Integrand size = 22, antiderivative size = 77 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {a (B c+A d)-(A b c-a B d) x}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {(2 A b c+a B d) x}{3 a^2 b \sqrt {a+b x^2}} \] Output:
-1/3*(a*(A*d+B*c)-(A*b*c-B*a*d)*x)/a/b/(b*x^2+a)^(3/2)+1/3*(2*A*b*c+B*a*d) *x/a^2/b/(b*x^2+a)^(1/2)
Time = 0.72 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-a^2 B c-a^2 A d+3 a A b c x+2 A b^2 c x^3+a b B d x^3}{3 a^2 b \left (a+b x^2\right )^{3/2}} \] Input:
Integrate[((A + B*x)*(c + d*x))/(a + b*x^2)^(5/2),x]
Output:
(-(a^2*B*c) - a^2*A*d + 3*a*A*b*c*x + 2*A*b^2*c*x^3 + a*b*B*d*x^3)/(3*a^2* b*(a + b*x^2)^(3/2))
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.17, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {675, 208}
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.
\(\displaystyle \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 675 |
\(\displaystyle \frac {(a B d+2 A b c) \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}-\frac {A d+B c}{3 b \left (a+b x^2\right )^{3/2}}+\frac {x (A b c-a B d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {x (a B d+2 A b c)}{3 a^2 b \sqrt {a+b x^2}}-\frac {A d+B c}{3 b \left (a+b x^2\right )^{3/2}}+\frac {x (A b c-a B d)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((A + B*x)*(c + d*x))/(a + b*x^2)^(5/2),x]
Output:
-1/3*(B*c + A*d)/(b*(a + b*x^2)^(3/2)) + ((A*b*c - a*B*d)*x)/(3*a*b*(a + b *x^2)^(3/2)) + ((2*A*b*c + a*B*d)*x)/(3*a^2*b*Sqrt[a + b*x^2])
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((d_) + (e_.)*(x_))*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[a*(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + (- Simp[(c*d*f - a*e*g)*x*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Simp[(a* e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)) Int[(a + c*x^2)^(p + 1), x], x]) / ; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && !(IntegerQ[p] && NiceSqrtQ [(-a)*c])
Time = 0.79 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.74
method | result | size |
gosper | \(-\frac {-2 A \,b^{2} c \,x^{3}-a B b d \,x^{3}-3 A a b c x +a^{2} A d +B \,a^{2} c}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b}\) | \(57\) |
trager | \(-\frac {-2 A \,b^{2} c \,x^{3}-a B b d \,x^{3}-3 A a b c x +a^{2} A d +B \,a^{2} c}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b}\) | \(57\) |
orering | \(-\frac {-2 A \,b^{2} c \,x^{3}-a B b d \,x^{3}-3 A a b c x +a^{2} A d +B \,a^{2} c}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b}\) | \(57\) |
default | \(A c \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )-\frac {A d +B c}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+B d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(113\) |
Input:
int((B*x+A)*(d*x+c)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(-2*A*b^2*c*x^3-B*a*b*d*x^3-3*A*a*b*c*x+A*a^2*d+B*a^2*c)/(b*x^2+a)^(3 /2)/a^2/b
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (3 \, A a b c x - B a^{2} c - A a^{2} d + {\left (2 \, A b^{2} c + B a b d\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/3*(3*A*a*b*c*x - B*a^2*c - A*a^2*d + (2*A*b^2*c + B*a*b*d)*x^3)*sqrt(b*x ^2 + a)/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)
Time = 8.73 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.25 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=A c \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + A d \left (\begin {cases} - \frac {1}{3 a b \sqrt {a + b x^{2}} + 3 b^{2} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + B c \left (\begin {cases} - \frac {1}{3 a b \sqrt {a + b x^{2}} + 3 b^{2} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {B d x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x**2+a)**(5/2),x)
Output:
A*c*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x **2/a)) + 2*b*x**3/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt (1 + b*x**2/a))) + A*d*Piecewise((-1/(3*a*b*sqrt(a + b*x**2) + 3*b**2*x**2 *sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(5/2)), True)) + B*c*Piecewise( (-1/(3*a*b*sqrt(a + b*x**2) + 3*b**2*x**2*sqrt(a + b*x**2)), Ne(b, 0)), (x **2/(2*a**(5/2)), True)) + B*d*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a** (3/2)*b*x**2*sqrt(1 + b*x**2/a))
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 \, A c x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A c x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {B d x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {B c}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {A d}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
2/3*A*c*x/(sqrt(b*x^2 + a)*a^2) + 1/3*A*c*x/((b*x^2 + a)^(3/2)*a) - 1/3*B* d*x/((b*x^2 + a)^(3/2)*b) + 1/3*B*d*x/(sqrt(b*x^2 + a)*a*b) - 1/3*B*c/((b* x^2 + a)^(3/2)*b) - 1/3*A*d/((b*x^2 + a)^(3/2)*b)
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (\frac {3 \, A c}{a} + \frac {{\left (2 \, A b^{2} c + B a b d\right )} x^{2}}{a^{2} b}\right )} x - \frac {B a^{2} c + A a^{2} d}{a^{2} b}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
1/3*((3*A*c/a + (2*A*b^2*c + B*a*b*d)*x^2/(a^2*b))*x - (B*a^2*c + A*a^2*d) /(a^2*b))/(b*x^2 + a)^(3/2)
Time = 6.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {a\,\left (B\,b\,d\,x^3+3\,A\,b\,c\,x\right )}{3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {2\,A\,b^2\,c\,x^3}{3\,{\left (b\,x^2+a\right )}^{3/2}}}{a^2\,b}-\frac {A\,d+B\,c}{3\,b\,{\left (b\,x^2+a\right )}^{3/2}} \] Input:
int(((A + B*x)*(c + d*x))/(a + b*x^2)^(5/2),x)
Output:
((a*(B*b*d*x^3 + 3*A*b*c*x))/(3*(a + b*x^2)^(3/2)) + (2*A*b^2*c*x^3)/(3*(a + b*x^2)^(3/2)))/(a^2*b) - (A*d + B*c)/(3*b*(a + b*x^2)^(3/2))
Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.06 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a^{2} d +3 \sqrt {b \,x^{2}+a}\, a b c x -\sqrt {b \,x^{2}+a}\, a b c +2 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{3}+\sqrt {b \,x^{2}+a}\, b^{2} d \,x^{3}-2 \sqrt {b}\, a^{2} c +\sqrt {b}\, a^{2} d -4 \sqrt {b}\, a b c \,x^{2}+2 \sqrt {b}\, a b d \,x^{2}-2 \sqrt {b}\, b^{2} c \,x^{4}+\sqrt {b}\, b^{2} d \,x^{4}}{3 a b \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((B*x+A)*(d*x+c)/(b*x^2+a)^(5/2),x)
Output:
( - sqrt(a + b*x**2)*a**2*d + 3*sqrt(a + b*x**2)*a*b*c*x - sqrt(a + b*x**2 )*a*b*c + 2*sqrt(a + b*x**2)*b**2*c*x**3 + sqrt(a + b*x**2)*b**2*d*x**3 - 2*sqrt(b)*a**2*c + sqrt(b)*a**2*d - 4*sqrt(b)*a*b*c*x**2 + 2*sqrt(b)*a*b*d *x**2 - 2*sqrt(b)*b**2*c*x**4 + sqrt(b)*b**2*d*x**4)/(3*a*b*(a**2 + 2*a*b* x**2 + b**2*x**4))