Integrand size = 22, antiderivative size = 70 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {a (B c+A d)-(A b c-a B d) x}{a b \sqrt {a+b x^2}}+\frac {B d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}} \] Output:
-(a*(A*d+B*c)-(A*b*c-B*a*d)*x)/a/b/(b*x^2+a)^(1/2)+B*d*arctanh(b^(1/2)*x/( b*x^2+a)^(1/2))/b^(3/2)
Time = 0.46 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-a B c-a A d+A b c x-a B d x}{a b \sqrt {a+b x^2}}-\frac {B d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}} \] Input:
Integrate[((A + B*x)*(c + d*x))/(a + b*x^2)^(3/2),x]
Output:
(-(a*B*c) - a*A*d + A*b*c*x - a*B*d*x)/(a*b*Sqrt[a + b*x^2]) - (B*d*Log[-( Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(3/2)
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {675, 224, 219}
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.
\(\displaystyle \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 675 |
\(\displaystyle \frac {B d \int \frac {1}{\sqrt {b x^2+a}}dx}{b}-\frac {A d+B c}{b \sqrt {a+b x^2}}+\frac {x (A b c-a B d)}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {B d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b}-\frac {A d+B c}{b \sqrt {a+b x^2}}+\frac {x (A b c-a B d)}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {A d+B c}{b \sqrt {a+b x^2}}+\frac {x (A b c-a B d)}{a b \sqrt {a+b x^2}}+\frac {B d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}\) |
Input:
Int[((A + B*x)*(c + d*x))/(a + b*x^2)^(3/2),x]
Output:
-((B*c + A*d)/(b*Sqrt[a + b*x^2])) + ((A*b*c - a*B*d)*x)/(a*b*Sqrt[a + b*x ^2]) + (B*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(3/2)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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 = 0.80 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {A c x}{\sqrt {b \,x^{2}+a}\, a}-\frac {A d +B c}{b \sqrt {b \,x^{2}+a}}+B d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(78\) |
Input:
int((B*x+A)*(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
A*c/(b*x^2+a)^(1/2)/a*x-(A*d+B*c)/b/(b*x^2+a)^(1/2)+B*d*(-x/b/(b*x^2+a)^(1 /2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.87 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (B a b d x^{2} + B a^{2} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (B a b c + A a b d - {\left (A b^{2} c - B a b d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, -\frac {{\left (B a b d x^{2} + B a^{2} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (B a b c + A a b d - {\left (A b^{2} c - B a b d\right )} x\right )} \sqrt {b x^{2} + a}}{a b^{3} x^{2} + a^{2} b^{2}}\right ] \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((B*a*b*d*x^2 + B*a^2*d)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqr t(b)*x - a) - 2*(B*a*b*c + A*a*b*d - (A*b^2*c - B*a*b*d)*x)*sqrt(b*x^2 + a ))/(a*b^3*x^2 + a^2*b^2), -((B*a*b*d*x^2 + B*a^2*d)*sqrt(-b)*arctan(sqrt(- b)*x/sqrt(b*x^2 + a)) + (B*a*b*c + A*a*b*d - (A*b^2*c - B*a*b*d)*x)*sqrt(b *x^2 + a))/(a*b^3*x^2 + a^2*b^2)]
Time = 4.16 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=A d \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {A c x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + B c \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + B d \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:
integrate((B*x+A)*(d*x+c)/(b*x**2+a)**(3/2),x)
Output:
A*d*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), Tru e)) + A*c*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + B*c*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)) + B*d*(asinh(sqrt(b)*x/sqr t(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a)))
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.16 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A c x}{\sqrt {b x^{2} + a} a} - \frac {B d x}{\sqrt {b x^{2} + a} b} + \frac {B d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {B c}{\sqrt {b x^{2} + a} b} - \frac {A d}{\sqrt {b x^{2} + a} b} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
A*c*x/(sqrt(b*x^2 + a)*a) - B*d*x/(sqrt(b*x^2 + a)*b) + B*d*arcsinh(b*x/sq rt(a*b))/b^(3/2) - B*c/(sqrt(b*x^2 + a)*b) - A*d/(sqrt(b*x^2 + a)*b)
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {B d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} + \frac {\frac {{\left (A b^{2} c - B a b d\right )} x}{a b^{2}} - \frac {B a b c + A a b d}{a b^{2}}}{\sqrt {b x^{2} + a}} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
-B*d*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + ((A*b^2*c - B*a*b*d) *x/(a*b^2) - (B*a*b*c + A*a*b*d)/(a*b^2))/sqrt(b*x^2 + a)
Time = 6.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B\,d\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{b^{3/2}}-\frac {B\,c}{b\,\sqrt {b\,x^2+a}}-\frac {\frac {A\,d}{b}-\frac {A\,c\,x}{a}}{\sqrt {b\,x^2+a}}-\frac {B\,d\,x}{b\,\sqrt {b\,x^2+a}} \] Input:
int(((A + B*x)*(c + d*x))/(a + b*x^2)^(3/2),x)
Output:
(B*d*log(b^(1/2)*x + (a + b*x^2)^(1/2)))/b^(3/2) - (B*c)/(b*(a + b*x^2)^(1 /2)) - ((A*d)/b - (A*c*x)/a)/(a + b*x^2)^(1/2) - (B*d*x)/(b*(a + b*x^2)^(1 /2))
Time = 0.34 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) (c+d x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a d +\sqrt {b \,x^{2}+a}\, b c x -\sqrt {b \,x^{2}+a}\, b c -\sqrt {b \,x^{2}+a}\, b d x +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a d +\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b d \,x^{2}+\sqrt {b}\, a c -\sqrt {b}\, a d +\sqrt {b}\, b c \,x^{2}-\sqrt {b}\, b d \,x^{2}}{b \left (b \,x^{2}+a \right )} \] Input:
int((B*x+A)*(d*x+c)/(b*x^2+a)^(3/2),x)
Output:
( - sqrt(a + b*x**2)*a*d + sqrt(a + b*x**2)*b*c*x - sqrt(a + b*x**2)*b*c - sqrt(a + b*x**2)*b*d*x + sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt( a))*a*d + sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b*d*x**2 + s qrt(b)*a*c - sqrt(b)*a*d + sqrt(b)*b*c*x**2 - sqrt(b)*b*d*x**2)/(b*(a + b* x**2))