Integrand size = 35, antiderivative size = 160 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x) \sqrt {d+e x}}-\frac {2 (2 b B d-A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac {2 b B (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x)} \] Output:
-2*(-a*e+b*d)*(-A*e+B*d)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)/(e*x+d)^(1/2)-2*(-A *b*e-B*a*e+2*B*b*d)*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)+2/3*b*B*(e *x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.53 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {(a+b x)^2} \left (3 A b e (2 d+e x)+3 a e (2 B d-A e+B e x)+b B \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 (a+b x) \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^(3/2),x]
Output:
(2*Sqrt[(a + b*x)^2]*(3*A*b*e*(2*d + e*x) + 3*a*e*(2*B*d - A*e + B*e*x) + b*B*(-8*d^2 - 4*d*e*x + e^2*x^2)))/(3*e^3*(a + b*x)*Sqrt[d + e*x])
Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1187, 27, 86, 2009}
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.
\(\displaystyle \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} (A+B x)}{(d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b (a+b x) (A+B x)}{(d+e x)^{3/2}}dx}{b (a+b x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) (A+B x)}{(d+e x)^{3/2}}dx}{a+b x}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b \sqrt {d+e x} B}{e^2}+\frac {-2 b B d+A b e+a B e}{e^2 \sqrt {d+e x}}+\frac {(a e-b d) (A e-B d)}{e^2 (d+e x)^{3/2}}\right )dx}{a+b x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 \sqrt {d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac {2 (b d-a e) (B d-A e)}{e^3 \sqrt {d+e x}}+\frac {2 b B (d+e x)^{3/2}}{3 e^3}\right )}{a+b x}\) |
Input:
Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(d + e*x)^(3/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)*(B*d - A*e))/(e^3*Sqrt[d + e*x]) - (2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[d + e*x])/e^3 + (2*b*B*(d + e*x )^(3/2))/(3*e^3)))/(a + b*x)
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 1.51 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49
method | result | size |
default | \(-\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (-B b \,e^{2} x^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right )}{3 e^{3} \sqrt {e x +d}}\) | \(79\) |
gosper | \(-\frac {2 \left (-B b \,e^{2} x^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \sqrt {e x +d}\, e^{3} \left (b x +a \right )}\) | \(89\) |
orering | \(-\frac {2 \left (-B b \,e^{2} x^{2}-3 A b \,e^{2} x -3 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}-6 A b d e -6 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \sqrt {e x +d}\, e^{3} \left (b x +a \right )}\) | \(89\) |
risch | \(\frac {2 \left (B b e x +3 A b e +3 B a e -5 B b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 e^{3} \left (b x +a \right )}-\frac {2 \left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{e^{3} \sqrt {e x +d}\, \left (b x +a \right )}\) | \(104\) |
Input:
int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3*csgn(b*x+a)*(-B*b*e^2*x^2-3*A*b*e^2*x-3*B*a*e^2*x+4*B*b*d*e*x+3*A*a*e ^2-6*A*b*d*e-6*B*a*d*e+8*B*b*d^2)/e^3/(e*x+d)^(1/2)
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \, {\left (B a + A b\right )} d e - {\left (4 \, B b d e - 3 \, {\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (e^{4} x + d e^{3}\right )}} \] Input:
integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x, algorithm="fricas")
Output:
2/3*(B*b*e^2*x^2 - 8*B*b*d^2 - 3*A*a*e^2 + 6*(B*a + A*b)*d*e - (4*B*b*d*e - 3*(B*a + A*b)*e^2)*x)*sqrt(e*x + d)/(e^4*x + d*e^3)
\[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)*((b*x+a)**2)**(1/2)/(e*x+d)**(3/2),x)
Output:
Integral((A + B*x)*sqrt((a + b*x)**2)/(d + e*x)**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (b e x + 2 \, b d - a e\right )} A}{\sqrt {e x + d} e^{2}} + \frac {2 \, {\left (b e^{2} x^{2} - 8 \, b d^{2} + 6 \, a d e - {\left (4 \, b d e - 3 \, a e^{2}\right )} x\right )} B}{3 \, \sqrt {e x + d} e^{3}} \] Input:
integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x, algorithm="maxima")
Output:
2*(b*e*x + 2*b*d - a*e)*A/(sqrt(e*x + d)*e^2) + 2/3*(b*e^2*x^2 - 8*b*d^2 + 6*a*d*e - (4*b*d*e - 3*a*e^2)*x)*B/(sqrt(e*x + d)*e^3)
Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (B b d^{2} \mathrm {sgn}\left (b x + a\right ) - B a d e \mathrm {sgn}\left (b x + a\right ) - A b d e \mathrm {sgn}\left (b x + a\right ) + A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{\sqrt {e x + d} e^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b e^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, \sqrt {e x + d} B b d e^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {e x + d} B a e^{7} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {e x + d} A b e^{7} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, e^{9}} \] Input:
integrate((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x, algorithm="giac")
Output:
-2*(B*b*d^2*sgn(b*x + a) - B*a*d*e*sgn(b*x + a) - A*b*d*e*sgn(b*x + a) + A *a*e^2*sgn(b*x + a))/(sqrt(e*x + d)*e^3) + 2/3*((e*x + d)^(3/2)*B*b*e^6*sg n(b*x + a) - 6*sqrt(e*x + d)*B*b*d*e^6*sgn(b*x + a) + 3*sqrt(e*x + d)*B*a* e^7*sgn(b*x + a) + 3*sqrt(e*x + d)*A*b*e^7*sgn(b*x + a))/e^9
Time = 11.72 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.68 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^2}{3\,e}-\frac {6\,A\,a\,e^2+16\,B\,b\,d^2-12\,A\,b\,d\,e-12\,B\,a\,d\,e}{3\,b\,e^3}+\frac {x\,\left (6\,A\,b\,e^2+6\,B\,a\,e^2-8\,B\,b\,d\,e\right )}{3\,b\,e^3}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \] Input:
int((((a + b*x)^2)^(1/2)*(A + B*x))/(d + e*x)^(3/2),x)
Output:
(((a + b*x)^2)^(1/2)*((2*B*x^2)/(3*e) - (6*A*a*e^2 + 16*B*b*d^2 - 12*A*b*d *e - 12*B*a*d*e)/(3*b*e^3) + (x*(6*A*b*e^2 + 6*B*a*e^2 - 8*B*b*d*e))/(3*b* e^3)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)
Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.39 \[ \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {\frac {2}{3} b^{2} e^{2} x^{2}+4 a b \,e^{2} x -\frac {8}{3} b^{2} d e x -2 a^{2} e^{2}+8 a b d e -\frac {16}{3} b^{2} d^{2}}{\sqrt {e x +d}\, e^{3}} \] Input:
int((B*x+A)*((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x)
Output:
(2*( - 3*a**2*e**2 + 12*a*b*d*e + 6*a*b*e**2*x - 8*b**2*d**2 - 4*b**2*d*e* x + b**2*e**2*x**2))/(3*sqrt(d + e*x)*e**3)