Integrand size = 26, antiderivative size = 119 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} f}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d e-c f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {c+d x}}\right )}{f \sqrt {b e-a f}} \] Output:
2*d^(1/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/f-2 *(-c*f+d*e)^(1/2)*arctanh((-c*f+d*e)^(1/2)*(b*x+a)^(1/2)/(-a*f+b*e)^(1/2)/ (d*x+c)^(1/2))/f/(-a*f+b*e)^(1/2)
Time = 0.49 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\frac {2 \left (-\frac {\sqrt {b} \sqrt {-d e+c f} \arctan \left (\frac {\sqrt {d} \left (-\sqrt {\frac {b}{d}} f \sqrt {a+b x} \sqrt {c+d x}+b (e+f x)\right )}{\sqrt {b} \sqrt {b e-a f} \sqrt {-d e+c f}}\right )}{\sqrt {b e-a f}}-\sqrt {d} \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )\right )}{\sqrt {\frac {b}{d}} \sqrt {d} f} \] Input:
Integrate[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]
Output:
(2*(-((Sqrt[b]*Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[d]*(-(Sqrt[b/d]*f*Sqrt[a + b*x]*Sqrt[c + d*x]) + b*(e + f*x)))/(Sqrt[b]*Sqrt[b*e - a*f]*Sqrt[-(d*e) + c*f])])/Sqrt[b*e - a*f]) - Sqrt[d]*Log[Sqrt[a + b*x] - Sqrt[b/d]*Sqrt[c + d*x]]))/(Sqrt[b/d]*Sqrt[d]*f)
Time = 0.23 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {140, 27, 66, 104, 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.
\(\displaystyle \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx\) |
\(\Big \downarrow \) 140 |
\(\displaystyle \int \frac {c-\frac {d e}{f}}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)}dx+\frac {d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \left (c-\frac {d e}{f}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)}dx+\frac {d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \left (c-\frac {d e}{f}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} (e+f x)}dx+\frac {2 d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{f}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle 2 \left (c-\frac {d e}{f}\right ) \int \frac {1}{b e-a f-\frac {(d e-c f) (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+\frac {2 d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \left (c-\frac {d e}{f}\right ) \text {arctanh}\left (\frac {\sqrt {a+b x} \sqrt {d e-c f}}{\sqrt {c+d x} \sqrt {b e-a f}}\right )}{\sqrt {b e-a f} \sqrt {d e-c f}}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} f}\) |
Input:
Int[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]
Output:
(2*Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt [b]*f) + (2*(c - (d*e)/f)*ArcTanh[(Sqrt[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b* e - a*f]*Sqrt[c + d*x])])/(Sqrt[b*e - a*f]*Sqrt[d*e - c*f])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(95)=190\).
Time = 0.33 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.52
method | result | size |
default | \(\frac {\left (\sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) d f -\ln \left (\frac {a d f x +b c f x -2 b d e x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, f +2 a c f -a d e -b c e}{f x +e}\right ) \sqrt {d b}\, c f +\ln \left (\frac {a d f x +b c f x -2 b d e x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, f +2 a c f -a d e -b c e}{f x +e}\right ) \sqrt {d b}\, d e \right ) \sqrt {b x +a}\, \sqrt {x d +c}}{\sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, \sqrt {d b}\, f^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}}\) | \(300\) |
Input:
int((d*x+c)^(1/2)/(b*x+a)^(1/2)/(f*x+e),x,method=_RETURNVERBOSE)
Output:
(((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2) *(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*d*f-ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b *x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e )/(f*x+e))*(d*b)^(1/2)*c*f+ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c ))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*( d*b)^(1/2)*d*e)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/((c*f-d*e)*(a*f-b*e)/f^2)^(1/2 )/(d*b)^(1/2)/f^2/((b*x+a)*(d*x+c))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (95) = 190\).
