Integrand size = 19, antiderivative size = 116 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}} \] Output:
-1/4*(-a*d+b*c)^2*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^( 3/2)/d^(3/2)+1/2*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b+1/4*(-a*d+b*c)*(b*x+a)^(1/2 )*(d*x+c)^(1/2)/b/d
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (b c+a d+2 b d x)}{4 b d}-\frac {(b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{3/2} d^{3/2}} \] Input:
Integrate[Sqrt[a + b*x]*Sqrt[c + d*x],x]
Output:
(Sqrt[a + b*x]*Sqrt[c + d*x]*(b*c + a*d + 2*b*d*x))/(4*b*d) - ((b*c - a*d) ^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(3/2)*d^ (3/2))
Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {60, 60, 66, 221}
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.
\(\displaystyle \int \sqrt {a+b x} \sqrt {c+d x} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 b}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}\) |
Input:
Int[Sqrt[a + b*x]*Sqrt[c + d*x],x]
Output:
((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*b) + ((b*c - a*d)*((Sqrt[a + b*x]*Sqrt[ c + d*x])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(4*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}}}{2 d}-\frac {\left (-a d +b c \right ) \left (\frac {\sqrt {d x +c}\, \sqrt {b x +a}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 d}\) | \(140\) |
Input:
int((b*x+a)^(1/2)*(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/d*(b*x+a)^(1/2)*(d*x+c)^(3/2)-1/4*(-a*d+b*c)/d*(1/b*(d*x+c)^(1/2)*(b*x +a)^(1/2)-1/2*(a*d-b*c)/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(1 /2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(b*d)^(1/2)+(b*d*x^2+(a*d+b*c)*x+a*c)^(1/2) )/(b*d)^(1/2))
Time = 0.09 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.59 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2} d^{2}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2} d^{2}}\right ] \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="fricas")
Output:
[1/16*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c ^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a) *sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d^2*x + b^2*c*d + a*b *d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^2), 1/8*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(2* b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^2)]
\[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\int \sqrt {a + b x} \sqrt {c + d x}\, dx \] Input:
integrate((b*x+a)**(1/2)*(d*x+c)**(1/2),x)
Output:
Integral(sqrt(a + b*x)*sqrt(c + d*x), x)
Exception generated. \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (90) = 180\).
Time = 0.15 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.00 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=-\frac {\frac {4 \, {\left (\frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} a {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} {\left | b \right |}}{b^{2}}}{4 \, b} \] Input:
integrate((b*x+a)^(1/2)*(d*x+c)^(1/2),x, algorithm="giac")
Output:
-1/4*(4*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + ( b*x + a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sq rt(b*x + a))*a*abs(b)/b^2 - (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*b*x + 2*a + (b*c*d - 5*a*d^2)/d^2)*sqrt(b*x + a) + (b^3*c^2 + 2*a*b^2*c*d - 3*a^ 2*b*d^2)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a *b*d)))/(sqrt(b*d)*d))*abs(b)/b^2)/b
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\left (\frac {x}{2}+\frac {a\,d+b\,c}{4\,b\,d}\right )\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}-\frac {\ln \left (a\,d+b\,c+2\,b\,d\,x+2\,\sqrt {b}\,\sqrt {d}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}\right )\,{\left (a\,d-b\,c\right )}^2}{8\,b^{3/2}\,d^{3/2}} \] Input:
int((a + b*x)^(1/2)*(c + d*x)^(1/2),x)
Output:
(x/2 + (a*d + b*c)/(4*b*d))*(a + b*x)^(1/2)*(c + d*x)^(1/2) - (log(a*d + b *c + 2*b*d*x + 2*b^(1/2)*d^(1/2)*(a + b*x)^(1/2)*(c + d*x)^(1/2))*(a*d - b *c)^2)/(8*b^(3/2)*d^(3/2))
Time = 0.15 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.69 \[ \int \sqrt {a+b x} \sqrt {c+d x} \, dx=\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{2}+\sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c d +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} d^{2} x -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{2}+2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c d -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2}}{4 b^{2} d^{2}} \] Input:
int((b*x+a)^(1/2)*(d*x+c)^(1/2),x)
Output:
(sqrt(c + d*x)*sqrt(a + b*x)*a*b*d**2 + sqrt(c + d*x)*sqrt(a + b*x)*b**2*c *d + 2*sqrt(c + d*x)*sqrt(a + b*x)*b**2*d**2*x - sqrt(d)*sqrt(b)*log((sqrt (d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*d**2 + 2* sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a *d - b*c))*a*b*c*d - sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)* sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**2)/(4*b**2*d**2)