \[ \int \frac {(A+B x) (c+d x)^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a^{2} b c d -2 \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{2}+3 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x -\sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2}+\sqrt {b \,x^{2}+a}\, a \,b^{2} d^{2} x^{3}-3 \sqrt {b \,x^{2}+a}\, a \,b^{2} d^{2} x^{2}+2 \sqrt {b \,x^{2}+a}\, b^{3} c^{2} x^{3}+2 \sqrt {b \,x^{2}+a}\, b^{3} c d \,x^{3}+\sqrt {b}\, a^{3} d^{2}-2 \sqrt {b}\, a^{2} b \,c^{2}+2 \sqrt {b}\, a^{2} b c d +2 \sqrt {b}\, a^{2} b \,d^{2} x^{2}-4 \sqrt {b}\, a \,b^{2} c^{2} x^{2}+4 \sqrt {b}\, a \,b^{2} c d \,x^{2}+\sqrt {b}\, a \,b^{2} d^{2} x^{4}-2 \sqrt {b}\, b^{3} c^{2} x^{4}+2 \sqrt {b}\, b^{3} c d \,x^{4}}{3 a \,b^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((B*x+A)*(d*x+c)^2/(b*x^2+a)^(5/2),x)
Output:
( - 2*sqrt(a + b*x**2)*a**2*b*c*d - 2*sqrt(a + b*x**2)*a**2*b*d**2 + 3*sqr t(a + b*x**2)*a*b**2*c**2*x - sqrt(a + b*x**2)*a*b**2*c**2 + sqrt(a + b*x* *2)*a*b**2*d**2*x**3 - 3*sqrt(a + b*x**2)*a*b**2*d**2*x**2 + 2*sqrt(a + b* x**2)*b**3*c**2*x**3 + 2*sqrt(a + b*x**2)*b**3*c*d*x**3 + sqrt(b)*a**3*d** 2 - 2*sqrt(b)*a**2*b*c**2 + 2*sqrt(b)*a**2*b*c*d + 2*sqrt(b)*a**2*b*d**2*x **2 - 4*sqrt(b)*a*b**2*c**2*x**2 + 4*sqrt(b)*a*b**2*c*d*x**2 + sqrt(b)*a*b **2*d**2*x**4 - 2*sqrt(b)*b**3*c**2*x**4 + 2*sqrt(b)*b**3*c*d*x**4)/(3*a*b **2*(a**2 + 2*a*b*x**2 + b**2*x**4))