Integrand size = 22, antiderivative size = 157 \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x)^2}-\frac {(4 b B d-3 A b e-a B e) \sqrt {d+e x}}{4 b (b d-a e)^2 (a+b x)}+\frac {e (4 b B d-3 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2}} \] Output:
-1/2*(A*b-B*a)*(e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)^2-1/4*(-3*A*b*e-B*a*e+4* B*b*d)*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/(b*x+a)+1/4*e*(-3*A*b*e-B*a*e+4*B*b*d) *arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(5/2)
Time = 0.63 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=\frac {\frac {\sqrt {b} \sqrt {d+e x} \left (A b (-2 b d+5 a e+3 b e x)-B \left (a^2 e+4 b^2 d x+a b (2 d-e x)\right )\right )}{(b d-a e)^2 (a+b x)^2}+\frac {e (-4 b B d+3 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}}{4 b^{3/2}} \] Input:
Integrate[(A + B*x)/((a + b*x)^3*Sqrt[d + e*x]),x]
Output:
((Sqrt[b]*Sqrt[d + e*x]*(A*b*(-2*b*d + 5*a*e + 3*b*e*x) - B*(a^2*e + 4*b^2 *d*x + a*b*(2*d - e*x))))/((b*d - a*e)^2*(a + b*x)^2) + (e*(-4*b*B*d + 3*A *b*e + a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(5/2))/(4*b^(3/2))
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {87, 52, 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.
\(\displaystyle \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-a B e-3 A b e+4 b B d) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {(-a B e-3 A b e+4 b B d) \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(-a B e-3 A b e+4 b B d) \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(-a B e-3 A b e+4 b B d) \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\) |
Input:
Int[(A + B*x)/((a + b*x)^3*Sqrt[d + e*x]),x]
Output:
-1/2*((A*b - a*B)*Sqrt[d + e*x])/(b*(b*d - a*e)*(a + b*x)^2) + ((4*b*B*d - 3*A*b*e - a*B*e)*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTanh[( Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(3/2))))/(4* b*(b*d - a*e))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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]
Time = 0.43 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\frac {3 \left (b x +a \right )^{2} \left (\left (A e -\frac {4 B d}{3}\right ) b +\frac {B a e}{3}\right ) e \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{4}+\frac {5 \sqrt {e x +d}\, \sqrt {\left (a e -d b \right ) b}\, \left (\frac {\left (3 A e x -2 d \left (2 B x +A \right )\right ) b^{2}}{5}+a \left (\left (\frac {B x}{5}+A \right ) e -\frac {2 B d}{5}\right ) b -\frac {B \,a^{2} e}{5}\right )}{4}}{\left (a e -d b \right )^{2} \sqrt {\left (a e -d b \right ) b}\, b \left (b x +a \right )^{2}}\) | \(148\) |
derivativedivides | \(2 e \left (\frac {\frac {\left (3 A b e +B a e -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 a^{2} e^{2}-16 a b d e +8 b^{2} d^{2}}+\frac {\left (5 A b e -B a e -4 B b d \right ) \sqrt {e x +d}}{8 \left (a e -d b \right ) b}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {\left (3 A b e +B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b \sqrt {\left (a e -d b \right ) b}}\right )\) | \(186\) |
default | \(2 e \left (\frac {\frac {\left (3 A b e +B a e -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 a^{2} e^{2}-16 a b d e +8 b^{2} d^{2}}+\frac {\left (5 A b e -B a e -4 B b d \right ) \sqrt {e x +d}}{8 \left (a e -d b \right ) b}}{\left (\left (e x +d \right ) b +a e -d b \right )^{2}}+\frac {\left (3 A b e +B a e -4 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -d b \right ) b}}\right )}{8 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b \sqrt {\left (a e -d b \right ) b}}\right )\) | \(186\) |
Input:
int((B*x+A)/(b*x+a)^3/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
5/4/(a*e-b*d)^2*(3/5*(b*x+a)^2*((A*e-4/3*B*d)*b+1/3*B*a*e)*e*arctan(b*(e*x +d)^(1/2)/((a*e-b*d)*b)^(1/2))+(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*(1/5*(3*A *e*x-2*d*(2*B*x+A))*b^2+a*((1/5*B*x+A)*e-2/5*B*d)*b-1/5*B*a^2*e))/((a*e-b* d)*b)^(1/2)/b/(b*x+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (137) = 274\).
