Integrand size = 33, antiderivative size = 107 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2 B \sqrt {d+e x}}{b^2}-\frac {(A b-a B) \sqrt {d+e x}}{b^2 (a+b x)}-\frac {(2 b B d+A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {b d-a e}} \] Output:
2*B*(e*x+d)^(1/2)/b^2-(A*b-B*a)*(e*x+d)^(1/2)/b^2/(b*x+a)-(A*b*e-3*B*a*e+2 *B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(5/2)/(-a*e+b*d) ^(1/2)
Time = 0.53 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {(-A b+3 a B+2 b B x) \sqrt {d+e x}}{b^2 (a+b x)}+\frac {(2 b B d+A b e-3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{5/2} \sqrt {-b d+a e}} \] Input:
Integrate[((A + B*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2),x]
Output:
((-(A*b) + 3*a*B + 2*b*B*x)*Sqrt[d + e*x])/(b^2*(a + b*x)) + ((2*b*B*d + A *b*e - 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(b^(5/ 2)*Sqrt[-(b*d) + a*e])
Time = 0.50 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1184, 27, 87, 60, 73, 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.
\(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^2 \int \frac {(A+B x) \sqrt {d+e x}}{b^2 (a+b x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^2}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \int \frac {\sqrt {d+e x}}{a+b x}dx}{2 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
Input:
Int[((A + B*x)*Sqrt[d + e*x])/(a^2 + 2*a*b*x + b^2*x^2),x]
Output:
-(((A*b - a*B)*(d + e*x)^(3/2))/(b*(b*d - a*e)*(a + b*x))) + ((2*b*B*d + A *b*e - 3*a*B*e)*((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b] *Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(3/2)))/(2*b*(b*d - a*e))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.52 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {-\left (\left (-2 B x +A \right ) b -3 B a \right ) \sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}+\left (b \left (A e +2 B d \right )-3 B a e \right ) \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{\sqrt {b \left (a e -b d \right )}\, b^{2} \left (b x +a \right )}\) | \(103\) |
risch | \(\frac {2 B \sqrt {e x +d}}{b^{2}}+\frac {\frac {2 \left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (A b e -3 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{\sqrt {b \left (a e -b d \right )}}}{b^{2}}\) | \(107\) |
derivativedivides | \(\frac {2 B \sqrt {e x +d}}{b^{2}}+\frac {\frac {2 \left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (A b e -3 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{\sqrt {b \left (a e -b d \right )}}}{b^{2}}\) | \(108\) |
default | \(\frac {2 B \sqrt {e x +d}}{b^{2}}+\frac {\frac {2 \left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (A b e -3 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{\sqrt {b \left (a e -b d \right )}}}{b^{2}}\) | \(108\) |
Input:
int((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)
Output:
(-((-2*B*x+A)*b-3*B*a)*(e*x+d)^(1/2)*(b*(a*e-b*d))^(1/2)+(b*(A*e+2*B*d)-3* B*a*e)*(b*x+a)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2)))/(b*(a*e-b*d))^ (1/2)/b^2/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (92) = 184\).
Time = 0.09 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.67 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\left [\frac {{\left (2 \, B a b d - {\left (3 \, B a^{2} - A a b\right )} e + {\left (2 \, B b^{2} d - {\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} d - {\left (3 \, B a^{2} b - A a b^{2}\right )} e + 2 \, {\left (B b^{3} d - B a b^{2} e\right )} x\right )} \sqrt {e x + d}}{2 \, {\left (a b^{4} d - a^{2} b^{3} e + {\left (b^{5} d - a b^{4} e\right )} x\right )}}, \frac {{\left (2 \, B a b d - {\left (3 \, B a^{2} - A a b\right )} e + {\left (2 \, B b^{2} d - {\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} d - {\left (3 \, B a^{2} b - A a b^{2}\right )} e + 2 \, {\left (B b^{3} d - B a b^{2} e\right )} x\right )} \sqrt {e x + d}}{a b^{4} d - a^{2} b^{3} e + {\left (b^{5} d - a b^{4} e\right )} x}\right ] \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas ")
Output:
[1/2*((2*B*a*b*d - (3*B*a^2 - A*a*b)*e + (2*B*b^2*d - (3*B*a*b - A*b^2)*e) *x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*s qrt(e*x + d))/(b*x + a)) + 2*((3*B*a*b^2 - A*b^3)*d - (3*B*a^2*b - A*a*b^2 )*e + 2*(B*b^3*d - B*a*b^2*e)*x)*sqrt(e*x + d))/(a*b^4*d - a^2*b^3*e + (b^ 5*d - a*b^4*e)*x), ((2*B*a*b*d - (3*B*a^2 - A*a*b)*e + (2*B*b^2*d - (3*B*a *b - A*b^2)*e)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e* x + d)/(b*e*x + b*d)) + ((3*B*a*b^2 - A*b^3)*d - (3*B*a^2*b - A*a*b^2)*e + 2*(B*b^3*d - B*a*b^2*e)*x)*sqrt(e*x + d))/(a*b^4*d - a^2*b^3*e + (b^5*d - a*b^4*e)*x)]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a + b x\right )^{2}}\, dx \] Input:
integrate((B*x+A)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2),x)
Output:
Integral((A + B*x)*sqrt(d + e*x)/(a + b*x)**2, x)
Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^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*e-b*d>0)', see `assume?` for m ore detail
Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, \sqrt {e x + d} B}{b^{2}} + \frac {{\left (2 \, B b d - 3 \, B a e + A b e\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {\sqrt {e x + d} B a e - \sqrt {e x + d} A b e}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b^{2}} \] Input:
integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")
Output:
2*sqrt(e*x + d)*B/b^2 + (2*B*b*d - 3*B*a*e + A*b*e)*arctan(sqrt(e*x + d)*b /sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^2) + (sqrt(e*x + d)*B*a*e - sqrt(e*x + d)*A*b*e)/(((e*x + d)*b - b*d + a*e)*b^2)
Time = 11.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {2\,B\,\sqrt {d+e\,x}}{b^2}-\frac {\left (A\,b\,e-B\,a\,e\right )\,\sqrt {d+e\,x}}{b^3\,\left (d+e\,x\right )-b^3\,d+a\,b^2\,e}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )\,\left (A\,b\,e-3\,B\,a\,e+2\,B\,b\,d\right )}{b^{5/2}\,\sqrt {a\,e-b\,d}} \] Input:
int(((A + B*x)*(d + e*x)^(1/2))/(a^2 + b^2*x^2 + 2*a*b*x),x)
Output:
(2*B*(d + e*x)^(1/2))/b^2 - ((A*b*e - B*a*e)*(d + e*x)^(1/2))/(b^3*(d + e* x) - b^3*d + a*b^2*e) + (atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)) *(A*b*e - 3*B*a*e + 2*B*b*d))/(b^(5/2)*(a*e - b*d)^(1/2))
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.48 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right )+2 \sqrt {e x +d}\, b}{b^{2}} \] Input:
int((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2),x)
Output:
(2*( - sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d))) + sqrt(d + e*x)*b))/b**2