Integrand size = 22, antiderivative size = 72 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {(2 (B c+A d)+B d x) \sqrt {a+b x^2}}{2 b}+\frac {(2 A b c-a B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \] Output:
1/2*(B*d*x+2*A*d+2*B*c)*(b*x^2+a)^(1/2)/b+1/2*(2*A*b*c-B*a*d)*arctanh(b^(1 /2)*x/(b*x^2+a)^(1/2))/b^(3/2)
Time = 0.59 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {(2 B c+2 A d+B d x) \sqrt {a+b x^2}}{2 b}+\frac {(-2 A b c+a B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{3/2}} \] Input:
Integrate[((A + B*x)*(c + d*x))/Sqrt[a + b*x^2],x]
Output:
((2*B*c + 2*A*d + B*d*x)*Sqrt[a + b*x^2])/(2*b) + ((-2*A*b*c + a*B*d)*Log[ -(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*b^(3/2))
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {676, 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.
\(\displaystyle \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {(2 A b c-a B d) \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}+\frac {\sqrt {a+b x^2} (A d+B c)}{b}+\frac {B d x \sqrt {a+b x^2}}{2 b}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {(2 A b c-a B d) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}+\frac {\sqrt {a+b x^2} (A d+B c)}{b}+\frac {B d x \sqrt {a+b x^2}}{2 b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 A b c-a B d)}{2 b^{3/2}}+\frac {\sqrt {a+b x^2} (A d+B c)}{b}+\frac {B d x \sqrt {a+b x^2}}{2 b}\) |
Input:
Int[((A + B*x)*(c + d*x))/Sqrt[a + b*x^2],x]
Output:
((B*c + A*d)*Sqrt[a + b*x^2])/b + (B*d*x*Sqrt[a + b*x^2])/(2*b) + ((2*A*b* c - a*B*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(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 _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Time = 0.81 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {\left (B d x +2 A d +2 B c \right ) \sqrt {b \,x^{2}+a}}{2 b}+\frac {\left (2 A b c -B a d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\) | \(61\) |
default | \(\frac {A c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {\left (A d +B c \right ) \sqrt {b \,x^{2}+a}}{b}+B d \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\) | \(85\) |
Input:
int((B*x+A)*(d*x+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(B*d*x+2*A*d+2*B*c)*(b*x^2+a)^(1/2)/b+1/2*(2*A*b*c-B*a*d)/b^(3/2)*ln(b ^(1/2)*x+(b*x^2+a)^(1/2))
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.97 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\left [-\frac {{\left (2 \, A b c - B a d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (B b d x + 2 \, B b c + 2 \, A b d\right )} \sqrt {b x^{2} + a}}{4 \, b^{2}}, -\frac {{\left (2 \, A b c - B a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (B b d x + 2 \, B b c + 2 \, A b d\right )} \sqrt {b x^{2} + a}}{2 \, b^{2}}\right ] \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
[-1/4*((2*A*b*c - B*a*d)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)* x - a) - 2*(B*b*d*x + 2*B*b*c + 2*A*b*d)*sqrt(b*x^2 + a))/b^2, -1/2*((2*A* b*c - B*a*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (B*b*d*x + 2*B* b*c + 2*A*b*d)*sqrt(b*x^2 + a))/b^2]
Time = 0.56 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {B d x}{2 b} + \frac {A d + B c}{b}\right ) + \left (A c - \frac {B a d}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {A c x + \frac {B d x^{3}}{3} + \frac {x^{2} \left (A d + B c\right )}{2}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x**2+a)**(1/2),x)
Output:
Piecewise((sqrt(a + b*x**2)*(B*d*x/(2*b) + (A*d + B*c)/b) + (A*c - B*a*d/( 2*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0) ), (x*log(x)/sqrt(b*x**2), True)), Ne(b, 0)), ((A*c*x + B*d*x**3/3 + x**2* (A*d + B*c)/2)/sqrt(a), True))
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.11 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} B d x}{2 \, b} + \frac {A c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {B a d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a} B c}{b} + \frac {\sqrt {b x^{2} + a} A d}{b} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
1/2*sqrt(b*x^2 + a)*B*d*x/b + A*c*arcsinh(b*x/sqrt(a*b))/sqrt(b) - 1/2*B*a *d*arcsinh(b*x/sqrt(a*b))/b^(3/2) + sqrt(b*x^2 + a)*B*c/b + sqrt(b*x^2 + a )*A*d/b
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} {\left (\frac {B d x}{b} + \frac {2 \, {\left (B b c + A b d\right )}}{b^{2}}\right )} - \frac {{\left (2 \, A b c - B a d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}}} \] Input:
integrate((B*x+A)*(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
1/2*sqrt(b*x^2 + a)*(B*d*x/b + 2*(B*b*c + A*b*d)/b^2) - 1/2*(2*A*b*c - B*a *d)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2)
Time = 6.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.89 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\left \{\begin {array}{cl} \frac {6\,A\,c\,x+3\,A\,d\,x^2+3\,B\,c\,x^2+2\,B\,d\,x^3}{6\,\sqrt {a}} & \text {\ if\ \ }b=0\\ \frac {A\,d\,\sqrt {b\,x^2+a}}{b}+\frac {B\,c\,\sqrt {b\,x^2+a}}{b}+\frac {A\,c\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}}-\frac {B\,a\,d\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{2\,b^{3/2}}+\frac {B\,d\,x\,\sqrt {b\,x^2+a}}{2\,b} & \text {\ if\ \ }b\neq 0 \end {array}\right . \] Input:
int(((A + B*x)*(c + d*x))/(a + b*x^2)^(1/2),x)
Output:
piecewise(b == 0, (6*A*c*x + 3*A*d*x^2 + 3*B*c*x^2 + 2*B*d*x^3)/(6*a^(1/2) ), b ~= 0, (A*d*(a + b*x^2)^(1/2))/b + (B*c*(a + b*x^2)^(1/2))/b + (A*c*lo g(b^(1/2)*x + (a + b*x^2)^(1/2)))/b^(1/2) - (B*a*d*log(2*b^(1/2)*x + 2*(a + b*x^2)^(1/2)))/(2*b^(3/2)) + (B*d*x*(a + b*x^2)^(1/2))/(2*b))
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {2 \sqrt {b \,x^{2}+a}\, a d +2 \sqrt {b \,x^{2}+a}\, b c +\sqrt {b \,x^{2}+a}\, b d x +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a c -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a d}{2 b} \] Input:
int((B*x+A)*(d*x+c)/(b*x^2+a)^(1/2),x)
Output:
(2*sqrt(a + b*x**2)*a*d + 2*sqrt(a + b*x**2)*b*c + sqrt(a + b*x**2)*b*d*x + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*c - sqrt(b)*log( (sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*d)/(2*b)