Time = 3.26 (sec) , antiderivative size = 1358, normalized size of antiderivative = 11.41 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(1/2)/(b*x+a)^(1/2)/(f*x+e),x, algorithm="fricas")
Output:
[1/2*(sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b ^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + sqrt((d*e - c*f)/(b*e - a*f))*log((8*a^2*c^2*f^2 + (b^2*c ^2 + 6*a*b*c*d + a^2*d^2)*e^2 - 8*(a*b*c^2 + a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*f^2)*x^2 - 4*(2*a^2*c*f^2 + (b^2*c + a*b*d)*e^2 - (3*a*b*c + a^2*d)*e*f + (2*b^2*d*e^ 2 - (b^2*c + 3*a*b*d)*e*f + (a*b*c + a^2*d)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt((d*e - c*f)/(b*e - a*f)) + 2*(4*(b^2*c*d + a*b*d^2)*e^2 - (3*b^ 2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*e*f + 4*(a*b*c^2 + a^2*c*d)*f^2)*x)/(f^2*x ^2 + 2*e*f*x + e^2)))/f, 1/2*(2*sqrt(-(d*e - c*f)/(b*e - a*f))*arctan(-1/2 *(2*a*c*f - (b*c + a*d)*e - (2*b*d*e - (b*c + a*d)*f)*x)*sqrt(b*x + a)*sqr t(d*x + c)*sqrt(-(d*e - c*f)/(b*e - a*f))/(a*c*d*e - a*c^2*f + (b*d^2*e - b*c*d*f)*x^2 + ((b*c*d + a*d^2)*e - (b*c^2 + a*c*d)*f)*x)) + sqrt(d/b)*log (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a* b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x))/f, -1/2*(2*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) - sqrt((d*e - c* f)/(b*e - a*f))*log((8*a^2*c^2*f^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e^2 - 8*(a*b*c^2 + a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*f^2)*x^2 - 4*(2*a^2*c*f^2 + (b^2*c + a*...
\[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\int \frac {\sqrt {c + d x}}{\sqrt {a + b x} \left (e + f x\right )}\, dx \] Input:
integrate((d*x+c)**(1/2)/(b*x+a)**(1/2)/(f*x+e),x)
Output:
Integral(sqrt(c + d*x)/(sqrt(a + b*x)*(e + f*x)), x)
Exception generated. \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(1/2)/(b*x+a)^(1/2)/(f*x+e),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)/f>0)', see `assume?` for more detai
Exception generated. \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x+c)^(1/2)/(b*x+a)^(1/2)/(f*x+e),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Time = 65.05 (sec) , antiderivative size = 39144, normalized size of antiderivative = 328.94 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\text {Too large to display} \] Input:
int((c + d*x)^(1/2)/((e + f*x)*(a + b*x)^(1/2)),x)
Output:
- (d^(1/2)*atan(((d^(1/2)*((65536*(24*a^(1/2)*b^8*c^(7/2)*d^5*e^4 - 24*a^( 3/2)*b^7*c^(5/2)*d^6*e^4 - 24*a^(5/2)*b^6*c^(3/2)*d^7*e^4 + 24*a^(7/2)*b^5 *c^(1/2)*d^8*e^4 - 40*a^(3/2)*b^7*c^(13/2)*d^2*f^4 + 8*a^(5/2)*b^6*c^(11/2 )*d^3*f^4 + 24*a^(7/2)*b^5*c^(9/2)*d^4*f^4 - 32*a^(9/2)*b^4*c^(7/2)*d^5*f^ 4 + 16*a^(11/2)*b^3*c^(5/2)*d^6*f^4 + 24*a^(1/2)*b^8*c^(15/2)*d*f^4 - 108* a^(1/2)*b^8*c^(9/2)*d^4*e^3*f + 176*a^(3/2)*b^7*c^(7/2)*d^5*e^3*f - 