Time = 0.15 (sec) , antiderivative size = 808, normalized size of antiderivative = 5.15 \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)/(b*x+a)^3/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
[-1/8*((4*B*a^2*b*d*e - (B*a^3 + 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (B*a*b^2 + 3*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (B*a^2*b + 3*A*a*b^2)*e^2)*x)*sqr t(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) + 2*(2*(B*a*b^3 + A*b^4)*d^2 - (B*a^2*b^2 + 7*A*a*b^3)*d *e - (B*a^3*b - 5*A*a^2*b^2)*e^2 + (4*B*b^4*d^2 - (5*B*a*b^3 + 3*A*b^4)*d* e + (B*a^2*b^2 + 3*A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^3 - 3*a^3*b^ 4*d^2*e + 3*a^4*b^3*d*e^2 - a^5*b^2*e^3 + (b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2 *b^5*d*e^2 - a^3*b^4*e^3)*x^2 + 2*(a*b^6*d^3 - 3*a^2*b^5*d^2*e + 3*a^3*b^4 *d*e^2 - a^4*b^3*e^3)*x), -1/4*((4*B*a^2*b*d*e - (B*a^3 + 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (B*a*b^2 + 3*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (B*a^2*b + 3*A*a*b^2)*e^2)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqr t(e*x + d)/(b*e*x + b*d)) + (2*(B*a*b^3 + A*b^4)*d^2 - (B*a^2*b^2 + 7*A*a* b^3)*d*e - (B*a^3*b - 5*A*a^2*b^2)*e^2 + (4*B*b^4*d^2 - (5*B*a*b^3 + 3*A*b ^4)*d*e + (B*a^2*b^2 + 3*A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^3 - 3* a^3*b^4*d^2*e + 3*a^4*b^3*d*e^2 - a^5*b^2*e^3 + (b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*x^2 + 2*(a*b^6*d^3 - 3*a^2*b^5*d^2*e + 3*a ^3*b^4*d*e^2 - a^4*b^3*e^3)*x)]
Timed out. \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)/(b*x+a)**3/(e*x+d)**(1/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)/(b*x+a)^3/(e*x+d)^(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*e-b*d>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.64 \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=-\frac {{\left (4 \, B b d e - B a e^{2} - 3 \, A b e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {e x + d} B b^{2} d^{2} e - {\left (e x + d\right )}^{\frac {3}{2}} B a b e^{2} - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} e^{2} + 3 \, \sqrt {e x + d} B a b d e^{2} + 5 \, \sqrt {e x + d} A b^{2} d e^{2} + \sqrt {e x + d} B a^{2} e^{3} - 5 \, \sqrt {e x + d} A a b e^{3}}{4 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \] Input:
integrate((B*x+A)/(b*x+a)^3/(e*x+d)^(1/2),x, algorithm="giac")
Output:
-1/4*(4*B*b*d*e - B*a*e^2 - 3*A*b*e^2)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*sqrt(-b^2*d + a*b*e)) - 1/4 *(4*(e*x + d)^(3/2)*B*b^2*d*e - 4*sqrt(e*x + d)*B*b^2*d^2*e - (e*x + d)^(3 /2)*B*a*b*e^2 - 3*(e*x + d)^(3/2)*A*b^2*e^2 + 3*sqrt(e*x + d)*B*a*b*d*e^2 + 5*sqrt(e*x + d)*A*b^2*d*e^2 + sqrt(e*x + d)*B*a^2*e^3 - 5*sqrt(e*x + d)* A*a*b*e^3)/((b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*((e*x + d)*b - b*d + a*e)^ 2)
Time = 0.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=\frac {\frac {{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e^2+B\,a\,e^2-4\,B\,b\,d\,e\right )}{4\,{\left (a\,e-b\,d\right )}^2}-\frac {\sqrt {d+e\,x}\,\left (B\,a\,e^2-5\,A\,b\,e^2+4\,B\,b\,d\,e\right )}{4\,b\,\left (a\,e-b\,d\right )}}{b^2\,{\left (d+e\,x\right )}^2-\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (d+e\,x\right )+a^2\,e^2+b^2\,d^2-2\,a\,b\,d\,e}+\frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,\sqrt {d+e\,x}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (3\,A\,b\,e^2+B\,a\,e^2-4\,B\,b\,d\,e\right )}\right )\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{4\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{5/2}} \] Input:
int((A + B*x)/((a + b*x)^3*(d + e*x)^(1/2)),x)
Output:
(((d + e*x)^(3/2)*(3*A*b*e^2 + B*a*e^2 - 4*B*b*d*e))/(4*(a*e - b*d)^2) - ( (d + e*x)^(1/2)*(B*a*e^2 - 5*A*b*e^2 + 4*B*b*d*e))/(4*b*(a*e - b*d)))/(b^2 *(d + e*x)^2 - (2*b^2*d - 2*a*b*e)*(d + e*x) + a^2*e^2 + b^2*d^2 - 2*a*b*d *e) + (e*atan((b^(1/2)*e*(d + e*x)^(1/2)*(3*A*b*e + B*a*e - 4*B*b*d))/((a* e - b*d)^(1/2)*(3*A*b*e^2 + B*a*e^2 - 4*B*b*d*e)))*(3*A*b*e + B*a*e - 4*B* b*d))/(4*b^(3/2)*(a*e - b*d)^(5/2))
Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}} \, dx=\frac {\sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) a e +\sqrt {b}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, b}{\sqrt {b}\, \sqrt {a e -b d}}\right ) b e x +\sqrt {e x +d}\, a b e -\sqrt {e x +d}\, b^{2} d}{b \left (a^{2} b \,e^{2} x -2 a \,b^{2} d e x +b^{3} d^{2} x +a^{3} e^{2}-2 a^{2} b d e +a \,b^{2} d^{2}\right )} \] Input:
int((B*x+A)/(b*x+a)^3/(e*x+d)^(1/2),x)
Output:
(sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d))) *a*e + sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*b*e*x + sqrt(d + e*x)*a*b*e - sqrt(d + e*x)*b**2*d)/(b*(a**3*e**2 - 2*a**2*b*d*e + a**2*b*e**2*x + a*b**2*d**2 - 2*a*b**2*d*e*x + b**3*d**2*x ))