72*a^( 5/2)*b^6*c^(5/2)*d^6*e^3*f + 48*a^(7/2)*b^5*c^(3/2)*d^7*e^3*f - 44*a^(9/2) *b^4*c^(1/2)*d^8*e^3*f - 108*a^(1/2)*b^8*c^(13/2)*d^2*e*f^3 + 208*a^(3/2)* b^7*c^(11/2)*d^3*e*f^3 - 136*a^(5/2)*b^6*c^(9/2)*d^4*e*f^3 + 48*a^(7/2)*b^ 5*c^(7/2)*d^5*e*f^3 + 20*a^(9/2)*b^4*c^(5/2)*d^6*e*f^3 - 32*a^(11/2)*b^3*c ^(3/2)*d^7*e*f^3 + 168*a^(1/2)*b^8*c^(11/2)*d^3*e^2*f^2 - 320*a^(3/2)*b^7* c^(9/2)*d^4*e^2*f^2 + 224*a^(5/2)*b^6*c^(7/2)*d^5*e^2*f^2 - 144*a^(7/2)*b^ 5*c^(5/2)*d^6*e^2*f^2 + 56*a^(9/2)*b^4*c^(3/2)*d^7*e^2*f^2 + 16*a^(11/2)*b ^3*c^(1/2)*d^8*e^2*f^2))/(d^17*e^8) + (2*d^(1/2)*((2*d^(1/2)*((65536*(44*a ^(5/2)*b^7*c^(11/2)*d^2*f^6 - 4*a^(1/2)*b^9*c^(15/2)*f^6 - 4*a^(7/2)*b^6*c ^(9/2)*d^3*f^6 - 24*a^(9/2)*b^5*c^(7/2)*d^4*f^6 + 28*a^(11/2)*b^4*c^(5/2)* d^5*f^6 - 16*a^(13/2)*b^3*c^(3/2)*d^6*f^6 - 24*a^(3/2)*b^8*c^(13/2)*d*f^6 + 51*a^(1/2)*b^9*c^(13/2)*d*e*f^5 + 16*a^(3/2)*b^8*c^(11/2)*d^2*e*f^5 - 15 0*a^(5/2)*b^7*c^(9/2)*d^3*e*f^5 + 116*a^(7/2)*b^6*c^(7/2)*d^4*e*f^5 - 85*a ^(9/2)*b^5*c^(5/2)*d^5*e*f^5 + 36*a^(11/2)*b^4*c^(3/2)*d^6*e*f^5 + 16*a...
Time = 0.24 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx=\frac {\sqrt {c f -d e}\, \sqrt {a f -b e}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {b}\, \sqrt {c f -d e}\, \sqrt {a f -b e}+a d f +b c f -2 b d e}+\sqrt {f}\, \sqrt {d}\, \sqrt {b x +a}+\sqrt {f}\, \sqrt {b}\, \sqrt {d x +c}\right ) b +\sqrt {c f -d e}\, \sqrt {a f -b e}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {b}\, \sqrt {c f -d e}\, \sqrt {a f -b e}+a d f +b c f -2 b d e}+\sqrt {f}\, \sqrt {d}\, \sqrt {b x +a}+\sqrt {f}\, \sqrt {b}\, \sqrt {d x +c}\right ) b -\sqrt {c f -d e}\, \sqrt {a f -b e}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {c f -d e}\, \sqrt {a f -b e}+2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}\, f +2 b d e +2 b d f x \right ) b +2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a f -2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b e}{b f \left (a f -b e \right )} \] Input:
int((d*x+c)^(1/2)/(b*x+a)^(1/2)/(f*x+e),x)
Output:
(sqrt(c*f - d*e)*sqrt(a*f - b*e)*log( - sqrt(2*sqrt(d)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c*f - 2*b*d*e) + sqrt(f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d*x))*b + sqrt(c*f - d*e)*sqrt(a*f - b*e) *log(sqrt(2*sqrt(d)*sqrt(b)*sqrt(c*f - d*e)*sqrt(a*f - b*e) + a*d*f + b*c* f - 2*b*d*e) + sqrt(f)*sqrt(d)*sqrt(a + b*x) + sqrt(f)*sqrt(b)*sqrt(c + d* x))*b - sqrt(c*f - d*e)*sqrt(a*f - b*e)*log(2*sqrt(d)*sqrt(b)*sqrt(c*f - d *e)*sqrt(a*f - b*e) + 2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x)*f + 2* b*d*e + 2*b*d*f*x)*b + 2*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt (b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*f - 2*sqrt(d)*sqrt(b)*log((sqrt(d)*s qrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b*e)/(b*f*(a*f - b* e))